本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{arctan(x)}^{cos(2x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {arctan(x)}^{cos(2x)}\right)}{dx}\\=&({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))\\=&-2{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x) + \frac{{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)arctan(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -2{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x) + \frac{{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)arctan(x)}\right)}{dx}\\=&-2({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))sin(2x) - \frac{2{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})sin(2x)}{(arctan(x))} - 2{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x)*2 + \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{cos(2x)}cos(2x)}{arctan(x)} + \frac{({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)arctan(x)} + \frac{{arctan(x)}^{cos(2x)}cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)}\\=&4{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x) - \frac{4{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)arctan(x)} - 4{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x) - \frac{2x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 4{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x) - \frac{4{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)arctan(x)} - 4{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x) - \frac{2x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)}\right)}{dx}\\=&4({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{2}(arctan(x))sin^{2}(2x) + \frac{4{arctan(x)}^{cos(2x)}*2ln(arctan(x))(\frac{(1)}{(1 + (x)^{2})})sin^{2}(2x)}{(arctan(x))} + 4{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))*2sin(2x)cos(2x)*2 - \frac{4(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{arctan(x)} - \frac{4({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})sin(2x)cos(2x)}{(x^{2} + 1)(arctan(x))arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x)*2cos(2x)}{(x^{2} + 1)arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)*-sin(2x)*2}{(x^{2} + 1)arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} - \frac{4(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{cos(2x)}sin(2x)}{arctan(x)} - \frac{4({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))sin(2x)}{(x^{2} + 1)arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}cos(2x)*2}{(x^{2} + 1)arctan(x)} - \frac{4{arctan(x)}^{cos(2x)}sin(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} - 4({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))cos(2x) - \frac{4{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})cos(2x)}{(arctan(x))} - 4{arctan(x)}^{cos(2x)}ln(arctan(x))*-sin(2x)*2 - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{cos(2x)}cos(2x)}{arctan(x)} - \frac{2{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x{arctan(x)}^{cos(2x)}cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{cos(2x)}cos^{2}(2x)}{arctan^{2}(x)} + \frac{({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{{arctan(x)}^{cos(2x)}*-2cos(2x)sin(2x)*2}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{{arctan(x)}^{cos(2x)}cos^{2}(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{cos(2x)}cos(2x)}{arctan^{2}(x)} - \frac{({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{{arctan(x)}^{cos(2x)}cos(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&24{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin(2x)cos(2x) + \frac{12{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{24{arctan(x)}^{cos(2x)}ln(arctan(x))sin^{2}(2x)}{(x^{2} + 1)arctan(x)} - 8{arctan(x)}^{cos(2x)}ln^{3}(arctan(x))sin^{3}(2x) + \frac{12x{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{cos(2x)}sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{cos(2x)}ln(arctan(x))cos^{2}(2x)}{(x^{2} + 1)arctan(x)} + \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{12x{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{12{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{6{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + 8{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x) + \frac{8x^{2}{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan(x)} - \frac{2{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{6x{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{{arctan(x)}^{cos(2x)}cos^{3}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 24{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin(2x)cos(2x) + \frac{12{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{24{arctan(x)}^{cos(2x)}ln(arctan(x))sin^{2}(2x)}{(x^{2} + 1)arctan(x)} - 8{arctan(x)}^{cos(2x)}ln^{3}(arctan(x))sin^{3}(2x) + \frac{12x{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{cos(2x)}sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{cos(2x)}ln(arctan(x))cos^{2}(2x)}{(x^{2} + 1)arctan(x)} + \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{12x{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{12{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{6{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + 8{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x) + \frac{8x^{2}{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan(x)} - \frac{2{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{6x{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{{arctan(x)}^{cos(2x)}cos^{3}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)}\right)}{dx}\\=&24({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{2}(arctan(x))sin(2x)cos(2x) + \frac{24{arctan(x)}^{cos(2x)}*2ln(arctan(x))(\frac{(1)}{(1 + (x)^{2})})sin(2x)cos(2x)}{(arctan(x))} + 24{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))cos(2x)*2cos(2x) + 24{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin(2x)*-sin(2x)*2 + \frac{12(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)cos(2x)}{arctan(x)} + \frac{12({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{2}(arctan(x))sin^{2}(2x)cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{12{arctan(x)}^{cos(2x)}*2ln(arctan(x))(\frac{(1)}{(1 + (x)^{2})})sin^{2}(2x)cos(2x)}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{12{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))*2sin(2x)cos(2x)*2cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{12{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)*-sin(2x)*2}{(x^{2} + 1)arctan(x)} + \frac{12{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} + \frac{24(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{cos(2x)}ln(arctan(x))sin^{2}(2x)}{arctan(x)} + \frac{24({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))sin^{2}(2x)}{(x^{2} + 1)arctan(x)} + \frac{24{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})sin^{2}(2x)}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{24{arctan(x)}^{cos(2x)}ln(arctan(x))*2sin(2x)cos(2x)*2}{(x^{2} + 1)arctan(x)} + \frac{24{arctan(x)}^{cos(2x)}ln(arctan(x))sin^{2}(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} - 8({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{3}(arctan(x))sin^{3}(2x) - \frac{8{arctan(x)}^{cos(2x)}*3ln^{2}(arctan(x))(\frac{(1)}{(1 + (x)^{2})})sin^{3}(2x)}{(arctan(x))} - 8{arctan(x)}^{cos(2x)}ln^{3}(arctan(x))*3sin^{2}(2x)cos(2x)*2 + \frac{12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{arctan(x)} + \frac{12{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{12x({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{12x{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})sin(2x)cos(2x)}{(x^{2} + 1)^{2}(arctan(x))arctan(x)} + \frac{12x{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x)*2cos(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{12x{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)*-sin(2x)*2}{(x^{2} + 1)^{2}arctan(x)} + \frac{12x{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos^{2}(2x)}{arctan^{2}(x)} - \frac{6({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))sin(2x)cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{6{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})sin(2x)cos^{2}(2x)}{(x^{2} + 1)^{2}(arctan(x))arctan^{2}(x)} - \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x)*2cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)*-2cos(2x)sin(2x)*2}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos^{2}(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{cos(2x)}sin(2x)cos(2x)}{arctan^{2}(x)} - \frac{12({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{cos(2x)}cos(2x)*2cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{cos(2x)}sin(2x)*-sin(2x)*2}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{cos(2x)}sin(2x)cos(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{12(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{cos(2x)}ln(arctan(x))cos^{2}(2x)}{arctan(x)} - \frac{12({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))cos^{2}(2x)}{(x^{2} + 1)arctan(x)} - \frac{12{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})cos^{2}(2x)}{(x^{2} + 1)(arctan(x))arctan(x)} - \frac{12{arctan(x)}^{cos(2x)}ln(arctan(x))*-2cos(2x)sin(2x)*2}{(x^{2} + 1)arctan(x)} - \frac{12{arctan(x)}^{cos(2x)}ln(arctan(x))cos^{2}(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} + \frac{6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{arctan^{2}(x)} + \frac{6({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})sin(2x)cos(2x)}{(x^{2} + 1)^{2}(arctan(x))arctan^{2}(x)} + \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x)*2cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)*-sin(2x)*2}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{cos(2x)}sin(2x)}{arctan(x)} + \frac{12{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{12x({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))sin(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{12x{arctan(x)}^{cos(2x)}cos(2x)*2}{(x^{2} + 1)^{2}arctan(x)} + \frac{12x{arctan(x)}^{cos(2x)}sin(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{12(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{cos(2x)}cos(2x)}{arctan(x)} - \frac{12({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)arctan(x)} - \frac{12{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)arctan(x)} - \frac{12{arctan(x)}^{cos(2x)}cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} + \frac{6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{cos(2x)}sin(2x)}{arctan^{2}(x)} + \frac{6({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))sin(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6{arctan(x)}^{cos(2x)}cos(2x)*2}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6{arctan(x)}^{cos(2x)}sin(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + 8({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))sin(2x) + \frac{8{arctan(x)}^{cos(2x)}(\frac{(1)}{(1 + (x)^{2})})sin(2x)}{(arctan(x))} + 8{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x)*2 + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}{arctan(x)}^{cos(2x)}cos(2x)}{arctan(x)} + \frac{8*2x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{2}({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{2}{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{2}{arctan(x)}^{cos(2x)}cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{cos(2x)}cos(2x)}{arctan(x)} - \frac{2({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{2{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)^{2}arctan(x)} - \frac{2{arctan(x)}^{cos(2x)}cos(2x)(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x{arctan(x)}^{cos(2x)}cos^{2}(2x)}{arctan^{2}(x)} - \frac{6{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{6x({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{6x{arctan(x)}^{cos(2x)}*-2cos(2x)sin(2x)*2}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{6x{arctan(x)}^{cos(2x)}cos^{2}(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x{arctan(x)}^{cos(2x)}cos(2x)}{arctan^{2}(x)} + \frac{6{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x{arctan(x)}^{cos(2x)}cos(2x)(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}}){arctan(x)}^{cos(2x)}cos^{3}(2x)}{arctan^{3}(x)} + \frac{({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos^{3}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{{arctan(x)}^{cos(2x)}*-3cos^{2}(2x)sin(2x)*2}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{{arctan(x)}^{cos(2x)}cos^{3}(2x)(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - \frac{3(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}}){arctan(x)}^{cos(2x)}cos^{2}(2x)}{arctan^{3}(x)} - \frac{3({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3{arctan(x)}^{cos(2x)}*-2cos(2x)sin(2x)*2}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3{arctan(x)}^{cos(2x)}cos^{2}(2x)(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{2(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}}){arctan(x)}^{cos(2x)}cos(2x)}{arctan^{3}(x)} + \frac{2({arctan(x)}^{cos(2x)}((-sin(2x)*2)ln(arctan(x)) + \frac{(cos(2x))((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))cos(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2{arctan(x)}^{cos(2x)}*-sin(2x)*2}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2{arctan(x)}^{cos(2x)}cos(2x)(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{16{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{224{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)arctan(x)} + 48{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))cos^{2}(2x) - 96{arctan(x)}^{cos(2x)}ln^{3}(arctan(x))sin^{2}(2x)cos(2x) - \frac{48x{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{32{arctan(x)}^{cos(2x)}ln^{3}(arctan(x))sin^{3}(2x)cos(2x)}{(x^{2} + 1)arctan(x)} + \frac{24{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{96{arctan(x)}^{cos(2x)}ln(arctan(x))sin^{2}(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{96{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin(2x)cos^{2}(2x)}{(x^{2} + 1)arctan(x)} - \frac{96{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{3}(2x)}{(x^{2} + 1)arctan(x)} - \frac{24{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x)cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{96x{arctan(x)}^{cos(2x)}ln(arctan(x))sin^{2}(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{48{arctan(x)}^{cos(2x)}sin^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{48{arctan(x)}^{cos(2x)}ln(arctan(x))sin^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - 64{arctan(x)}^{cos(2x)}ln^{2}(arctan(x))sin^{2}(2x) - \frac{64x^{2}{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{3}arctan(x)} + 16{arctan(x)}^{cos(2x)}ln^{4}(arctan(x))sin^{4}(2x) + \frac{48x{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{96x{arctan(x)}^{cos(2x)}sin(2x)cos(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{48x{arctan(x)}^{cos(2x)}ln(arctan(x))cos^{2}(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{48x{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{8{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos^{3}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{24{arctan(x)}^{cos(2x)}sin(2x)cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{24{arctan(x)}^{cos(2x)}ln(arctan(x))cos^{3}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{24{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{48{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{48{arctan(x)}^{cos(2x)}sin(2x)cos(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{24{arctan(x)}^{cos(2x)}ln(arctan(x))cos^{2}(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{16{arctan(x)}^{cos(2x)}ln(arctan(x))sin(2x)cos(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{64x^{2}{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{3}arctan(x)} + \frac{16{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{2}arctan(x)} + \frac{48x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan(x)} - \frac{48x{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{32{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)arctan(x)} + \frac{24{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{16{arctan(x)}^{cos(2x)}sin(2x)}{(x^{2} + 1)^{3}arctan^{3}(x)} + 16{arctan(x)}^{cos(2x)}ln(arctan(x))cos(2x) - \frac{48x^{3}{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{4}arctan(x)} + \frac{24x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan(x)} + \frac{44x^{2}{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{4}arctan^{2}(x)} - \frac{44x^{2}{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{4}arctan^{2}(x)} - \frac{8{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{8{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{12x{arctan(x)}^{cos(2x)}cos^{3}(2x)}{(x^{2} + 1)^{4}arctan^{3}(x)} + \frac{36x{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{4}arctan^{3}(x)} - \frac{24x{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{4}arctan^{3}(x)} + \frac{{arctan(x)}^{cos(2x)}cos^{4}(2x)}{(x^{2} + 1)^{4}arctan^{4}(x)} - \frac{6{arctan(x)}^{cos(2x)}cos^{3}(2x)}{(x^{2} + 1)^{4}arctan^{4}(x)} + \frac{11{arctan(x)}^{cos(2x)}cos^{2}(2x)}{(x^{2} + 1)^{4}arctan^{4}(x)} - \frac{6{arctan(x)}^{cos(2x)}cos(2x)}{(x^{2} + 1)^{4}arctan^{4}(x)}\\ \end{split}\end{equation} \]

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