Integrand size = 26, antiderivative size = 347 \[ \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=-\frac {7 i \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {7 i \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {5 \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{2 \sqrt {3} a d}+\frac {7 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}-\frac {5 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{6 a d}+\frac {7 i \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {7 i \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}+\frac {5 \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{12 a d}-\frac {5}{4 a d \tan ^{\frac {4}{3}}(c+d x)}+\frac {7 i}{2 a d \sqrt [3]{\tan (c+d x)}}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))} \]
7/12*I*arctan(-3^(1/2)+2*tan(d*x+c)^(1/3))/a/d+7/12*I*arctan(3^(1/2)+2*tan (d*x+c)^(1/3))/a/d+7/6*I*arctan(tan(d*x+c)^(1/3))/a/d-5/6*ln(1+tan(d*x+c)^ (2/3))/a/d+5/12*ln(1-tan(d*x+c)^(2/3)+tan(d*x+c)^(4/3))/a/d+5/6*arctan(1/3 *(1-2*tan(d*x+c)^(2/3))*3^(1/2))/a/d*3^(1/2)+7/24*I*ln(1-3^(1/2)*tan(d*x+c )^(1/3)+tan(d*x+c)^(2/3))/a/d*3^(1/2)-7/24*I*ln(1+3^(1/2)*tan(d*x+c)^(1/3) +tan(d*x+c)^(2/3))/a/d*3^(1/2)-5/4/a/d/tan(d*x+c)^(4/3)+7/2*I/a/d/tan(d*x+ c)^(1/3)+1/2/d/tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))
Time = 5.84 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {\frac {2}{a+i a \tan (c+d x)}-\frac {7 \tan (c+d x) \left (-6 i+\log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}-\log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}-(-1)^{2/3} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}+(-1)^{2/3} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}+\sqrt [3]{-1} \log \left (1-(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}-\sqrt [3]{-1} \log \left (1+(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}\right )+5 \left (3+2 \log \left (1+\sqrt [3]{\tan ^2(c+d x)}\right ) \tan ^2(c+d x)^{2/3}+2 (-1)^{2/3} \log \left (1-\sqrt [3]{-1} \sqrt [3]{\tan ^2(c+d x)}\right ) \tan ^2(c+d x)^{2/3}-2 \sqrt [3]{-1} \log \left (1+(-1)^{2/3} \sqrt [3]{\tan ^2(c+d x)}\right ) \tan ^2(c+d x)^{2/3}\right )}{3 a}}{4 d \tan ^{\frac {4}{3}}(c+d x)} \]
(2/(a + I*a*Tan[c + d*x]) - (7*Tan[c + d*x]*(-6*I + Log[1 - I*(Tan[c + d*x ]^2)^(1/6)]*(Tan[c + d*x]^2)^(1/6) - Log[1 + I*(Tan[c + d*x]^2)^(1/6)]*(Ta n[c + d*x]^2)^(1/6) - (-1)^(2/3)*Log[1 - (-1)^(1/6)*(Tan[c + d*x]^2)^(1/6) ]*(Tan[c + d*x]^2)^(1/6) + (-1)^(2/3)*Log[1 + (-1)^(1/6)*(Tan[c + d*x]^2)^ (1/6)]*(Tan[c + d*x]^2)^(1/6) + (-1)^(1/3)*Log[1 - (-1)^(5/6)*(Tan[c + d*x ]^2)^(1/6)]*(Tan[c + d*x]^2)^(1/6) - (-1)^(1/3)*Log[1 + (-1)^(5/6)*(Tan[c + d*x]^2)^(1/6)]*(Tan[c + d*x]^2)^(1/6)) + 5*(3 + 2*Log[1 + (Tan[c + d*x]^ 2)^(1/3)]*(Tan[c + d*x]^2)^(2/3) + 2*(-1)^(2/3)*Log[1 - (-1)^(1/3)*(Tan[c + d*x]^2)^(1/3)]*(Tan[c + d*x]^2)^(2/3) - 2*(-1)^(1/3)*Log[1 + (-1)^(2/3)* (Tan[c + d*x]^2)^(1/3)]*(Tan[c + d*x]^2)^(2/3)))/(3*a))/(4*d*Tan[c + d*x]^ (4/3))
Time = 1.02 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.80, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 4035, 27, 3042, 4012, 25, 3042, 4012, 3042, 4021, 3042, 3957, 266, 807, 750, 16, 824, 27, 216, 1142, 25, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^{7/3} (a+i a \tan (c+d x))}dx\) |
| \(\Big \downarrow \) 4035 |
| \(\displaystyle \frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac {\int -\frac {10 a-7 i a \tan (c+d x)}{3 \tan ^{\frac {7}{3}}(c+d x)}dx}{2 a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {10 a-7 i a \tan (c+d x)}{\tan ^{\frac {7}{3}}(c+d x)}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {10 a-7 i a \tan (c+d x)}{\tan (c+d x)^{7/3}}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\int -\frac {10 \tan (c+d x) a+7 i a}{\tan ^{\frac {4}{3}}(c+d x)}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}-\int \frac {10 \tan (c+d x) a+7 i a}{\tan ^{\frac {4}{3}}(c+d x)}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}-\int \frac {10 \tan (c+d x) a+7 i a}{\tan (c+d x)^{4/3}}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {-\int \frac {10 a-7 i a \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}}dx-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\int \frac {10 a-7 i a \tan (c+d x)}{\sqrt [3]{\tan (c+d x)}}dx-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 4021 |
| \(\displaystyle \frac {7 i a \int \tan ^{\frac {2}{3}}(c+d x)dx-10 a \int \frac {1}{\sqrt [3]{\tan (c+d x)}}dx-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-10 a \int \frac {1}{\sqrt [3]{\tan (c+d x)}}dx+7 i a \int \tan (c+d x)^{2/3}dx-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {-\frac {10 a \int \frac {1}{\sqrt [3]{\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}+\frac {7 i a \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {30 a \int \frac {\sqrt [3]{\tan (c+d x)}}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}+\frac {21 i a \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {-\frac {15 a \int \frac {1}{\tan (c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)}{d}+\frac {21 i a \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {-\frac {15 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)\right )}{d}+\frac {21 i a \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {21 i a \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {15 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {\frac {21 i a \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int -\frac {1-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{2 \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int -\frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}{2 \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}\right )}{d}-\frac {15 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {21 i a \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )}{d}-\frac {15 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {21 i a \left (-\frac {1}{6} \int \frac {1-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {15 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {21 i a \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {15 a \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{2} \int \left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )d\tan ^{\frac {2}{3}}(c+d x)\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {21 i a \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {15 a \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {21 i a \left (\frac {1}{6} \left (-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {15 a \left (\frac {1}{3} \left (\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)-3 \int \frac {1}{-2 \tan ^{\frac {2}{3}}(c+d x)-2}d\left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {21 i a \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {15 a \left (\frac {1}{3} \left (\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {15 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}+\frac {21 i a \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {15 a}{2 d \tan ^{\frac {4}{3}}(c+d x)}+\frac {21 i a}{d \sqrt [3]{\tan (c+d x)}}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {4}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
((-15*a*(ArcTan[(-1 + 2*Tan[c + d*x]^(2/3))/Sqrt[3]]/Sqrt[3] + Log[1 + Tan [c + d*x]^(2/3)]/3))/d + ((21*I)*a*(ArcTan[Tan[c + d*x]^(1/3)]/3 + (-ArcTa n[Sqrt[3] - 2*Tan[c + d*x]^(1/3)] + (Sqrt[3]*Log[1 - Sqrt[3]*Tan[c + d*x]^ (1/3) + Tan[c + d*x]^(2/3)])/2)/6 + (ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3) ] - (Sqrt[3]*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/2)/ 6))/d - (15*a)/(2*d*Tan[c + d*x]^(4/3)) + ((21*I)*a)/(d*Tan[c + d*x]^(1/3) ))/(6*a^2) + 1/(2*d*Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x]))
3.3.39.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Tan[e + f*x])^m, x], x] + Simp[d/b Int [(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^ 2 + d^2, 0] && !IntegerQ[2*m]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-a)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b* c - a*d)*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a*(b*c - a*d)) Int[(c + d *Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0]
Time = 0.61 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.61
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{8}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{4}+\frac {i}{6 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+6 i}-\frac {17 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{12}-\frac {3}{4 \tan \left (d x +c \right )^{\frac {4}{3}}}+\frac {3 i}{\tan \left (d x +c \right )^{\frac {1}{3}}}-\frac {-4 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )-2}{12 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}+\frac {17 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{24}-\frac {17 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{4}}{d a}\) | \(210\) |
| default | \(\frac {\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{8}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{4}+\frac {i}{6 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+6 i}-\frac {17 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{12}-\frac {3}{4 \tan \left (d x +c \right )^{\frac {4}{3}}}+\frac {3 i}{\tan \left (d x +c \right )^{\frac {1}{3}}}-\frac {-4 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )-2}{12 \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}+\frac {17 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{24}-\frac {17 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{4}}{d a}\) | \(210\) |
1/d/a*(1/8*ln(I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)+1/4*I*3^(1/2)*arctanh (1/3*(I+2*tan(d*x+c)^(1/3))*3^(1/2))+1/6*I/(tan(d*x+c)^(1/3)+I)-17/12*ln(t an(d*x+c)^(1/3)+I)-3/4/tan(d*x+c)^(4/3)+3*I/tan(d*x+c)^(1/3)-1/12*(-4*I*ta n(d*x+c)^(1/3)-2)/(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)+17/24*ln(-I*tan (d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)-17/12*I*3^(1/2)*arctanh(1/3*(-I+2*tan(d* x+c)^(1/3))*3^(1/2))-1/4*ln(tan(d*x+c)^(1/3)-I))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 787 vs. \(2 (270) = 540\).
Time = 0.28 (sec) , antiderivative size = 787, normalized size of antiderivative = 2.27 \[ \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\text {Too large to display} \]
-1/24*(3*(sqrt(3)*(-I*a*d*e^(6*I*d*x + 6*I*c) + 2*I*a*d*e^(4*I*d*x + 4*I*c ) - I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) - e^(6*I*d*x + 6*I*c) + 2 *e^(4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log(1/2*sqrt(3)*a*d*sqrt(1/(a^ 2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) + 1/2*I) + 3*(sqrt(3)*(I*a*d*e^(6*I*d*x + 6*I*c) - 2*I*a*d*e^(4*I*d*x + 4*I *c) + I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) - e^(6*I*d*x + 6*I*c) + 2*e^(4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log(-1/2*sqrt(3)*a*d*sqrt(1/ (a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3 ) + 1/2*I) + 17*(3*sqrt(1/3)*(I*a*d*e^(6*I*d*x + 6*I*c) - 2*I*a*d*e^(4*I*d *x + 4*I*c) + I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) - e^(6*I*d*x + 6*I*c) + 2*e^(4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log(3/2*sqrt(1/3)*a* d*sqrt(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) - 1/2*I) + 17*(3*sqrt(1/3)*(-I*a*d*e^(6*I*d*x + 6*I*c) + 2*I*a* d*e^(4*I*d*x + 4*I*c) - I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) - e^( 6*I*d*x + 6*I*c) + 2*e^(4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log(-3/2*s qrt(1/3)*a*d*sqrt(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) - 1/2*I) + 34*(e^(6*I*d*x + 6*I*c) - 2*e^(4*I*d*x + 4*I*c) + e^(2*I*d*x + 2*I*c))*log(((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d* x + 2*I*c) + 1))^(1/3) + I) + 6*(e^(6*I*d*x + 6*I*c) - 2*e^(4*I*d*x + 4*I* c) + e^(2*I*d*x + 2*I*c))*log(((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x...
\[ \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=- \frac {i \int \frac {1}{\tan ^{\frac {10}{3}}{\left (c + d x \right )} - i \tan ^{\frac {7}{3}}{\left (c + d x \right )}}\, dx}{a} \]
Exception generated. \[ \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
Time = 0.74 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {17 i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right )}{24 \, a d} - \frac {i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right )}{8 \, a d} + \frac {\log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{8 \, a d} + \frac {17 \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{24 \, a d} - \frac {17 \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{12 \, a d} - \frac {\log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{4 \, a d} - \frac {3 \, {\left (-4 i \, \tan \left (d x + c\right ) + 1\right )}}{4 \, a d \tan \left (d x + c\right )^{\frac {4}{3}}} + \frac {i \, \tan \left (d x + c\right )^{\frac {2}{3}}}{2 \, a d {\left (\tan \left (d x + c\right ) - i\right )}} \]
17/24*I*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) + I)/(sqrt(3) + 2*tan (d*x + c)^(1/3) - I))/(a*d) - 1/8*I*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c) ^(1/3) - I)/(sqrt(3) + 2*tan(d*x + c)^(1/3) + I))/(a*d) + 1/8*log(tan(d*x + c)^(2/3) + I*tan(d*x + c)^(1/3) - 1)/(a*d) + 17/24*log(tan(d*x + c)^(2/3 ) - I*tan(d*x + c)^(1/3) - 1)/(a*d) - 17/12*log(tan(d*x + c)^(1/3) + I)/(a *d) - 1/4*log(tan(d*x + c)^(1/3) - I)/(a*d) - 3/4*(-4*I*tan(d*x + c) + 1)/ (a*d*tan(d*x + c)^(4/3)) + 1/2*I*tan(d*x + c)^(2/3)/(a*d*(tan(d*x + c) - I ))
Time = 7.38 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.76 \[ \int \frac {1}{\tan ^{\frac {7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\ln \left (52020\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}-\left (-514944\,a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{1/3}+a^3\,d^3\,703584{}\mathrm {i}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{2/3}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{1/3}+\frac {17\,\ln \left (52020\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}-\frac {289\,\left (-729504\,a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (-\frac {1}{a^3\,d^3}\right )}^{1/3}+a^3\,d^3\,703584{}\mathrm {i}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{2/3}}{144}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{1/3}}{12}-\frac {\frac {3}{4\,a\,d}+\frac {7\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,a\,d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,9{}\mathrm {i}}{4\,a\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^{4/3}+{\mathrm {tan}\left (c+d\,x\right )}^{7/3}\,1{}\mathrm {i}}+\frac {17\,\ln \left (52020\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}-\frac {289\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,703584{}\mathrm {i}-364752\,a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{2/3}}{576}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{1/3}}{24}-\frac {17\,\ln \left (52020\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}-\frac {289\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (a^3\,d^3\,703584{}\mathrm {i}+364752\,a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{2/3}}{576}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {1}{a^3\,d^3}\right )}^{1/3}}{24}+\ln \left (52020\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}-{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left (a^3\,d^3\,703584{}\mathrm {i}-514944\,a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{2/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{1/3}-\ln \left (52020\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}-{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left (a^3\,d^3\,703584{}\mathrm {i}+514944\,a^4\,d^4\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{2/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {1}{64\,a^3\,d^3}\right )}^{1/3} \]
log(52020*a*d*tan(c + d*x)^(1/3) - (a^3*d^3*703584i - 514944*a^4*d^4*tan(c + d*x)^(1/3)*(-1/(64*a^3*d^3))^(1/3))*(-1/(64*a^3*d^3))^(2/3))*(-1/(64*a^ 3*d^3))^(1/3) + (17*log(52020*a*d*tan(c + d*x)^(1/3) - (289*(a^3*d^3*70358 4i - 729504*a^4*d^4*tan(c + d*x)^(1/3)*(-1/(a^3*d^3))^(1/3))*(-1/(a^3*d^3) )^(2/3))/144)*(-1/(a^3*d^3))^(1/3))/12 - (3/(4*a*d) - (tan(c + d*x)*9i)/(4 *a*d) + (7*tan(c + d*x)^2)/(2*a*d))/(tan(c + d*x)^(4/3) + tan(c + d*x)^(7/ 3)*1i) + (17*log(52020*a*d*tan(c + d*x)^(1/3) - (289*(3^(1/2)*1i - 1)^2*(a ^3*d^3*703584i - 364752*a^4*d^4*tan(c + d*x)^(1/3)*(3^(1/2)*1i - 1)*(-1/(a ^3*d^3))^(1/3))*(-1/(a^3*d^3))^(2/3))/576)*(3^(1/2)*1i - 1)*(-1/(a^3*d^3)) ^(1/3))/24 - (17*log(52020*a*d*tan(c + d*x)^(1/3) - (289*(3^(1/2)*1i + 1)^ 2*(a^3*d^3*703584i + 364752*a^4*d^4*tan(c + d*x)^(1/3)*(3^(1/2)*1i + 1)*(- 1/(a^3*d^3))^(1/3))*(-1/(a^3*d^3))^(2/3))/576)*(3^(1/2)*1i + 1)*(-1/(a^3*d ^3))^(1/3))/24 + log(52020*a*d*tan(c + d*x)^(1/3) - ((3^(1/2)*1i)/2 - 1/2) ^2*(a^3*d^3*703584i - 514944*a^4*d^4*tan(c + d*x)^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(-1/(64*a^3*d^3))^(1/3))*(-1/(64*a^3*d^3))^(2/3))*((3^(1/2)*1i)/2 - 1 /2)*(-1/(64*a^3*d^3))^(1/3) - log(52020*a*d*tan(c + d*x)^(1/3) - ((3^(1/2) *1i)/2 + 1/2)^2*(a^3*d^3*703584i + 514944*a^4*d^4*tan(c + d*x)^(1/3)*((3^( 1/2)*1i)/2 + 1/2)*(-1/(64*a^3*d^3))^(1/3))*(-1/(64*a^3*d^3))^(2/3))*((3^(1 /2)*1i)/2 + 1/2)*(-1/(64*a^3*d^3))^(1/3)