\(\int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx\) [243]

Optimal result

Integrand size = 26, antiderivative size = 335 \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {\arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {i \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {\arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {i \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {\log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {\log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {i \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{18 a^2 d}+\frac {\sqrt [3]{\tan (c+d x)}}{3 a^2 d (1+i \tan (c+d x))}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2} \] Output:

-1/72*arctan(-3^(1/2)+2*tan(d*x+c)^(1/3))/a^2/d-1/72*arctan(3^(1/2)+2*tan( 
d*x+c)^(1/3))/a^2/d+1/9*I*arctan(1/3*(1-2*tan(d*x+c)^(2/3))*3^(1/2))*3^(1/ 
2)/a^2/d-1/36*arctan(tan(d*x+c)^(1/3))/a^2/d+1/9*I*ln(1+tan(d*x+c)^(2/3))/ 
a^2/d+1/144*ln(1-3^(1/2)*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3))*3^(1/2)/a^2/d- 
1/144*ln(1+3^(1/2)*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3))*3^(1/2)/a^2/d-1/18*I 
*ln(1-tan(d*x+c)^(2/3)+tan(d*x+c)^(4/3))/a^2/d+1/3*tan(d*x+c)^(1/3)/a^2/d/ 
(1+I*tan(d*x+c))-1/4*tan(d*x+c)^(1/3)/d/(a+I*a*tan(d*x+c))^2
 

Mathematica [A] (warning: unable to verify)

Time = 3.71 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.12 \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\frac {\sqrt [3]{\tan (c+d x)} \left (18 \tan ^2(c+d x)+\frac {(a+i a \tan (c+d x)) \left (6 a \tan ^2(c+d x)^{2/3}+18 i a \tan (c+d x) \tan ^2(c+d x)^{2/3}+(a+i a \tan (c+d x)) \left (8 i \left (\log \left (1+\sqrt [3]{\tan ^2(c+d x)}\right )-\sqrt [3]{-1} \log \left (1-\sqrt [3]{-1} \sqrt [3]{\tan ^2(c+d x)}\right )+(-1)^{2/3} \log \left (1+(-1)^{2/3} \sqrt [3]{\tan ^2(c+d x)}\right )\right ) \tan (c+d x)+\left (-i \log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right )+i \log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right )+\sqrt [6]{-1} \left (-(-1)^{2/3} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )+(-1)^{2/3} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )-\log \left (1-(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )+\log \left (1+(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )\right )\right ) \sqrt {\tan ^2(c+d x)}\right )\right )}{a^2 \tan ^2(c+d x)^{2/3}}\right )}{72 d (a+i a \tan (c+d x))^2} \] Input:

Integrate[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^2,x]
 

Output:

(Tan[c + d*x]^(1/3)*(18*Tan[c + d*x]^2 + ((a + I*a*Tan[c + d*x])*(6*a*(Tan 
[c + d*x]^2)^(2/3) + (18*I)*a*Tan[c + d*x]*(Tan[c + d*x]^2)^(2/3) + (a + I 
*a*Tan[c + d*x])*((8*I)*(Log[1 + (Tan[c + d*x]^2)^(1/3)] - (-1)^(1/3)*Log[ 
1 - (-1)^(1/3)*(Tan[c + d*x]^2)^(1/3)] + (-1)^(2/3)*Log[1 + (-1)^(2/3)*(Ta 
n[c + d*x]^2)^(1/3)])*Tan[c + d*x] + ((-I)*Log[1 - I*(Tan[c + d*x]^2)^(1/6 
)] + I*Log[1 + I*(Tan[c + d*x]^2)^(1/6)] + (-1)^(1/6)*(-((-1)^(2/3)*Log[1 
- (-1)^(1/6)*(Tan[c + d*x]^2)^(1/6)]) + (-1)^(2/3)*Log[1 + (-1)^(1/6)*(Tan 
[c + d*x]^2)^(1/6)] - Log[1 - (-1)^(5/6)*(Tan[c + d*x]^2)^(1/6)] + Log[1 + 
 (-1)^(5/6)*(Tan[c + d*x]^2)^(1/6)]))*Sqrt[Tan[c + d*x]^2])))/(a^2*(Tan[c 
+ d*x]^2)^(2/3))))/(72*d*(a + I*a*Tan[c + d*x])^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.85, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 4041, 27, 3042, 4079, 27, 3042, 4021, 3042, 3957, 266, 753, 27, 216, 807, 821, 16, 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.

\(\displaystyle \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (c+d x)^{4/3}}{(a+i a \tan (c+d x))^2}dx\)

\(\Big \downarrow \) 4041

\(\displaystyle -\frac {\int -\frac {a-7 i a \tan (c+d x)}{3 \tan ^{\frac {2}{3}}(c+d x) (i \tan (c+d x) a+a)}dx}{4 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a-7 i a \tan (c+d x)}{\tan ^{\frac {2}{3}}(c+d x) (i \tan (c+d x) a+a)}dx}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a-7 i a \tan (c+d x)}{\tan (c+d x)^{2/3} (i \tan (c+d x) a+a)}dx}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 4079

\(\displaystyle \frac {\frac {\int -\frac {2 \left (8 i \tan (c+d x) a^2+a^2\right )}{3 \tan ^{\frac {2}{3}}(c+d x)}dx}{2 a^2}+\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\int \frac {8 i \tan (c+d x) a^2+a^2}{\tan ^{\frac {2}{3}}(c+d x)}dx}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\int \frac {8 i \tan (c+d x) a^2+a^2}{\tan (c+d x)^{2/3}}dx}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 4021

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {a^2 \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)}dx+8 i a^2 \int \sqrt [3]{\tan (c+d x)}dx}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {a^2 \int \frac {1}{\tan (c+d x)^{2/3}}dx+8 i a^2 \int \sqrt [3]{\tan (c+d x)}dx}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {a^2 \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}+\frac {8 i a^2 \int \frac {\sqrt [3]{\tan (c+d x)}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}+\frac {24 i a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 753

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{2 \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{2 \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}\right )}{d}+\frac {24 i a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )}{d}+\frac {24 i a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {24 i a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {12 i a^2 \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\tan (c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 821

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {12 i a^2 \left (\frac {1}{3} \int \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)\right )}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {12 i a^2 \left (\frac {1}{3} \int \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {12 i a^2 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{2} \int \left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {12 i a^2 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}-\sqrt {3}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {12 i a^2 \left (\frac {1}{3} \left (-3 \int \frac {1}{-2 \tan ^{\frac {2}{3}}(c+d x)-2}d\left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {12 i a^2 \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {4 \sqrt [3]{\tan (c+d x)}}{d (1+i \tan (c+d x))}-\frac {\frac {3 a^2 \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )+\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}+\frac {12 i a^2 \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}}{3 a^2}}{12 a^2}-\frac {\sqrt [3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}\)

Input:

Int[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^2,x]
 

Output:

(-1/3*(((12*I)*a^2*(ArcTan[(-1 + 2*Tan[c + d*x]^(2/3))/Sqrt[3]]/Sqrt[3] - 
Log[1 + Tan[c + d*x]^(2/3)]/3))/d + (3*a^2*(ArcTan[Tan[c + d*x]^(1/3)]/3 + 
 (-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 + (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)/a^2 + (4*Tan[c + d*x]^(1/3))/(d*(1 + I*Tan[c + d*x])))/(12* 
a^2) - Tan[c + d*x]^(1/3)/(4*d*(a + I*a*Tan[c + d*x])^2)
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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])
 

rule 217

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])
 

rule 266

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]
 

rule 753

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ 
b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k 
 - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ 
(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* 
x^2), x]; 2*(r^2/(a*n))   Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n))   Sum[u, 
{k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a 
/b]
 

rule 807

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]
 

rule 821

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 
1)   Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) 
 Int[(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]
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3957

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]
 

rule 4021

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]
 

rule 4041

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m* 
((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m)), x] + Simp[1/(2*a^2*m)   Int[(a + 
b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n 
 - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Ta 
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] 
 && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (In 
tegerQ[m] || IntegersQ[2*m, 2*n])
 

rule 4079

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( 
b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d))   Int[(a + b*Tan[e + f*x])^(m 
 + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m 
- b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free 
Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] 
 && LtQ[m, 0] &&  !GtQ[n, 0]
 

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {-\frac {6 \tan \left (d x +c \right )+4 i \tan \left (d x +c \right )^{\frac {2}{3}}+4 \tan \left (d x +c \right )^{\frac {1}{3}}-4 i}{72 \left (-i \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}-1\right )^{2}}-\frac {7 i \ln \left (-i \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}-1\right )}{144}+\frac {7 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )}{72}-\frac {i \ln \left (i \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}-1\right )}{16}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )}{8}+\frac {i \ln \left (\tan \left (d x +c \right )^{\frac {1}{3}}-i\right )}{8}-\frac {i}{36 \left (\tan \left (d x +c \right )^{\frac {1}{3}}+i\right )^{2}}+\frac {7 i \ln \left (\tan \left (d x +c \right )^{\frac {1}{3}}+i\right )}{72}+\frac {1}{12 \tan \left (d x +c \right )^{\frac {1}{3}}+12 i}}{d \,a^{2}}\)	\(225\)
default	\(\frac {-\frac {6 \tan \left (d x +c \right )+4 i \tan \left (d x +c \right )^{\frac {2}{3}}+4 \tan \left (d x +c \right )^{\frac {1}{3}}-4 i}{72 \left (-i \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}-1\right )^{2}}-\frac {7 i \ln \left (-i \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}-1\right )}{144}+\frac {7 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )}{72}-\frac {i \ln \left (i \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}-1\right )}{16}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )}{8}+\frac {i \ln \left (\tan \left (d x +c \right )^{\frac {1}{3}}-i\right )}{8}-\frac {i}{36 \left (\tan \left (d x +c \right )^{\frac {1}{3}}+i\right )^{2}}+\frac {7 i \ln \left (\tan \left (d x +c \right )^{\frac {1}{3}}+i\right )}{72}+\frac {1}{12 \tan \left (d x +c \right )^{\frac {1}{3}}+12 i}}{d \,a^{2}}\)	\(225\)

Input:

int(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d/a^2*(-1/72*(6*tan(d*x+c)+4*I*tan(d*x+c)^(2/3)+4*tan(d*x+c)^(1/3)-4*I)/ 
(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)^2-7/144*I*ln(-I*tan(d*x+c)^(1/3)+ 
tan(d*x+c)^(2/3)-1)+7/72*3^(1/2)*arctanh(1/3*(-I+2*tan(d*x+c)^(1/3))*3^(1/ 
2))-1/16*I*ln(I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)-1/8*3^(1/2)*arctanh(1 
/3*(I+2*tan(d*x+c)^(1/3))*3^(1/2))+1/8*I*ln(tan(d*x+c)^(1/3)-I)-1/36*I/(ta 
n(d*x+c)^(1/3)+I)^2+7/72*I*ln(tan(d*x+c)^(1/3)+I)+1/12/(tan(d*x+c)^(1/3)+I 
))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.56 \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx =\text {Too large to display} \] Input:

integrate(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
 

Output:

-1/144*(9*(sqrt(3)*a^2*d*sqrt(1/(a^4*d^2))*e^(4*I*d*x + 4*I*c) + I*e^(4*I* 
d*x + 4*I*c))*log(1/2*sqrt(3)*a^2*d*sqrt(1/(a^4*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) - 9*(sqrt(3)*a^2*d*s 
qrt(1/(a^4*d^2))*e^(4*I*d*x + 4*I*c) - I*e^(4*I*d*x + 4*I*c))*log(-1/2*sqr 
t(3)*a^2*d*sqrt(1/(a^4*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) - 7*(3*sqrt(1/3)*a^2*d*sqrt(1/(a^4*d^2))*e^(4 
*I*d*x + 4*I*c) - I*e^(4*I*d*x + 4*I*c))*log(3/2*sqrt(1/3)*a^2*d*sqrt(1/(a 
^4*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) + 7*(3*sqrt(1/3)*a^2*d*sqrt(1/(a^4*d^2))*e^(4*I*d*x + 4*I*c) + I* 
e^(4*I*d*x + 4*I*c))*log(-3/2*sqrt(1/3)*a^2*d*sqrt(1/(a^4*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) - 14*I*e^( 
4*I*d*x + 4*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) - 18*I*e^(4*I*d*x + 4*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) - 3*((-I*e^(2*I*d*x + 2*I*c) + I)/ 
(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*(5*e^(4*I*d*x + 4*I*c) + 2*e^(2*I*d*x + 2 
*I*c) - 3))*e^(-4*I*d*x - 4*I*c)/(a^2*d)
 

Sympy [F]

\[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {\tan ^{\frac {4}{3}}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \] Input:

integrate(tan(d*x+c)**(4/3)/(a+I*a*tan(d*x+c))**2,x)
 

Output:

-Integral(tan(c + d*x)**(4/3)/(tan(c + d*x)**2 - 2*I*tan(c + d*x) - 1), x) 
/a**2
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.58 \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {7 \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right ) - 9 \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right ) - \frac {12 \, {\left (-4 i \, \tan \left (d x + c\right )^{\frac {4}{3}} - \tan \left (d x + c\right )^{\frac {1}{3}}\right )}}{{\left (\tan \left (d x + c\right ) - i\right )}^{2}} + 9 i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right ) + 7 i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right ) - 14 i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right ) - 18 i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{144 \, a^{2} d} \] Input:

integrate(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
 

Output:

-1/144*(7*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) + I)/(sqrt(3) + 2*t 
an(d*x + c)^(1/3) - I)) - 9*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) - 
 I)/(sqrt(3) + 2*tan(d*x + c)^(1/3) + I)) - 12*(-4*I*tan(d*x + c)^(4/3) - 
tan(d*x + c)^(1/3))/(tan(d*x + c) - I)^2 + 9*I*log(tan(d*x + c)^(2/3) + I* 
tan(d*x + c)^(1/3) - 1) + 7*I*log(tan(d*x + c)^(2/3) - I*tan(d*x + c)^(1/3 
) - 1) - 14*I*log(tan(d*x + c)^(1/3) + I) - 18*I*log(tan(d*x + c)^(1/3) - 
I))/(a^2*d)
 

Mupad [B] (verification not implemented)

Time = 2.41 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.95 \[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:

int(tan(c + d*x)^(4/3)/(a + a*tan(c + d*x)*1i)^2,x)
 

Output:

log(((a^6*d^3*49408i)/3 + 3538944*a^10*d^5*tan(c + d*x)^(1/3)*(-1i/(512*a^ 
6*d^3))^(2/3))*(-1i/(512*a^6*d^3))^(1/3) + (a^4*d^2*tan(c + d*x)^(1/3)*145 
60i)/3)*(-1i/(512*a^6*d^3))^(1/3) - ((tan(c + d*x)^(1/3)*1i)/(12*a^2*d) - 
tan(c + d*x)^(4/3)/(3*a^2*d))/(2*tan(c + d*x) + tan(c + d*x)^2*1i - 1i) + 
log(((a^6*d^3*49408i)/3 + 3538944*a^10*d^5*tan(c + d*x)^(1/3)*(-343i/(3732 
48*a^6*d^3))^(2/3))*(-343i/(373248*a^6*d^3))^(1/3) + (a^4*d^2*tan(c + d*x) 
^(1/3)*14560i)/3)*(-343i/(373248*a^6*d^3))^(1/3) + (log(((3^(1/2)*1i - 1)* 
((a^6*d^3*49408i)/3 + 884736*a^10*d^5*tan(c + d*x)^(1/3)*(3^(1/2)*1i - 1)^ 
2*(-1i/(512*a^6*d^3))^(2/3))*(-1i/(512*a^6*d^3))^(1/3))/2 + (a^4*d^2*tan(c 
 + d*x)^(1/3)*14560i)/3)*(3^(1/2)*1i - 1)*(-1i/(512*a^6*d^3))^(1/3))/2 - ( 
log(((3^(1/2)*1i + 1)*((a^6*d^3*49408i)/3 + 884736*a^10*d^5*tan(c + d*x)^( 
1/3)*(3^(1/2)*1i + 1)^2*(-1i/(512*a^6*d^3))^(2/3))*(-1i/(512*a^6*d^3))^(1/ 
3))/2 - (a^4*d^2*tan(c + d*x)^(1/3)*14560i)/3)*(3^(1/2)*1i + 1)*(-1i/(512* 
a^6*d^3))^(1/3))/2 + (log(((3^(1/2)*1i - 1)*((a^6*d^3*49408i)/3 + 884736*a 
^10*d^5*tan(c + d*x)^(1/3)*(3^(1/2)*1i - 1)^2*(-343i/(373248*a^6*d^3))^(2/ 
3))*(-343i/(373248*a^6*d^3))^(1/3))/2 + (a^4*d^2*tan(c + d*x)^(1/3)*14560i 
)/3)*(3^(1/2)*1i - 1)*(-343i/(373248*a^6*d^3))^(1/3))/2 - (log(((3^(1/2)*1 
i + 1)*((a^6*d^3*49408i)/3 + 884736*a^10*d^5*tan(c + d*x)^(1/3)*(3^(1/2)*1 
i + 1)^2*(-343i/(373248*a^6*d^3))^(2/3))*(-343i/(373248*a^6*d^3))^(1/3))/2 
 - (a^4*d^2*tan(c + d*x)^(1/3)*14560i)/3)*(3^(1/2)*1i + 1)*(-343i/(3732...
 

Reduce [F]

\[ \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\int \frac {\tan \left (d x +c \right )^{\frac {4}{3}}}{\tan \left (d x +c \right )^{2}-2 \tan \left (d x +c \right ) i -1}d x}{a^{2}} \] Input:

int(tan(d*x+c)^(4/3)/(a+I*a*tan(d*x+c))^2,x)
 

Output:

( - int((tan(c + d*x)**(1/3)*tan(c + d*x))/(tan(c + d*x)**2 - 2*tan(c + d* 
x)*i - 1),x))/a**2