\(\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\) [286]

Optimal result

Integrand size = 26, antiderivative size = 321 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {i a^{4/3} x}{2^{2/3}}+\frac {11 a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} d}-\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}+\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac {11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac {11 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{6 d}+\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {2 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{3 d}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d} \] Output:

1/2*I*a^(4/3)*x*2^(1/3)+11/9*3^(1/2)*a^(4/3)*arctan(1/3*(a^(1/3)+2*(a+I*a* 
tan(d*x+c))^(1/3))*3^(1/2)/a^(1/3))/d-2^(1/3)*3^(1/2)*a^(4/3)*arctan(1/3*( 
a^(1/3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))*3^(1/2)/a^(1/3))/d+1/2*a^(4/3)*l 
n(cos(d*x+c))*2^(1/3)/d+11/18*a^(4/3)*ln(tan(d*x+c))/d-11/6*a^(4/3)*ln(a^( 
1/3)-(a+I*a*tan(d*x+c))^(1/3))/d+3/2*a^(4/3)*ln(2^(1/3)*a^(1/3)-(a+I*a*tan 
(d*x+c))^(1/3))*2^(1/3)/d-2/3*I*a*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/3)/d-1/ 
2*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(4/3)/d
 

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.15 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {-22 \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )+18 \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+22 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )-18 \sqrt [3]{2} a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )-11 a^{4/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )+9 \sqrt [3]{2} a^{4/3} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )+21 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}+9 a \cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{18 d} \] Input:

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

Output:

-1/18*(-22*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3 
))/(Sqrt[3]*a^(1/3))] + 18*2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(1 + (2^(2/3)*(a 
 + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]] + 22*a^(4/3)*Log[a^(1/3) - ( 
a + I*a*Tan[c + d*x])^(1/3)] - 18*2^(1/3)*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a 
 + I*a*Tan[c + d*x])^(1/3)] - 11*a^(4/3)*Log[a^(2/3) + a^(1/3)*(a + I*a*Ta 
n[c + d*x])^(1/3) + (a + I*a*Tan[c + d*x])^(2/3)] + 9*2^(1/3)*a^(4/3)*Log[ 
2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + I*a*Tan[c + d*x])^(1/3) + (a + I*a* 
Tan[c + d*x])^(2/3)] + (21*I)*a*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3) 
+ 9*a*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3))/d
 

Rubi [A] (warning: unable to verify)

Time = 1.22 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.91, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {3042, 4044, 27, 3042, 4076, 27, 3042, 4083, 3042, 3962, 69, 16, 1082, 217, 4082, 69, 16, 1082, 217}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4044

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4076

\(\displaystyle \frac {\int -\frac {1}{3} \cot (c+d x) \sqrt [3]{i \tan (c+d x) a+a} \left (7 i \tan (c+d x) a^2+11 a^2\right )dx-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{3} \int \cot (c+d x) \sqrt [3]{i \tan (c+d x) a+a} \left (7 i \tan (c+d x) a^2+11 a^2\right )dx-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {1}{3} \int \frac {\sqrt [3]{i \tan (c+d x) a+a} \left (7 i \tan (c+d x) a^2+11 a^2\right )}{\tan (c+d x)}dx-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 4083

\(\displaystyle \frac {\frac {1}{3} \left (-18 i a^2 \int \sqrt [3]{i \tan (c+d x) a+a}dx-11 a \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}dx\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (-18 i a^2 \int \sqrt [3]{i \tan (c+d x) a+a}dx-11 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 3962

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {18 a^3 \int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}d(i a \tan (c+d x))}{d}-11 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {18 a^3 \left (\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2\ 2^{2/3} a^{2/3}}+\frac {3 \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}-11 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {18 a^3 \left (\frac {3 \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}-11 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {18 a^3 \left (-\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}-11 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{3} \left (-11 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx-\frac {18 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 4082

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {11 a^3 \int \frac {\cot (c+d x)}{(i \tan (c+d x) a+a)^{2/3}}d\tan (c+d x)}{d}-\frac {18 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {11 a^3 \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{i \tan (c+d x) a+a}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}\right )}{d}-\frac {18 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {11 a^3 \left (-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 a^{2/3}}\right )}{d}-\frac {18 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {11 a^3 \left (\frac {3 \int \frac {1}{-(i \tan (c+d x) a+a)^{2/3}-3}d\left (\frac {2 \sqrt [3]{i \tan (c+d x) a+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 a^{2/3}}\right )}{d}-\frac {18 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{3} \left (-\frac {11 a^3 \left (-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 a^{2/3}}\right )}{d}-\frac {18 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )-\frac {2 i a^2 \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{3 a}-\frac {\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}\)

Input:

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

Output:

-1/2*(Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3))/d + (((-18*a^3*((I*Sqrt 
[3]*ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(2/3)*a^(2/3)) - (3*Log[2^(1/3)* 
a^(1/3) - I*a*Tan[c + d*x]])/(2*2^(2/3)*a^(2/3)) + Log[a - I*a*Tan[c + d*x 
]]/(2*2^(2/3)*a^(2/3))))/d - (11*a^3*(-((Sqrt[3]*ArcTan[(1 + (2*(a + I*a*T 
an[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3)) - Log[Tan[c + d*x]]/(2*a^( 
2/3)) + (3*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(2/3))))/d)/3 
 - ((2*I)*a^2*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/d)/(3*a)
 

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 27

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

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 3042

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

rule 3962

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d   S 
ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b 
, c, d, n}, x] && EqQ[a^2 + b^2, 0]
 

rule 4044

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

rule 4076

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^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n 
+ 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Simp[a/(d*(b*c + a*d)*(n + 1))   Int[ 
(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n 
 - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b 
*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] 
 && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]
 

rule 4082

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[b*(B/f)   Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], 
x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
 

rule 4083

Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)]))/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[( 
A*b + a*B)/(b*c + a*d)   Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[(B*c - A 
*d)/(b*c + a*d)   Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*T 
an[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - 
 a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
 

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}-\frac {7 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{6 d \tan \left (d x +c \right )^{2}}+\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{3 d \tan \left (d x +c \right )^{2}}-\frac {11 a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 d}+\frac {11 a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{18 d}+\frac {11 a^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 d}\)	\(301\)
default	\(\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}-\frac {7 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{6 d \tan \left (d x +c \right )^{2}}+\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{3 d \tan \left (d x +c \right )^{2}}-\frac {11 a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 d}+\frac {11 a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{18 d}+\frac {11 a^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 d}\)	\(301\)

Input:

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

Output:

1/d*a^(4/3)*2^(1/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))-1/2/d*a^( 
4/3)*2^(1/3)*ln((a+I*a*tan(d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I*a*tan(d*x+c) 
)^(1/3)+2^(2/3)*a^(2/3))-1/d*a^(4/3)*2^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2 
^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))-7/6/d/tan(d*x+c)^2*(a+I*a*tan( 
d*x+c))^(4/3)+2/3/d*a/tan(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/3)-11/9/d*a^(4/3) 
*ln((a+I*a*tan(d*x+c))^(1/3)-a^(1/3))+11/18/d*a^(4/3)*ln((a+I*a*tan(d*x+c) 
)^(2/3)+a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+a^(2/3))+11/9/d*a^(4/3)*3^(1/2)*a 
rctan(1/3*3^(1/2)*(2/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (240) = 480\).

Time = 0.11 (sec) , antiderivative size = 744, normalized size of antiderivative = 2.32 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/18*(6*2^(1/3)*(5*a*e^(4*I*d*x + 4*I*c) + 3*a*e^(2*I*d*x + 2*I*c) - 2*a)* 
(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - 9*2^(1/3)*(( 
I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) + 2*(-I*sqrt(3)*d - d)*e^(2*I*d*x + 2 
*I*c) + I*sqrt(3)*d + d)*(a^4/d^3)^(1/3)*log(1/2*(2*2^(1/3)*a*(a/(e^(2*I*d 
*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) + 2^(1/3)*(I*sqrt(3)*d + d 
)*(a^4/d^3)^(1/3))/a) - 9*2^(1/3)*((-I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) 
+ 2*(I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(a^4/d^3)^(1/ 
3)*log(1/2*(2*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 
 2/3*I*c) + 2^(1/3)*(-I*sqrt(3)*d + d)*(a^4/d^3)^(1/3))/a) + 18*2^(1/3)*(d 
*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*(a^4/d^3)^(1/3)*log((2 
^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - 2^( 
1/3)*(a^4/d^3)^(1/3)*d)/a) - 11*((I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) + 2 
*(-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*(-a^4/d^3)^(1/3 
)*log(1/2*(2*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 
2/3*I*c) - (I*sqrt(3)*d + d)*(-a^4/d^3)^(1/3))/a) - 11*((-I*sqrt(3)*d + d) 
*e^(4*I*d*x + 4*I*c) + 2*(I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3) 
*d + d)*(-a^4/d^3)^(1/3)*log(1/2*(2*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1) 
)^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - (-I*sqrt(3)*d + d)*(-a^4/d^3)^(1/3))/a) 
+ 22*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*(-a^4/d^3)^(1/3 
)*log((2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3...
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.95 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {{\left (\frac {18 \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} - \frac {22 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} + \frac {9 \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} - \frac {18 \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} - \frac {11 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} + \frac {22 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} - \frac {3 \, {\left (7 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + a^{2}}\right )} a^{2}}{18 \, d} \] Input:

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

Output:

-1/18*(18*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2* 
(I*a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(2/3) - 22*sqrt(3)*arctan(1/3*sqr 
t(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(2/3) + 9*2^(1/ 
3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I 
*a*tan(d*x + c) + a)^(2/3))/a^(2/3) - 18*2^(1/3)*log(-2^(1/3)*a^(1/3) + (I 
*a*tan(d*x + c) + a)^(1/3))/a^(2/3) - 11*log((I*a*tan(d*x + c) + a)^(2/3) 
+ (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(2/3) + 22*log((I*a*ta 
n(d*x + c) + a)^(1/3) - a^(1/3))/a^(2/3) - 3*(7*(I*a*tan(d*x + c) + a)^(4/ 
3) - 4*(I*a*tan(d*x + c) + a)^(1/3)*a)/((I*a*tan(d*x + c) + a)^2 - 2*(I*a* 
tan(d*x + c) + a)*a + a^2))*a^2/d
 

Giac [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.88 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {i \, {\left (-18 i \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) + 22 i \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - 9 i \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) + 18 i \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) + 11 i \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 22 i \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - \frac {3 \, {\left (7 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a - 4 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{2}\right )}}{a^{2} \tan \left (d x + c\right )^{2}}\right )} a}{18 \, d} \] Input:

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

Output:

-1/18*I*(-18*I*sqrt(3)*2^(1/3)*a^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) 
*a^(1/3) + 2*(I*a*tan(d*x + c) + a)^(1/3))/a^(1/3)) + 22*I*sqrt(3)*a^(1/3) 
*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3)) - 
9*I*2^(1/3)*a^(1/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^( 
1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(2/3)) + 18*I*2^(1/3)*a^(1/3)*log(-2 
^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3)) + 11*I*a^(1/3)*log((I*a*tan 
(d*x + c) + a)^(2/3) + (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3)) - 2 
2*I*a^(1/3)*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3)) - 3*(7*I*(I*a*tan( 
d*x + c) + a)^(4/3)*a - 4*I*(I*a*tan(d*x + c) + a)^(1/3)*a^2)/(a^2*tan(d*x 
 + c)^2))*a/d
 

Mupad [B] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.43 \[ \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx =\text {Too large to display} \] Input:

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

Output:

(11*log(d*(-a^4/d^3)^(1/3) + a*(a*(tan(c + d*x)*1i + 1))^(1/3))*(-a^4/d^3) 
^(1/3))/9 - ((2*a^3*(a + a*tan(c + d*x)*1i)^(1/3))/3 - (7*a^2*(a + a*tan(c 
 + d*x)*1i)^(4/3))/6)/(d*(a + a*tan(c + d*x)*1i)^2 + a^2*d - 2*a*d*(a + a* 
tan(c + d*x)*1i)) + log(a*(a*(tan(c + d*x)*1i + 1))^(1/3) - 2^(1/3)*d*(a^4 
/d^3)^(1/3))*((2*a^4)/d^3)^(1/3) - (11*log(d*(-a^4/d^3)^(1/3) - 2*a*(a*(ta 
n(c + d*x)*1i + 1))^(1/3) + 3^(1/2)*d*(-a^4/d^3)^(1/3)*1i)*(3^(1/2)*1i + 1 
)*(-a^4/d^3)^(1/3))/18 + (11*log(2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) - d*( 
-a^4/d^3)^(1/3) + 3^(1/2)*d*(-a^4/d^3)^(1/3)*1i)*(3^(1/2)*1i - 1)*(-a^4/d^ 
3)^(1/3))/18 + log(2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) + 2^(1/3)*d*(a^4/d^ 
3)^(1/3) - 2^(1/3)*3^(1/2)*d*(a^4/d^3)^(1/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(( 
2*a^4)/d^3)^(1/3) - log(2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) + 2^(1/3)*d*(a 
^4/d^3)^(1/3) + 2^(1/3)*3^(1/2)*d*(a^4/d^3)^(1/3)*1i)*((3^(1/2)*1i)/2 + 1/ 
2)*((2*a^4)/d^3)^(1/3)
 

Reduce [F]

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

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

Output:

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