\(\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\) [285]

Optimal result

Integrand size = 26, antiderivative size = 315 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {a^{4/3} x}{2^{2/3}}-\frac {4 i a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}+\frac {i \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac {2 i a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{d}-\frac {3 i a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}+\frac {i a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d} \] Output:

1/2*a^(4/3)*x*2^(1/3)-4/3*I*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))*3^(1/2)/d+I*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*I*a^(4/3 
)*ln(cos(d*x+c))*2^(1/3)/d-2/3*I*a^(4/3)*ln(tan(d*x+c))/d+2*I*a^(4/3)*ln(a 
^(1/3)-(a+I*a*tan(d*x+c))^(1/3))/d-3/2*I*a^(4/3)*ln(2^(1/3)*a^(1/3)-(a+I*a 
*tan(d*x+c))^(1/3))*2^(1/3)/d+I*a*(a+I*a*tan(d*x+c))^(1/3)/d-cot(d*x+c)*(a 
+I*a*tan(d*x+c))^(4/3)/d
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {i \left (8 \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )-6 \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-8 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+6 \sqrt [3]{2} a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+4 a^{4/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-3 \sqrt [3]{2} a^{4/3} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-6 i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}\right )}{6 d} \] Input:

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

Output:

((-1/6*I)*(8*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1 
/3))/(Sqrt[3]*a^(1/3))] - 6*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]] - 8*a^(4/3)*Log[a^(1/3) - ( 
a + I*a*Tan[c + d*x])^(1/3)] + 6*2^(1/3)*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a 
+ I*a*Tan[c + d*x])^(1/3)] + 4*a^(4/3)*Log[a^(2/3) + a^(1/3)*(a + I*a*Tan[ 
c + d*x])^(1/3) + (a + I*a*Tan[c + d*x])^(2/3)] - 3*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*Ta 
n[c + d*x])^(2/3)] - (6*I)*a*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3)))/d
 

Rubi [A] (warning: unable to verify)

Time = 1.20 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.90, 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, 4077, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4044

\(\displaystyle \frac {\int \frac {1}{3} \cot (c+d x) (i \tan (c+d x) a+a)^{4/3} (\tan (c+d x) a+4 i a)dx}{a}-\frac {\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4077

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4083

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3962

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

-((Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3))/d) + (2*(((3*I)*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 + ((2*I)*a^3*(-((Sqrt[3]*ArcTan[(1 + (2*(a + I* 
a*Tan[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*I)*a^2*(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 4077

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*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + 
n))), x] + Simp[1/(d*(m + n))   Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan 
[e + f*x])^n*Simp[a*A*d*(m + n) + B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - 
 a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a 
, b, c, d, e, f, A, B, n}, 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 = 1.15 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {3 i a^{3} \left (-\frac {2 \left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right )}{a}-\frac {-\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{3 a \tan \left (d x +c \right )}-\frac {4 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )}{d}\)	\(278\)
default	\(\frac {3 i a^{3} \left (-\frac {2 \left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right )}{a}-\frac {-\frac {i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{3 a \tan \left (d x +c \right )}-\frac {4 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )}{d}\)	\(278\)

Input:

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

Output:

3*I/d*a^3*(-2/a*(1/6*2^(1/3)/a^(2/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a 
^(1/3))-1/12*2^(1/3)/a^(2/3)*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))-1/6*2^(1/3)/a^(2/3)*3^(1/2)*arcta 
n(1/3*3^(1/2)*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1)))-1/a*(-1/3*I*( 
a+I*a*tan(d*x+c))^(1/3)/a/tan(d*x+c)-4/9/a^(2/3)*ln((a+I*a*tan(d*x+c))^(1/ 
3)-a^(1/3))+2/9/a^(2/3)*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))+4/9/a^(2/3)*3^(1/2)*arctan(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. 622 vs. \(2 (232) = 464\).

Time = 0.09 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.97 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(2*2^(1/3)*(I*a*e^(2*I*d*x + 2*I*c) + I*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)*e^(2*I*d*x + 2*I*c) 
 - I*sqrt(3)*d + d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/8*(8*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) - 3*(sqrt(3)*d + I*d)* 
(-64/27*I*a^4/d^3)^(1/3))/a) - ((-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I 
*sqrt(3)*d + d)*(-64/27*I*a^4/d^3)^(1/3)*log(1/8*(8*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) + 3*(sqrt(3)*d - I*d)*(-64 
/27*I*a^4/d^3)^(1/3))/a) - 2*(d*e^(2*I*d*x + 2*I*c) - d)*(-64/27*I*a^4/d^3 
)^(1/3)*log(1/4*(4*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) + 3*I*(-64/27*I*a^4/d^3)^(1/3)*d)/a) - ((I*sqrt(3)*d - d)*e 
^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(2*I*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) + (sqrt(3 
)*d + I*d)*(2*I*a^4/d^3)^(1/3))/a) - ((-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I* 
c) + I*sqrt(3)*d + d)*(2*I*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) - (sqrt(3)*d - I*d)*(2*I* 
a^4/d^3)^(1/3))/a) - 2*(d*e^(2*I*d*x + 2*I*c) - d)*(2*I*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) - 
I*(2*I*a^4/d^3)^(1/3)*d)/a))/(d*e^(2*I*d*x + 2*I*c) - d)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.82 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {i \, {\left (6 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 8 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 3 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 6 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 4 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 8 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {6 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{\tan \left (d x + c\right )}\right )} a}{6 \, d} \] Input:

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

Output:

1/6*I*(6*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)) - 8*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)) + 3*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)) - 6*2^(1/3)*a^(1/3)*log(-2^(1/3)*a^(1/3) 
 + (I*a*tan(d*x + c) + a)^(1/3)) - 4*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)) + 8*a^(1/3)*log((I*a 
*tan(d*x + c) + a)^(1/3) - a^(1/3)) + 6*I*(I*a*tan(d*x + c) + a)^(1/3)/tan 
(d*x + c))*a/d
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.82 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {i \, {\left (6 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 8 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 3 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 6 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 4 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 8 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {6 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{\tan \left (d x + c\right )}\right )} a}{6 \, d} \] Input:

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

Output:

1/6*I*(6*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)) - 8*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)) + 3*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)) - 6*2^(1/3)*a^(1/3)*log(-2^(1/3)*a^(1/3) 
 + (I*a*tan(d*x + c) + a)^(1/3)) - 4*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)) + 8*a^(1/3)*log((I*a 
*tan(d*x + c) + a)^(1/3) - a^(1/3)) + 6*I*(I*a*tan(d*x + c) + a)^(1/3)/tan 
(d*x + c))*a/d
 

Mupad [B] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 855, normalized size of antiderivative = 2.71 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\text {Too large to display} \] Input:

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

Output:

log(((1458*a^7*d^6*((a^4*2i)/d^3)^(1/3) - a^8*d^5*(a + a*tan(c + d*x)*1i)^ 
(1/3)*810i)*((a^4*2i)/d^3)^(2/3) - a^11*d^3*792i)*((a^4*2i)/d^3)^(1/3) - 3 
744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3))*((a^4*2i)/d^3)^(1/3) + log(((1 
458*a^7*d^6*(-(a^4*64i)/(27*d^3))^(1/3) - a^8*d^5*(a + a*tan(c + d*x)*1i)^ 
(1/3)*810i)*(-(a^4*64i)/(27*d^3))^(2/3) - a^11*d^3*792i)*(-(a^4*64i)/(27*d 
^3))^(1/3) - 3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3))*(-(a^4*64i)/(27* 
d^3))^(1/3) + (log(3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3) + ((3^(1/2) 
*1i - 1)*(a^11*d^3*792i + ((3^(1/2)*1i - 1)^2*(a^8*d^5*(a + a*tan(c + d*x) 
*1i)^(1/3)*810i - 729*a^7*d^6*(3^(1/2)*1i - 1)*((a^4*2i)/d^3)^(1/3))*((a^4 
*2i)/d^3)^(2/3))/4)*((a^4*2i)/d^3)^(1/3))/2)*(3^(1/2)*1i - 1)*((a^4*2i)/d^ 
3)^(1/3))/2 - (log(3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3) - ((3^(1/2) 
*1i + 1)*(a^11*d^3*792i + ((3^(1/2)*1i + 1)^2*(a^8*d^5*(a + a*tan(c + d*x) 
*1i)^(1/3)*810i + 729*a^7*d^6*(3^(1/2)*1i + 1)*((a^4*2i)/d^3)^(1/3))*((a^4 
*2i)/d^3)^(2/3))/4)*((a^4*2i)/d^3)^(1/3))/2)*(3^(1/2)*1i + 1)*((a^4*2i)/d^ 
3)^(1/3))/2 + (log(3744*a^12*d^2*(a + a*tan(c + d*x)*1i)^(1/3) + ((3^(1/2) 
*1i - 1)*(a^11*d^3*792i + ((3^(1/2)*1i - 1)^2*(a^8*d^5*(a + a*tan(c + d*x) 
*1i)^(1/3)*810i - 729*a^7*d^6*(3^(1/2)*1i - 1)*(-(a^4*64i)/(27*d^3))^(1/3) 
)*(-(a^4*64i)/(27*d^3))^(2/3))/4)*(-(a^4*64i)/(27*d^3))^(1/3))/2)*(3^(1/2) 
*1i - 1)*(-(a^4*64i)/(27*d^3))^(1/3))/2 - (log(3744*a^12*d^2*(a + a*tan(c 
+ d*x)*1i)^(1/3) - ((3^(1/2)*1i + 1)*(a^11*d^3*792i + ((3^(1/2)*1i + 1)...
 

Reduce [F]

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

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