\(\int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx\) [278]

Optimal result

Integrand size = 26, antiderivative size = 327 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {8 \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} d}-\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac {4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac {4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d} \] Output:

1/4*I*a^(1/3)*x*2^(1/3)+8/9*3^(1/2)*a^(1/3)*arctan(1/3*(a^(1/3)+2*(a+I*a*t 
an(d*x+c))^(1/3))*3^(1/2)/a^(1/3))/d-1/2*3^(1/2)*a^(1/3)*arctan(1/3*(a^(1/ 
3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))*3^(1/2)/a^(1/3))*2^(1/3)/d+1/4*a^(1/3 
)*ln(cos(d*x+c))*2^(1/3)/d+4/9*a^(1/3)*ln(tan(d*x+c))/d-4/3*a^(1/3)*ln(a^( 
1/3)-(a+I*a*tan(d*x+c))^(1/3))/d+3/4*a^(1/3)*ln(2^(1/3)*a^(1/3)-(a+I*a*tan 
(d*x+c))^(1/3))*2^(1/3)/d-1/6*I*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))^(1/3)/d
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.12 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {-32 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )+18 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+32 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )-18 \sqrt [3]{2} \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )-16 \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )+9 \sqrt [3]{2} \sqrt [3]{a} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )+6 i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}+18 \cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{36 d} \] Input:

Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(1/3),x]
 

Output:

-1/36*(-32*Sqrt[3]*a^(1/3)*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^(1/3)*ArcTan[(1 + (2^(2/3)*(a 
 + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]] + 32*a^(1/3)*Log[a^(1/3) - ( 
a + I*a*Tan[c + d*x])^(1/3)] - 18*2^(1/3)*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a 
 + I*a*Tan[c + d*x])^(1/3)] - 16*a^(1/3)*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^(1/3)*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)] + (6*I)*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3) + 1 
8*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3))/d
 

Rubi [A] (warning: unable to verify)

Time = 1.18 (sec) , antiderivative size = 293, 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, 4081, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4044

\(\displaystyle \frac {\int \frac {1}{3} \cot ^2(c+d x) (i a-5 a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}dx}{2 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \cot ^2(c+d x) (i a-5 a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}dx}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(i a-5 a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)^2}dx}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4081

\(\displaystyle \frac {\frac {\int -\frac {2}{3} \cot (c+d x) \sqrt [3]{i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+8 a^2\right )dx}{a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {2 \int \cot (c+d x) \sqrt [3]{i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+8 a^2\right )dx}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 \int \frac {\sqrt [3]{i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+8 a^2\right )}{\tan (c+d x)}dx}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4083

\(\displaystyle \frac {-\frac {2 \left (9 i a^2 \int \sqrt [3]{i \tan (c+d x) a+a}dx+8 a \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}dx\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 \left (9 i a^2 \int \sqrt [3]{i \tan (c+d x) a+a}dx+8 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3962

\(\displaystyle \frac {-\frac {2 \left (\frac {9 a^3 \int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}d(i a \tan (c+d x))}{d}+8 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {-\frac {2 \left (\frac {9 a^3 \left (\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2\ 2^{2/3} a^{2/3}}+\frac {3 \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}+8 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {-\frac {2 \left (\frac {9 a^3 \left (\frac {3 \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}+8 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {-\frac {2 \left (\frac {9 a^3 \left (-\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}+8 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {-\frac {2 \left (8 a \int \frac {(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}{\tan (c+d x)}dx+\frac {9 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4082

\(\displaystyle \frac {-\frac {2 \left (\frac {8 a^3 \int \frac {\cot (c+d x)}{(i \tan (c+d x) a+a)^{2/3}}d\tan (c+d x)}{d}+\frac {9 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {-\frac {2 \left (\frac {8 a^3 \left (-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{i \tan (c+d x) a+a}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 a^{2/3}}-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}\right )}{d}+\frac {9 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {-\frac {2 \left (\frac {8 a^3 \left (-\frac {3 \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 a^{2/3}}\right )}{d}+\frac {9 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {-\frac {2 \left (\frac {8 a^3 \left (\frac {3 \int \frac {1}{-(i \tan (c+d x) a+a)^{2/3}-3}d\left (\frac {2 \sqrt [3]{i \tan (c+d x) a+a}}{\sqrt [3]{a}}+1\right )}{a^{2/3}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 a^{2/3}}\right )}{d}+\frac {9 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {-\frac {2 \left (\frac {8 a^3 \left (-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {\log (\tan (c+d x))}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 a^{2/3}}\right )}{d}+\frac {9 a^3 \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\right )}{3 a}-\frac {i a \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{d}}{6 a}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}\)

Input:

Int[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(1/3),x]
 

Output:

-1/2*(Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3))/d + ((-2*((9*a^3*((I*Sq 
rt[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 + (8*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*a) - (I*a*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/d)/(6*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 4081

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

\[\int \cot \left (d x +c \right )^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}d x\]

Input:

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

Output:

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

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. 678 vs. \(2 (239) = 478\).

Time = 0.10 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.07 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/3),x, algorithm="fricas")
 

Output:

1/18*(6*2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*(2*e^(4*I*d*x + 4*I*c) 
 + 3*e^(2*I*d*x + 2*I*c) + 1)*e^(2/3*I*d*x + 2/3*I*c) - 9*(1/4)^(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/d^3)^(1/3)*log((1/4)^(1/3)*(I*sqrt(3)*d + d)*(a/ 
d^3)^(1/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/ 
3*I*c)) - 9*(1/4)^(1/3)*((-I*sqrt(3)*d + d)*e^(4*I*d*x + 4*I*c) + 2*(I*sqr 
t(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*(a/d^3)^(1/3)*log((1/4) 
^(1/3)*(-I*sqrt(3)*d + d)*(a/d^3)^(1/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) 
+ 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 18*(1/4)^(1/3)*(d*e^(4*I*d*x + 4*I* 
c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*(a/d^3)^(1/3)*log(-2*(1/4)^(1/3)*d*(a/d^ 
3)^(1/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3* 
I*c)) - 8*((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/d^3)^(1/3)*log(2^(1/3)*(a/(e^(2*I 
*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - 1/2*(I*sqrt(3)*d + d)* 
(-a/d^3)^(1/3)) - 8*((-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/d^3)^(1/3)*log(2^(1/3)* 
(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - 1/2*(-I*sqrt 
(3)*d + d)*(-a/d^3)^(1/3)) + 16*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 
2*I*c) + d)*(-a/d^3)^(1/3)*log(2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) 
*e^(2/3*I*d*x + 2/3*I*c) + d*(-a/d^3)^(1/3)))/(d*e^(4*I*d*x + 4*I*c) - ...
 

Sympy [F]

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

integrate(cot(d*x+c)**3*(a+I*a*tan(d*x+c))**(1/3),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.94 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {a^{2} {\left (\frac {18 \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} - \frac {6 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} + a^{3}} - \frac {32 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {9 \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} - \frac {18 \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {5}{3}}} - \frac {16 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} + \frac {32 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {5}{3}}}\right )}}{36 \, d} \] Input:

integrate(cot(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/3),x, algorithm="maxima")
 

Output:

-1/36*a^2*(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^(5/3) - 6*((I*a*tan(d*x + c) 
+ a)^(4/3) + 2*(I*a*tan(d*x + c) + a)^(1/3)*a)/((I*a*tan(d*x + c) + a)^2*a 
 - 2*(I*a*tan(d*x + c) + a)*a^2 + a^3) - 32*sqrt(3)*arctan(1/3*sqrt(3)*(2* 
(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(5/3) + 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^(5/3) - 18*2^(1/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d 
*x + c) + a)^(1/3))/a^(5/3) - 16*log((I*a*tan(d*x + c) + a)^(2/3) + (I*a*t 
an(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(5/3) + 32*log((I*a*tan(d*x + 
c) + a)^(1/3) - a^(1/3))/a^(5/3))/d
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.86 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {18 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 32 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 9 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 18 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 16 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 32 \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {6 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{2}\right )}}{a^{2} \tan \left (d x + c\right )^{2}}}{36 \, d} \] Input:

integrate(cot(d*x+c)^3*(a+I*a*tan(d*x+c))^(1/3),x, algorithm="giac")
 

Output:

-1/36*(18*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)) - 32*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*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*2^(1/3)*a^(1/3)*log(-2^(1/3)*a^(1 
/3) + (I*a*tan(d*x + c) + a)^(1/3)) - 16*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)) + 32*a^(1/3)*log 
((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3)) + 6*((I*a*tan(d*x + c) + a)^(4/3) 
*a + 2*(I*a*tan(d*x + c) + a)^(1/3)*a^2)/(a^2*tan(d*x + c)^2))/d
 

Mupad [B] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.28 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {8\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3}+\frac {\frac {a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}{6}+\frac {a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{3}}{d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2+a^2\,d-2\,a\,d\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {4\,\ln \left (d\,{\left (-\frac {a}{d^3}\right )}^{1/3}-2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\frac {4\,\ln \left (2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-d\,{\left (-\frac {a}{d^3}\right )}^{1/3}+\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}-\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3}-\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3} \] Input:

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

Output:

(8*log((a*(tan(c + d*x)*1i + 1))^(1/3) + d*(-a/d^3)^(1/3))*(-a/d^3)^(1/3)) 
/9 + log((a*(tan(c + d*x)*1i + 1))^(1/3) - 2^(1/3)*d*(a/d^3)^(1/3))*(a/(4* 
d^3))^(1/3) + ((a*(a + a*tan(c + d*x)*1i)^(4/3))/6 + (a^2*(a + a*tan(c + d 
*x)*1i)^(1/3))/3)/(d*(a + a*tan(c + d*x)*1i)^2 + a^2*d - 2*a*d*(a + a*tan( 
c + d*x)*1i)) - (4*log(d*(-a/d^3)^(1/3) - 2*(a*(tan(c + d*x)*1i + 1))^(1/3 
) + 3^(1/2)*d*(-a/d^3)^(1/3)*1i)*(3^(1/2)*1i + 1)*(-a/d^3)^(1/3))/9 + (4*l 
og(2*(a*(tan(c + d*x)*1i + 1))^(1/3) - d*(-a/d^3)^(1/3) + 3^(1/2)*d*(-a/d^ 
3)^(1/3)*1i)*(3^(1/2)*1i - 1)*(-a/d^3)^(1/3))/9 + log((a*(tan(c + d*x)*1i 
+ 1))^(1/3) + (2^(1/3)*d*(a/d^3)^(1/3))/2 - (2^(1/3)*3^(1/2)*d*(a/d^3)^(1/ 
3)*1i)/2)*((3^(1/2)*1i)/2 - 1/2)*(a/(4*d^3))^(1/3) - log((a*(tan(c + d*x)* 
1i + 1))^(1/3) + (2^(1/3)*d*(a/d^3)^(1/3))/2 + (2^(1/3)*3^(1/2)*d*(a/d^3)^ 
(1/3)*1i)/2)*((3^(1/2)*1i)/2 + 1/2)*(a/(4*d^3))^(1/3)
 

Reduce [F]

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

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

Output:

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