\(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx\) [1136]

Optimal result

Integrand size = 30, antiderivative size = 446 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 (c-i d)^{5/2} f}+\frac {\left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{11/2} f}+\frac {d \left (6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right )}{48 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^{3/2}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}+\frac {i c-4 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}+\frac {2 c^2+11 i c d-30 d^2}{16 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right )}{16 a^3 (c-i d)^2 (c+i d)^5 f \sqrt {c+d \tan (e+f x)}} \] Output:

-1/8*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^3/(c-I*d)^(5/2)/f+1 
/16*(2*I*c^3-16*c^2*d-61*I*c*d^2+152*d^3)*arctanh((c+d*tan(f*x+e))^(1/2)/( 
c+I*d)^(1/2))/a^3/(c+I*d)^(11/2)/f+1/48*d*(6*c^3+33*I*c^2*d-83*c*d^2+154*I 
*d^3)/a^3/(c-I*d)/(c+I*d)^4/f/(c+d*tan(f*x+e))^(3/2)-1/6/(I*c-d)/f/(a+I*a* 
tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2)+1/8*(I*c-4*d)/a/(c+I*d)^2/f/(a+I*a*ta 
n(f*x+e))^2/(c+d*tan(f*x+e))^(3/2)+1/16*(2*c^2+11*I*c*d-30*d^2)/(I*c-d)^3/ 
f/(a^3+I*a^3*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2)+1/16*d*(2*c^4+11*I*c^3*d-2 
6*c^2*d^2+253*I*c*d^3+150*d^4)/a^3/(c-I*d)^2/(c+I*d)^5/f/(c+d*tan(f*x+e))^ 
(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.69 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {-\frac {i \left (2 (c+i d)^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )+\left (-2 c^4-14 i c^3 d+45 c^2 d^2+91 i c d^3+152 d^4\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )\right )}{(c-i d) (c+i d)^2}+\frac {8 (c+i d)}{(-i+\tan (e+f x))^3}+\frac {6 i (c+4 i d)}{(-i+\tan (e+f x))^2}+\frac {-6 c^2-33 i c d+90 d^2}{(c+i d) (-i+\tan (e+f x))}}{48 a^3 (c+i d)^2 f (c+d \tan (e+f x))^{3/2}} \] Input:

Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)),x]
 

Output:

-1/48*(((-I)*(2*(c + I*d)^4*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e 
+ f*x])/(c - I*d)] + (-2*c^4 - (14*I)*c^3*d + 45*c^2*d^2 + (91*I)*c*d^3 + 
152*d^4)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]) 
)/((c - I*d)*(c + I*d)^2) + (8*(c + I*d))/(-I + Tan[e + f*x])^3 + ((6*I)*( 
c + (4*I)*d))/(-I + Tan[e + f*x])^2 + (-6*c^2 - (33*I)*c*d + 90*d^2)/((c + 
 I*d)*(-I + Tan[e + f*x])))/(a^3*(c + I*d)^2*f*(c + d*Tan[e + f*x])^(3/2))
 

Rubi [A] (warning: unable to verify)

Time = 2.60 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.633, Rules used = {3042, 4042, 27, 3042, 4079, 3042, 4079, 25, 3042, 4012, 3042, 4012, 3042, 4022, 3042, 4020, 25, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 4042

\(\displaystyle -\frac {\int -\frac {3 (a (2 i c-5 d)+3 i a d \tan (e+f x))}{2 (i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))^{5/2}}dx}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (2 i c-5 d)+3 i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (2 i c-5 d)+3 i a d \tan (e+f x)}{(i \tan (e+f x) a+a)^2 (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4079

\(\displaystyle \frac {-\frac {\int \frac {\left (4 c^2+15 i d c-32 d^2\right ) a^2+7 (c+4 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {\left (4 c^2+15 i d c-32 d^2\right ) a^2+7 (c+4 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^{5/2}}dx}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4079

\(\displaystyle \frac {-\frac {\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac {\int -\frac {\left (4 i c^3-22 d c^2-67 i d^2 c+154 d^3\right ) a^3+5 d \left (2 i c^2-11 d c-30 i d^2\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (4 i c^3-22 d c^2-67 i d^2 c+154 d^3\right ) a^3+5 d \left (2 i c^2-11 d c-30 i d^2\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (4 i c^3-22 d c^2-67 i d^2 c+154 d^3\right ) a^3+5 d \left (2 i c^2-11 d c-30 i d^2\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{5/2}}dx}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (4 i c^4-22 d c^3-57 i d^2 c^2+99 d^3 c-150 i d^4\right ) a^3+d \left (6 i c^3-33 d c^2-83 i d^2 c-154 d^3\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (4 i c^4-22 d c^3-57 i d^2 c^2+99 d^3 c-150 i d^4\right ) a^3+d \left (6 i c^3-33 d c^2-83 i d^2 c-154 d^3\right ) \tan (e+f x) a^3}{(c+d \tan (e+f x))^{3/2}}dx}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\int \frac {\left (4 i c^5-22 d c^4-51 i d^2 c^3+66 d^3 c^2-233 i d^4 c-154 d^5\right ) a^3+d \left (2 i c^4-11 d c^3-26 i d^2 c^2-253 d^3 c+150 i d^4\right ) \tan (e+f x) a^3}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\int \frac {\left (4 i c^5-22 d c^4-51 i d^2 c^3+66 d^3 c^2-233 i d^4 c-154 d^5\right ) a^3+d \left (2 i c^4-11 d c^3-26 i d^2 c^2-253 d^3 c+150 i d^4\right ) \tan (e+f x) a^3}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4022

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+2 a^3 (-d+i c)^5 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+2 a^3 (-d+i c)^5 \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\frac {2 i a^3 (-d+i c)^5 \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{f}-\frac {i a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{f}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\frac {i a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{f}-\frac {2 i a^3 (-d+i c)^5 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{f}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {\frac {2 a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {4 a^3 (-d+i c)^5 \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}+\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {\frac {a^2 \left (2 i c^2-11 c d-30 i d^2\right )}{f (c+i d) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac {\frac {\frac {\frac {2 a^3 (c-i d)^2 \left (2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right ) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {4 a^3 (-d+i c)^5 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}}{c^2+d^2}+\frac {2 a^3 d \left (2 i c^4-11 c^3 d-26 i c^2 d^2-253 c d^3+150 i d^4\right )}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{c^2+d^2}+\frac {2 a^3 d \left (6 i c^3-33 c^2 d-83 i c d^2-154 d^3\right )}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}}{2 a^2 (-d+i c)}}{4 a^2 (-d+i c)}-\frac {a (c+4 i d)}{2 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}}{4 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}\)

Input:

Int[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)),x]
 

Output:

-1/6*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)) + 
 (-1/2*(a*(c + (4*I)*d))/((c + I*d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[ 
e + f*x])^(3/2)) - ((a^2*((2*I)*c^2 - 11*c*d - (30*I)*d^2))/((c + I*d)*f*( 
a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + ((2*a^3*d*((6*I)*c^3 - 
 33*c^2*d - (83*I)*c*d^2 - 154*d^3))/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x]) 
^(3/2)) + (((4*a^3*(I*c - d)^5*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c 
 - I*d]*f) + (2*a^3*(c - I*d)^2*((2*I)*c^3 - 16*c^2*d - (61*I)*c*d^2 + 152 
*d^3)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f))/(c^2 + d^2) + 
 (2*a^3*d*((2*I)*c^4 - 11*c^3*d - (26*I)*c^2*d^2 - 253*c*d^3 + (150*I)*d^4 
))/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]))/(c^2 + d^2))/(2*a^2*(I*c - d) 
))/(4*a^2*(I*c - d)))/(4*a^2*(I*c - d))
 

Defintions of rubi rules used

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

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

rule 4012

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

rule 4020

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

rule 4022

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

rule 4042

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(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))   In 
t[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m 
 + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, 
e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] 
 && LtQ[m, 0] && (IntegerQ[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 = 0.40 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.67

method	result	size
derivativedivides	\(\frac {2 d^{4} \left (\frac {i \left (\frac {\frac {d \left (2 i c^{8}-82 i c^{6} d^{2}-116 i c^{4} d^{4}+22 i c^{2} d^{6}+54 i d^{8}-19 d \,c^{7}+85 d^{3} c^{5}+227 c^{3} d^{5}+123 c \,d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{9}-166 i c^{7} d^{2}-44 i c^{5} d^{4}+422 i c^{3} d^{6}+297 i c \,d^{8}-33 c^{8} d +282 c^{6} d^{3}+572 c^{4} d^{5}+166 c^{2} d^{7}-91 d^{9}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{10}-146 i c^{8} d^{2}+192 i c^{6} d^{4}+760 i c^{4} d^{6}+350 i c^{2} d^{8}-70 i d^{10}-25 c^{9} d +340 c^{7} d^{3}+458 c^{5} d^{5}-204 c^{3} d^{7}-297 c \,d^{9}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (20 i c^{8} d -250 i c^{6} d^{3}-408 i c^{4} d^{5}+14 i c^{2} d^{7}+152 i d^{9}+2 c^{9}-91 c^{7} d^{2}+177 c^{5} d^{4}+635 c^{3} d^{6}+365 c \,d^{8}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 d^{4} \left (i d +c \right )^{6} \left (i d -c \right )^{2}}+\frac {\left (i c^{6}-15 i c^{4} d^{2}+15 i c^{2} d^{4}-i d^{6}-6 c^{5} d +20 c^{3} d^{3}-6 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{6} d^{4}}-\frac {-5 i c^{2}-3 i d^{2}+2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{6} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-i c^{3}-i c \,d^{2}+c^{2} d +d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{6} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f \,a^{3}}\)	\(743\)
default	\(\frac {2 d^{4} \left (\frac {i \left (\frac {\frac {d \left (2 i c^{8}-82 i c^{6} d^{2}-116 i c^{4} d^{4}+22 i c^{2} d^{6}+54 i d^{8}-19 d \,c^{7}+85 d^{3} c^{5}+227 c^{3} d^{5}+123 c \,d^{7}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}-\frac {2 d \left (3 i c^{9}-166 i c^{7} d^{2}-44 i c^{5} d^{4}+422 i c^{3} d^{6}+297 i c \,d^{8}-33 c^{8} d +282 c^{6} d^{3}+572 c^{4} d^{5}+166 c^{2} d^{7}-91 d^{9}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {d \left (2 i c^{10}-146 i c^{8} d^{2}+192 i c^{6} d^{4}+760 i c^{4} d^{6}+350 i c^{2} d^{8}-70 i d^{10}-25 c^{9} d +340 c^{7} d^{3}+458 c^{5} d^{5}-204 c^{3} d^{7}-297 c \,d^{9}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{6 i c^{2} d -2 i d^{3}+2 c^{3}-6 c \,d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (20 i c^{8} d -250 i c^{6} d^{3}-408 i c^{4} d^{5}+14 i c^{2} d^{7}+152 i d^{9}+2 c^{9}-91 c^{7} d^{2}+177 c^{5} d^{4}+635 c^{3} d^{6}+365 c \,d^{8}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}\right )}{16 d^{4} \left (i d +c \right )^{6} \left (i d -c \right )^{2}}+\frac {\left (i c^{6}-15 i c^{4} d^{2}+15 i c^{2} d^{4}-i d^{6}-6 c^{5} d +20 c^{3} d^{3}-6 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{6} d^{4}}-\frac {-5 i c^{2}-3 i d^{2}+2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{6} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {-i c^{3}-i c \,d^{2}+c^{2} d +d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{6} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\right )}{f \,a^{3}}\)	\(743\)

Input:

int(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/f/a^3*d^4*(1/16*I/d^4/(c+I*d)^6/(I*d-c)^2*((1/2*d*(2*I*c^8-82*I*c^6*d^2- 
116*I*c^4*d^4+22*I*c^2*d^6+54*I*d^8-19*d*c^7+85*d^3*c^5+227*c^3*d^5+123*c* 
d^7)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)-2/3*d*(-33*c^8*d 
+282*c^6*d^3+572*c^4*d^5+166*c^2*d^7-91*d^9+3*I*c^9-166*I*c^7*d^2-44*I*c^5 
*d^4+422*I*c^3*d^6+297*I*c*d^8)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x 
+e))^(3/2)+1/2*d*(2*I*c^10-146*I*c^8*d^2+192*I*c^6*d^4+760*I*c^4*d^6+350*I 
*c^2*d^8-70*I*d^10-25*c^9*d+340*c^7*d^3+458*c^5*d^5-204*c^3*d^7-297*c*d^9) 
/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2))/(-d*tan(f*x+e)+I*d) 
^3-1/2*(-91*c^7*d^2+177*c^5*d^4+635*c^3*d^6+365*c*d^8+20*I*c^8*d-250*I*c^6 
*d^3-408*I*c^4*d^5+14*I*c^2*d^7+152*I*d^9+2*c^9)/(3*I*c^2*d-I*d^3+c^3-3*c* 
d^2)/(-c-I*d)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-c-I*d)^(1/2)))+1/16/(I 
*d-c)^(5/2)/(c+I*d)^6*(I*c^6-I*d^6+20*c^3*d^3-6*c*d^5+15*I*c^2*d^4-15*I*c^ 
4*d^2-6*c^5*d)/d^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))-1/(I*d-c)^ 
2/(c+I*d)^6*(-5*I*c^2-3*I*d^2+2*c*d)/(c+d*tan(f*x+e))^(1/2)-1/3/(I*d-c)^2/ 
(c+I*d)^6*(-I*c^3-I*c*d^2+c^2*d+d^3)/(c+d*tan(f*x+e))^(3/2))
 

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. 3803 vs. \(2 (364) = 728\).

Time = 16.92 (sec) , antiderivative size = 3803, normalized size of antiderivative = 8.53 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="fric 
as")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+I*a*tan(f*x+e))**3/(c+d*tan(f*x+e))**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxi 
ma")
 

Output:

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (364) = 728\).

Time = 0.87 (sec) , antiderivative size = 978, normalized size of antiderivative = 2.19 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac 
")
 

Output:

-1/16*I*(sqrt(2)*(2*c^3 + 16*I*c^2*d - 61*c*d^2 - 152*I*d^3)*arctan(2*(sqr 
t(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(sqrt( 
2)*c*sqrt(-c + sqrt(c^2 + d^2)) + I*sqrt(2)*sqrt(-c + sqrt(c^2 + d^2))*d - 
 sqrt(2)*sqrt(c^2 + d^2)*sqrt(-c + sqrt(c^2 + d^2))))/((c^5 + 5*I*c^4*d - 
10*c^3*d^2 - 10*I*c^2*d^3 + 5*c*d^4 + I*d^5)*sqrt(-c + sqrt(c^2 + d^2))*(I 
*d/(c - sqrt(c^2 + d^2)) + 1)) - 2*sqrt(2)*arctan(2*(sqrt(d*tan(f*x + e) + 
 c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(sqrt(2)*c*sqrt(-c + sqr 
t(c^2 + d^2)) - I*sqrt(2)*sqrt(-c + sqrt(c^2 + d^2))*d - sqrt(2)*sqrt(c^2 
+ d^2)*sqrt(-c + sqrt(c^2 + d^2))))/((c^2 - 2*I*c*d - d^2)*sqrt(-c + sqrt( 
c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) + 32*(6*(d*tan(f*x + e) + c) 
^4*c^4*d - 12*(d*tan(f*x + e) + c)^3*c^5*d + 6*(d*tan(f*x + e) + c)^2*c^6* 
d + 33*I*(d*tan(f*x + e) + c)^4*c^3*d^2 - 84*I*(d*tan(f*x + e) + c)^3*c^4* 
d^2 + 51*I*(d*tan(f*x + e) + c)^2*c^5*d^2 - 78*(d*tan(f*x + e) + c)^4*c^2* 
d^3 + 256*(d*tan(f*x + e) + c)^3*c^3*d^3 - 198*(d*tan(f*x + e) + c)^2*c^4* 
d^3 + 759*I*(d*tan(f*x + e) + c)^4*c*d^4 - 1856*I*(d*tan(f*x + e) + c)^3*c 
^2*d^4 + 1446*I*(d*tan(f*x + e) + c)^2*c^3*d^4 - 384*(I*d*tan(f*x + e) + I 
*c)*c^4*d^4 - 32*I*c^5*d^4 + 450*(d*tan(f*x + e) + c)^4*d^5 + 844*(d*tan(f 
*x + e) + c)^3*c*d^5 - 2334*(d*tan(f*x + e) + c)^2*c^2*d^5 + 960*(d*tan(f* 
x + e) + c)*c^3*d^5 + 96*c^4*d^5 - 1196*I*(d*tan(f*x + e) + c)^3*d^6 + 243 
*I*(d*tan(f*x + e) + c)^2*c*d^6 - 576*(-I*d*tan(f*x + e) - I*c)*c^2*d^6...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \] Input:

int(1/((a + a*tan(e + f*x)*1i)^3*(c + d*tan(e + f*x))^(5/2)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (a +i a \tan \left (f x +e \right )\right )^{3} \left (d \tan \left (f x +e \right )+c \right )^{\frac {5}{2}}}d x \] Input:

int(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x)
 

Output:

int(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x)