\(\int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx\) [1134]

Optimal result

Integrand size = 30, antiderivative size = 267 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a (c-i d)^{5/2} f}+\frac {(i c-6 d) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a (c+i d)^{7/2} f}+\frac {d (3 i c+7 d)}{6 a (i c-d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac {d \left (c^2-14 i c d-5 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 f \sqrt {c+d \tan (e+f x)}} \] Output:

-1/2*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a/(c-I*d)^(5/2)/f+1/2 
*(I*c-6*d)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a/(c+I*d)^(7/2)/f 
+1/6*d*(3*I*c+7*d)/a/(I*c-d)/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)-1/2/(I*c-d 
)/f/(a+I*a*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2)+1/2*d*(c^2-14*I*c*d-5*d^2)/a 
/(c-I*d)^2/(c+I*d)^3/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 = 0.78 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \left (-\frac {(c+i d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{c-i d}+\frac {(c+6 i d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{c+i d}+\frac {3 i}{-i+\tan (e+f x)}\right )}{6 a (c+i d) f (c+d \tan (e+f x))^{3/2}} \] Input:

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

Output:

((-1/6*I)*(-(((c + I*d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f* 
x])/(c - I*d)])/(c - I*d)) + ((c + (6*I)*d)*Hypergeometric2F1[-3/2, 1, -1/ 
2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*d) + (3*I)/(-I + Tan[e + f*x])) 
)/(a*(c + I*d)*f*(c + d*Tan[e + f*x])^(3/2))
 

Rubi [F]

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4035

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

Input:

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

Output:

$Aborted
 

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 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 4035

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-a)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b* 
c - a*d)*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a*(b*c - a*d))   Int[(c + d 
*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*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] &&  !GtQ[n, 0]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {2 d^{2} \left (-\frac {i c^{3}+i c \,d^{2}-c^{2} d -d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {3 i c^{2}+i d^{2}-2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{4} \sqrt {c +d \tan \left (f x +e \right )}}+\frac {\left (i c^{4}-6 i c^{2} d^{2}+i d^{4}-4 c^{3} d +4 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{4 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{4} d^{2}}+\frac {-\frac {\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) d \sqrt {c +d \tan \left (f x +e \right )}}{\left (i d +c \right ) \left (-d \tan \left (f x +e \right )+i d \right )}-\frac {\left (i c^{5}+2 i c^{3} d^{2}+i c \,d^{4}-6 c^{4} d -12 c^{2} d^{3}-6 d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{\left (i d +c \right ) \sqrt {-i d -c}}}{4 \left (i d -c \right )^{2} \left (i d +c \right )^{4} d^{2}}\right )}{f a}\)	\(364\)
default	\(\frac {2 d^{2} \left (-\frac {i c^{3}+i c \,d^{2}-c^{2} d -d^{3}}{3 \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {3 i c^{2}+i d^{2}-2 c d}{\left (i d -c \right )^{2} \left (i d +c \right )^{4} \sqrt {c +d \tan \left (f x +e \right )}}+\frac {\left (i c^{4}-6 i c^{2} d^{2}+i d^{4}-4 c^{3} d +4 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{4 \left (i d -c \right )^{\frac {5}{2}} \left (i d +c \right )^{4} d^{2}}+\frac {-\frac {\left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) d \sqrt {c +d \tan \left (f x +e \right )}}{\left (i d +c \right ) \left (-d \tan \left (f x +e \right )+i d \right )}-\frac {\left (i c^{5}+2 i c^{3} d^{2}+i c \,d^{4}-6 c^{4} d -12 c^{2} d^{3}-6 d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{\left (i d +c \right ) \sqrt {-i d -c}}}{4 \left (i d -c \right )^{2} \left (i d +c \right )^{4} d^{2}}\right )}{f a}\)	\(364\)

Input:

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

Output:

2/f/a*d^2*(-1/3/(I*d-c)^2/(c+I*d)^4*(I*c^3+I*c*d^2-c^2*d-d^3)/(c+d*tan(f*x 
+e))^(3/2)-1/(I*d-c)^2/(c+I*d)^4*(3*I*c^2+I*d^2-2*c*d)/(c+d*tan(f*x+e))^(1 
/2)+1/4/(I*d-c)^(5/2)/(c+I*d)^4*(-6*I*c^2*d^2+I*d^4-4*c^3*d+4*c*d^3+I*c^4) 
/d^2*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))+1/4/(I*d-c)^2/(c+I*d)^4/ 
d^2*(-(c^4+2*c^2*d^2+d^4)*d/(c+I*d)*(c+d*tan(f*x+e))^(1/2)/(-d*tan(f*x+e)+ 
I*d)-(-6*c^4*d-12*c^2*d^3-6*d^5+I*c^5+2*I*c^3*d^2+I*c*d^4)/(c+I*d)/(-c-I*d 
)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-c-I*d)^(1/2))))
 

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. 2659 vs. \(2 (213) = 426\).

Time = 1.62 (sec) , antiderivative size = 2659, normalized size of antiderivative = 9.96 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(6*((I*a*c^7 + a*c^6*d + 3*I*a*c^5*d^2 + 3*a*c^4*d^3 + 3*I*a*c^3*d^4 
+ 3*a*c^2*d^5 + I*a*c*d^6 + a*d^7)*f*e^(6*I*f*x + 6*I*e) + 2*(I*a*c^7 - a* 
c^6*d + 3*I*a*c^5*d^2 - 3*a*c^4*d^3 + 3*I*a*c^3*d^4 - 3*a*c^2*d^5 + I*a*c* 
d^6 - a*d^7)*f*e^(4*I*f*x + 4*I*e) + (I*a*c^7 - 3*a*c^6*d - I*a*c^5*d^2 - 
5*a*c^4*d^3 - 5*I*a*c^3*d^4 - a*c^2*d^5 - 3*I*a*c*d^6 + a*d^7)*f*e^(2*I*f* 
x + 2*I*e))*sqrt(1/4*I/((-I*a^2*c^5 - 5*a^2*c^4*d + 10*I*a^2*c^3*d^2 + 10* 
a^2*c^2*d^3 - 5*I*a^2*c*d^4 - a^2*d^5)*f^2))*log(-2*(2*((I*a*c^3 + 3*a*c^2 
*d - 3*I*a*c*d^2 - a*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a*c^3 + 3*a*c^2*d - 3 
*I*a*c*d^2 - a*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^( 
2*I*f*x + 2*I*e) + 1))*sqrt(1/4*I/((-I*a^2*c^5 - 5*a^2*c^4*d + 10*I*a^2*c^ 
3*d^2 + 10*a^2*c^2*d^3 - 5*I*a^2*c*d^4 - a^2*d^5)*f^2)) - (c - I*d)*e^(2*I 
*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + 6*((-I*a*c^7 - a*c^6*d - 3*I*a* 
c^5*d^2 - 3*a*c^4*d^3 - 3*I*a*c^3*d^4 - 3*a*c^2*d^5 - I*a*c*d^6 - a*d^7)*f 
*e^(6*I*f*x + 6*I*e) + 2*(-I*a*c^7 + a*c^6*d - 3*I*a*c^5*d^2 + 3*a*c^4*d^3 
 - 3*I*a*c^3*d^4 + 3*a*c^2*d^5 - I*a*c*d^6 + a*d^7)*f*e^(4*I*f*x + 4*I*e) 
+ (-I*a*c^7 + 3*a*c^6*d + I*a*c^5*d^2 + 5*a*c^4*d^3 + 5*I*a*c^3*d^4 + a*c^ 
2*d^5 + 3*I*a*c*d^6 - a*d^7)*f*e^(2*I*f*x + 2*I*e))*sqrt(1/4*I/((-I*a^2*c^ 
5 - 5*a^2*c^4*d + 10*I*a^2*c^3*d^2 + 10*a^2*c^2*d^3 - 5*I*a^2*c*d^4 - a^2* 
d^5)*f^2))*log(-2*(2*((-I*a*c^3 - 3*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f*e^(2* 
I*f*x + 2*I*e) + (-I*a*c^3 - 3*a*c^2*d + 3*I*a*c*d^2 + a*d^3)*f)*sqrt((...
 

Sympy [F]

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

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

Output:

-I*Integral(1/(c**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x) - I*c**2*sqrt(c 
+ d*tan(e + f*x)) + 2*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2 - 2*I*c 
*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x) + d**2*sqrt(c + d*tan(e + f*x))*t 
an(e + f*x)**3 - I*d**2*sqrt(c + d*tan(e + f*x))*tan(e + f*x)**2), x)/a
 

Maxima [F(-2)]

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

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

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. 519 vs. \(2 (213) = 426\).

Time = 0.70 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i \, {\left (\frac {6 \, \sqrt {2} {\left (-i \, c + 6 \, d\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{\sqrt {2} c \sqrt {-c + \sqrt {c^{2} + d^{2}}} + i \, \sqrt {2} \sqrt {-c + \sqrt {c^{2} + d^{2}}} d - \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {-c + \sqrt {c^{2} + d^{2}}}}\right )}{-2 \, {\left (i \, c^{3} - 3 \, c^{2} d - 3 i \, c d^{2} + d^{3}\right )} \sqrt {-c + \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {12 \, \sqrt {d \tan \left (f x + e\right ) + c} d}{-4 \, {\left (i \, c^{3} - 3 \, c^{2} d - 3 i \, c d^{2} + d^{3}\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}} - \frac {3 \, \sqrt {2} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{\sqrt {2} c \sqrt {-c + \sqrt {c^{2} + d^{2}}} - i \, \sqrt {2} \sqrt {-c + \sqrt {c^{2} + d^{2}}} d - \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {-c + \sqrt {c^{2} + d^{2}}}}\right )}{{\left (c^{2} - 2 i \, c d - d^{2}\right )} \sqrt {-c + \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {4 \, {\left (9 \, {\left (d \tan \left (f x + e\right ) + c\right )} c d^{2} + c^{2} d^{2} - 3 \, {\left (i \, d \tan \left (f x + e\right ) + i \, c\right )} d^{3} + d^{4}\right )}}{{\left (c^{5} + i \, c^{4} d + 2 \, c^{3} d^{2} + 2 i \, c^{2} d^{3} + c d^{4} + i \, d^{5}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\right )}}{6 \, a f} \] Input:

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

Output:

-1/6*I*(6*sqrt(2)*(-I*c + 6*d)*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 + 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))))/((-2*I*c^3 + 6*c^2*d + 6*I*c*d^2 - 2*d^3)*sqrt(-c 
+ sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 12*sqrt(d*tan(f*x + 
e) + c)*d/((-4*I*c^3 + 12*c^2*d + 12*I*c*d^2 - 4*d^3)*(d*tan(f*x + e) - I* 
d)) - 3*sqrt(2)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqr 
t(d*tan(f*x + e) + c))/(sqrt(2)*c*sqrt(-c + sqrt(c^2 + d^2)) - I*sqrt(2)*s 
qrt(-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)) + 4*(9*(d*tan(f*x + e) + c)*c*d^2 + c^2*d^2 - 3*(I*d*ta 
n(f*x + e) + I*c)*d^3 + d^4)/((c^5 + I*c^4*d + 2*c^3*d^2 + 2*I*c^2*d^3 + c 
*d^4 + I*d^5)*(d*tan(f*x + e) + c)^(3/2)))/(a*f)
 

Mupad [B] (verification not implemented)

Time = 90.26 (sec) , antiderivative size = 69981, normalized size of antiderivative = 262.10 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

log((a*f*(139*c*d^7 - d^8*30i + c^2*d^6*180i - 62*c^3*d^5 + c^4*d^4*10i - 
c^5*d^3))/2 - (((a*f*(208*a^2*c^2*d^11*f^2 - a^2*c*d^12*f^2*320i - 112*a^2 
*d^13*f^2 - a^2*c^3*d^10*f^2*640i + 1312*a^2*c^4*d^9*f^2 + 1568*a^2*c^6*d^ 
7*f^2 + a^2*c^7*d^6*f^2*640i + 592*a^2*c^8*d^5*f^2 + a^2*c^9*d^4*f^2*320i 
+ 16*a^2*c^10*d^3*f^2))/2 - 2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^ 
2*c^2*f^2 + a^2*c*d*f^2*2i)*((4480*c^2*d^9 - 560*d^11 - c*d^10*2800i + c^3 
*d^8*4480i - 560*c^4*d^7 - c^5*d^6*112i - 224*c^6*d^5 + c^7*d^4*32i - a^2* 
c^10*f^2*(((3920*c*d^12 - 16240*c^3*d^10 + 5712*c^5*d^8 + 304*c^7*d^6 + 32 
*c^9*d^4)/(a^2*c^12*f^2 + a^2*d^12*f^2 + 6*a^2*c^2*d^10*f^2 + 15*a^2*c^4*d 
^8*f^2 + 20*a^2*c^6*d^6*f^2 + 15*a^2*c^8*d^4*f^2 + 6*a^2*c^10*d^2*f^2) + ( 
(10640*c^2*d^11 - 560*d^13 - 14000*c^4*d^9 + 560*c^6*d^7 + 160*c^8*d^5)*1i 
)/(a^2*c^12*f^2 + a^2*d^12*f^2 + 6*a^2*c^2*d^10*f^2 + 15*a^2*c^4*d^8*f^2 + 
 20*a^2*c^6*d^6*f^2 + 15*a^2*c^8*d^4*f^2 + 6*a^2*c^10*d^2*f^2))^2 - 4*(256 
*d^6 + 256*c^2*d^4)*(((60*c*d^7 + 10*c^3*d^5)*1i)/(a^4*c^12*f^4 + a^4*d^12 
*f^4 + 6*a^4*c^2*d^10*f^4 + 15*a^4*c^4*d^8*f^4 + 20*a^4*c^6*d^6*f^4 + 15*a 
^4*c^8*d^4*f^4 + 6*a^4*c^10*d^2*f^4) + (36*d^8 - 13*c^2*d^6 + c^4*d^4)/(a^ 
4*c^12*f^4 + a^4*d^12*f^4 + 6*a^4*c^2*d^10*f^4 + 15*a^4*c^4*d^8*f^4 + 20*a 
^4*c^6*d^6*f^4 + 15*a^4*c^8*d^4*f^4 + 6*a^4*c^10*d^2*f^4)))^(1/2)*1i + a^2 
*d^10*f^2*(((3920*c*d^12 - 16240*c^3*d^10 + 5712*c^5*d^8 + 304*c^7*d^6 + 3 
2*c^9*d^4)/(a^2*c^12*f^2 + a^2*d^12*f^2 + 6*a^2*c^2*d^10*f^2 + 15*a^2*c...
 

Reduce [F]

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

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

Output:

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