\(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx\) [1094]

Optimal result

Integrand size = 28, antiderivative size = 357 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=\frac {\left (c^5+5 i c^4 d-10 c^3 d^2-10 i c^2 d^3-35 c d^4+25 i d^5\right ) x}{8 a^3 (c-i d)^2 (c+i d)^5}+\frac {(5 c-3 i d) d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (i c-d)^5 (c-i d)^2 f}+\frac {d \left (c^3+5 i c^2 d-11 c d^2+25 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}+\frac {3 i c-11 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}+\frac {c^2+5 i c d-12 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))} \] Output:

1/8*(c^5+5*I*c^4*d-10*c^3*d^2-10*I*c^2*d^3-35*c*d^4+25*I*d^5)*x/a^3/(c-I*d 
)^2/(c+I*d)^5+(5*c-3*I*d)*d^4*ln(c*cos(f*x+e)+d*sin(f*x+e))/a^3/(I*c-d)^5/ 
(c-I*d)^2/f+1/8*d*(c^3+5*I*c^2*d-11*c*d^2+25*I*d^3)/a^3/(c-I*d)/(c+I*d)^4/ 
f/(c+d*tan(f*x+e))-1/6/(I*c-d)/f/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))+1/2 
4*(3*I*c-11*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))+1/8*(c^ 
2+5*I*c*d-12*d^2)/(I*c-d)^3/f/(a^3+I*a^3*tan(f*x+e))/(c+d*tan(f*x+e))
 

Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=-\frac {\frac {8 (c+i d)}{(-i+\tan (e+f x))^3 (c+d \tan (e+f x))}+\frac {6 i c-22 d}{(-i+\tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {6 \left (c^2+5 i c d-12 d^2\right )}{(c+i d) (-i+\tan (e+f x)) (c+d \tan (e+f x))}-\frac {3 \left (\frac {2 \left (c^2+5 i c d-12 d^2\right ) ((-i c-d) \log (i-\tan (e+f x))+i (c+i d) \log (i+\tan (e+f x))+2 d \log (c+d \tan (e+f x)))}{c^2+d^2}+\left (c^3+5 i c^2 d-11 c d^2+25 i d^3\right ) \left (\frac {i \log (i-\tan (e+f x))}{(c+i d)^2}-\frac {i \log (i+\tan (e+f x))}{(c-i d)^2}+\frac {2 d \left (-2 c \log (c+d \tan (e+f x))+\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}\right )\right )}{c+i d}}{48 a^3 (c+i d)^2 f} \] Input:

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

Output:

-1/48*((8*(c + I*d))/((-I + Tan[e + f*x])^3*(c + d*Tan[e + f*x])) + ((6*I) 
*c - 22*d)/((-I + Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (6*(c^2 + (5*I)* 
c*d - 12*d^2))/((c + I*d)*(-I + Tan[e + f*x])*(c + d*Tan[e + f*x])) - (3*( 
(2*(c^2 + (5*I)*c*d - 12*d^2)*(((-I)*c - d)*Log[I - Tan[e + f*x]] + I*(c + 
 I*d)*Log[I + Tan[e + f*x]] + 2*d*Log[c + d*Tan[e + f*x]]))/(c^2 + d^2) + 
(c^3 + (5*I)*c^2*d - 11*c*d^2 + (25*I)*d^3)*((I*Log[I - Tan[e + f*x]])/(c 
+ I*d)^2 - (I*Log[I + Tan[e + f*x]])/(c - I*d)^2 + (2*d*(-2*c*Log[c + d*Ta 
n[e + f*x]] + (c^2 + d^2)/(c + d*Tan[e + f*x])))/(c^2 + d^2)^2)))/(c + I*d 
))/(a^3*(c + I*d)^2*f)
 

Rubi [A] (verified)

Time = 1.87 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {3042, 4042, 25, 3042, 4079, 27, 3042, 4079, 27, 3042, 4012, 3042, 4014, 3042, 4013}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4079

\(\displaystyle \frac {-\frac {\int \frac {3 \left (\left (2 c^2+7 i d c-13 d^2\right ) a^2+(3 c+11 i d) d \tan (e+f x) a^2\right )}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^2}dx}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \int \frac {\left (2 c^2+7 i d c-13 d^2\right ) a^2+(3 c+11 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^2}dx}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 \int \frac {\left (2 c^2+7 i d c-13 d^2\right ) a^2+(3 c+11 i d) d \tan (e+f x) a^2}{(i \tan (e+f x) a+a) (c+d \tan (e+f x))^2}dx}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4079

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int -\frac {2 \left (\left (i c^3-5 d c^2-13 i d^2 c+25 d^3\right ) a^3+2 d \left (i c^2-5 d c-12 i d^2\right ) \tan (e+f x) a^3\right )}{(c+d \tan (e+f x))^2}dx}{2 a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (\frac {\int \frac {a^3 \left (i c^3-5 d c^2-13 i d^2 c+25 d^3\right )-2 a^3 d \left (5 c d-i \left (c^2-12 d^2\right )\right ) \tan (e+f x)}{(c+d \tan (e+f x))^2}dx}{a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 \left (\frac {\int \frac {a^3 \left (i c^3-5 d c^2-13 i d^2 c+25 d^3\right )-2 a^3 d \left (5 c d-i \left (c^2-12 d^2\right )\right ) \tan (e+f x)}{(c+d \tan (e+f x))^2}dx}{a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {-\frac {3 \left (\frac {\frac {\int \frac {\left (i c^4-5 d c^3-11 i d^2 c^2+15 d^3 c-24 i d^4\right ) a^3+d \left (i c^3-5 d c^2-11 i d^2 c-25 d^3\right ) \tan (e+f x) a^3}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^3 d \left (i c^3-5 c^2 d-11 i c d^2-25 d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 \left (\frac {\frac {\int \frac {\left (i c^4-5 d c^3-11 i d^2 c^2+15 d^3 c-24 i d^4\right ) a^3+d \left (i c^3-5 d c^2-11 i d^2 c-25 d^3\right ) \tan (e+f x) a^3}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^3 d \left (i c^3-5 c^2 d-11 i c d^2-25 d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4014

\(\displaystyle \frac {-\frac {3 \left (\frac {\frac {\frac {8 a^3 d^4 (5 c-3 i d) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^3 x \left (i c^5-5 c^4 d-10 i c^3 d^2+10 c^2 d^3-35 i c d^4-25 d^5\right )}{c^2+d^2}}{c^2+d^2}+\frac {a^3 d \left (i c^3-5 c^2 d-11 i c d^2-25 d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 \left (\frac {\frac {\frac {8 a^3 d^4 (5 c-3 i d) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^3 x \left (i c^5-5 c^4 d-10 i c^3 d^2+10 c^2 d^3-35 i c d^4-25 d^5\right )}{c^2+d^2}}{c^2+d^2}+\frac {a^3 d \left (i c^3-5 c^2 d-11 i c d^2-25 d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4013

\(\displaystyle \frac {-\frac {3 \left (\frac {\frac {a^3 d \left (i c^3-5 c^2 d-11 i c d^2-25 d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}+\frac {\frac {8 a^3 d^4 (5 c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )}+\frac {a^3 x \left (i c^5-5 c^4 d-10 i c^3 d^2+10 c^2 d^3-35 i c d^4-25 d^5\right )}{c^2+d^2}}{c^2+d^2}}{a^2 (-d+i c)}-\frac {a^2 \left (c^2+5 i c d-12 d^2\right )}{f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}\right )}{4 a^2 (-d+i c)}-\frac {a (3 c+11 i d)}{4 f (c+i d) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}}{6 a^2 (-d+i c)}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}\)

Input:

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

Output:

-1/6*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])) + (-1/4 
*(a*(3*c + (11*I)*d))/((c + I*d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + 
 f*x])) - (3*(-((a^2*(c^2 + (5*I)*c*d - 12*d^2))/((I*c - d)*f*(a + I*a*Tan 
[e + f*x])*(c + d*Tan[e + f*x]))) + (((a^3*(I*c^5 - 5*c^4*d - (10*I)*c^3*d 
^2 + 10*c^2*d^3 - (35*I)*c*d^4 - 25*d^5)*x)/(c^2 + d^2) + (8*a^3*(5*c - (3 
*I)*d)*d^4*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)*f))/(c^2 + d 
^2) + (a^3*d*(I*c^3 - 5*c^2*d - (11*I)*c*d^2 - 25*d^3))/((c^2 + d^2)*f*(c 
+ d*Tan[e + f*x])))/(a^2*(I*c - d))))/(4*a^2*(I*c - d)))/(6*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 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 4013

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

rule 4014

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

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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (327 ) = 654\).

Time = 0.84 (sec) , antiderivative size = 830, normalized size of antiderivative = 2.32

method	result	size
derivativedivides	\(-\frac {49 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {i d^{4} c^{2}}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {5 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5}}-\frac {49 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{5}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {17 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {3 d^{5} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5}}+\frac {7 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {23 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {7 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {23 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{5}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {i d^{6}}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}+\frac {7 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {23 i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {7 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{5}}+\frac {11 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{5}}-\frac {5 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{32 f \,a^{3} \left (i d -c \right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{16 f \,a^{3} \left (i d -c \right )^{2}}\)	\(830\)
default	\(-\frac {49 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {i d^{4} c^{2}}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {5 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5}}-\frac {49 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{5}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {17 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {3 d^{5} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5}}+\frac {7 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {23 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {7 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {23 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{5}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {i d^{6}}{f \,a^{3} \left (i d -c \right )^{2} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}+\frac {7 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {23 i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{5}}+\frac {7 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{5}}+\frac {11 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{5}}-\frac {5 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{32 f \,a^{3} \left (i d -c \right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{16 f \,a^{3} \left (i d -c \right )^{2}}\)	\(830\)
risch	\(-\frac {x}{8 a^{3} \left (2 i c d -c^{2}+d^{2}\right )}-\frac {7 \,{\mathrm e}^{-2 i \left (f x +e \right )} c d}{8 a^{3} \left (i d +c \right )^{2} \left (2 i c d +c^{2}-d^{2}\right ) f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c^{2}}{16 a^{3} \left (i d +c \right )^{2} \left (2 i c d +c^{2}-d^{2}\right ) f}-\frac {23 i {\mathrm e}^{-2 i \left (f x +e \right )} d^{2}}{16 a^{3} \left (i d +c \right )^{2} \left (2 i c d +c^{2}-d^{2}\right ) f}-\frac {7 \,{\mathrm e}^{-4 i \left (f x +e \right )} d}{32 a^{3} \left (i d +c \right ) \left (2 i c d +c^{2}-d^{2}\right ) f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c}{32 a^{3} \left (i d +c \right ) \left (2 i c d +c^{2}-d^{2}\right ) f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{48 a^{3} \left (2 i c d +c^{2}-d^{2}\right ) f}-\frac {10 d^{4} c x}{a^{3} \left (3 i c^{6} d +5 i c^{4} d^{3}+i c^{2} d^{5}-i d^{7}+c^{7}-c^{5} d^{2}-5 c^{3} d^{4}-3 c \,d^{6}\right )}-\frac {10 d^{4} c e}{a^{3} f \left (3 i c^{6} d +5 i c^{4} d^{3}+i c^{2} d^{5}-i d^{7}+c^{7}-c^{5} d^{2}-5 c^{3} d^{4}-3 c \,d^{6}\right )}+\frac {6 i d^{5} x}{a^{3} \left (3 i c^{6} d +5 i c^{4} d^{3}+i c^{2} d^{5}-i d^{7}+c^{7}-c^{5} d^{2}-5 c^{3} d^{4}-3 c \,d^{6}\right )}+\frac {6 i d^{5} e}{a^{3} f \left (3 i c^{6} d +5 i c^{4} d^{3}+i c^{2} d^{5}-i d^{7}+c^{7}-c^{5} d^{2}-5 c^{3} d^{4}-3 c \,d^{6}\right )}-\frac {2 i d^{5}}{\left (-i c +d \right )^{2} f \,a^{3} \left (-2 i c d -c^{2}+d^{2}\right ) \left (i c +d \right )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )} d -d +i {\mathrm e}^{2 i \left (f x +e \right )} c +i c \right )}-\frac {5 i d^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) c}{a^{3} f \left (3 i c^{6} d +5 i c^{4} d^{3}+i c^{2} d^{5}-i d^{7}+c^{7}-c^{5} d^{2}-5 c^{3} d^{4}-3 c \,d^{6}\right )}-\frac {3 d^{5} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{a^{3} f \left (3 i c^{6} d +5 i c^{4} d^{3}+i c^{2} d^{5}-i d^{7}+c^{7}-c^{5} d^{2}-5 c^{3} d^{4}-3 c \,d^{6}\right )}\)	\(836\)
norman	\(\text {Expression too large to display}\)	\(1282\)

Input:

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

Output:

1/16/f/a^3/(I*d-c)^2*arctan(tan(f*x+e))+I/f/a^3*d^6/(I*d-c)^2/(c+I*d)^5/(c 
+d*tan(f*x+e))+7/8*I/f/a^3/(c+I*d)^5/(-I+tan(f*x+e))*c^2*d-1/2*I/f/a^3/(c+ 
I*d)^5/(-I+tan(f*x+e))^3*c^2*d+23/32*I/f/a^3/(c+I*d)^5*ln(1+tan(f*x+e)^2)* 
c*d^2+7/16*I/f/a^3/(c+I*d)^5*arctan(tan(f*x+e))*c^2*d+11/8*I/f/a^3/(c+I*d) 
^5/(-I+tan(f*x+e))^2*c*d^2+I/f/a^3*d^4/(I*d-c)^2/(c+I*d)^5/(c+d*tan(f*x+e) 
)*c^2-5*I/f/a^3*d^4/(I*d-c)^2/(c+I*d)^5*ln(c+d*tan(f*x+e))*c-49/16*I/f/a^3 
/(c+I*d)^5*arctan(tan(f*x+e))*d^3+1/6*I/f/a^3/(c+I*d)^5/(-I+tan(f*x+e))^3* 
d^3-17/8*I/f/a^3/(c+I*d)^5/(-I+tan(f*x+e))*d^3-1/8*I/f/a^3/(c+I*d)^5/(-I+t 
an(f*x+e))^2*c^3-3/f/a^3*d^5/(I*d-c)^2/(c+I*d)^5*ln(c+d*tan(f*x+e))+7/32/f 
/a^3/(c+I*d)^5*ln(1+tan(f*x+e)^2)*c^2*d+1/2/f/a^3/(c+I*d)^5/(-I+tan(f*x+e) 
)^3*c*d^2-23/8/f/a^3/(c+I*d)^5/(-I+tan(f*x+e))*c*d^2+7/8/f/a^3/(c+I*d)^5/( 
-I+tan(f*x+e))^2*c^2*d-23/16/f/a^3/(c+I*d)^5*arctan(tan(f*x+e))*c*d^2-1/32 
*I/f/a^3/(c+I*d)^5*ln(1+tan(f*x+e)^2)*c^3-49/32/f/a^3/(c+I*d)^5*ln(1+tan(f 
*x+e)^2)*d^3+1/16/f/a^3/(c+I*d)^5*arctan(tan(f*x+e))*c^3-5/8/f/a^3/(c+I*d) 
^5/(-I+tan(f*x+e))^2*d^3-1/6/f/a^3/(c+I*d)^5/(-I+tan(f*x+e))^3*c^3+1/8/f/a 
^3/(c+I*d)^5/(-I+tan(f*x+e))*c^3+1/32*I/f/a^3/(I*d-c)^2*ln(1+tan(f*x+e)^2)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=-\frac {-2 i \, c^{6} + 4 \, c^{5} d - 2 i \, c^{4} d^{2} + 8 \, c^{3} d^{3} + 2 i \, c^{2} d^{4} + 4 \, c d^{5} + 2 i \, d^{6} - 12 \, {\left (c^{6} + 4 i \, c^{5} d - 5 \, c^{4} d^{2} - 85 \, c^{2} d^{4} + 124 i \, c d^{5} + 49 \, d^{6}\right )} f x e^{\left (8 i \, f x + 8 i \, e\right )} - 6 \, {\left (3 i \, c^{6} - 8 \, c^{5} d + 5 i \, c^{4} d^{2} - 40 \, c^{3} d^{3} + 25 i \, c^{2} d^{4} + 55 i \, d^{6} + 2 \, {\left (c^{6} + 6 i \, c^{5} d - 15 \, c^{4} d^{2} - 20 i \, c^{3} d^{3} - 65 \, c^{2} d^{4} - 26 i \, c d^{5} - 49 \, d^{6}\right )} f x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 3 \, {\left (9 i \, c^{6} - 32 \, c^{5} d - 21 i \, c^{4} d^{2} - 64 \, c^{3} d^{3} - 69 i \, c^{2} d^{4} - 32 \, c d^{5} - 39 i \, d^{6}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-11 i \, c^{6} + 30 \, c^{5} d - 3 i \, c^{4} d^{2} + 60 \, c^{3} d^{3} + 27 i \, c^{2} d^{4} + 30 \, c d^{5} + 19 i \, d^{6}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 96 \, {\left ({\left (-5 i \, c^{2} d^{4} - 8 \, c d^{5} + 3 i \, d^{6}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-5 i \, c^{2} d^{4} + 2 \, c d^{5} - 3 i \, d^{6}\right )} e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right )}{96 \, {\left ({\left (a^{3} c^{8} + 2 i \, a^{3} c^{7} d + 2 \, a^{3} c^{6} d^{2} + 6 i \, a^{3} c^{5} d^{3} + 6 i \, a^{3} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{6} + 2 i \, a^{3} c d^{7} - a^{3} d^{8}\right )} f e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (a^{3} c^{8} + 4 i \, a^{3} c^{7} d - 4 \, a^{3} c^{6} d^{2} + 4 i \, a^{3} c^{5} d^{3} - 10 \, a^{3} c^{4} d^{4} - 4 i \, a^{3} c^{3} d^{5} - 4 \, a^{3} c^{2} d^{6} - 4 i \, a^{3} c d^{7} + a^{3} d^{8}\right )} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )}} \] Input:

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

Output:

-1/96*(-2*I*c^6 + 4*c^5*d - 2*I*c^4*d^2 + 8*c^3*d^3 + 2*I*c^2*d^4 + 4*c*d^ 
5 + 2*I*d^6 - 12*(c^6 + 4*I*c^5*d - 5*c^4*d^2 - 85*c^2*d^4 + 124*I*c*d^5 + 
 49*d^6)*f*x*e^(8*I*f*x + 8*I*e) - 6*(3*I*c^6 - 8*c^5*d + 5*I*c^4*d^2 - 40 
*c^3*d^3 + 25*I*c^2*d^4 + 55*I*d^6 + 2*(c^6 + 6*I*c^5*d - 15*c^4*d^2 - 20* 
I*c^3*d^3 - 65*c^2*d^4 - 26*I*c*d^5 - 49*d^6)*f*x)*e^(6*I*f*x + 6*I*e) - 3 
*(9*I*c^6 - 32*c^5*d - 21*I*c^4*d^2 - 64*c^3*d^3 - 69*I*c^2*d^4 - 32*c*d^5 
 - 39*I*d^6)*e^(4*I*f*x + 4*I*e) + (-11*I*c^6 + 30*c^5*d - 3*I*c^4*d^2 + 6 
0*c^3*d^3 + 27*I*c^2*d^4 + 30*c*d^5 + 19*I*d^6)*e^(2*I*f*x + 2*I*e) - 96*( 
(-5*I*c^2*d^4 - 8*c*d^5 + 3*I*d^6)*e^(8*I*f*x + 8*I*e) + (-5*I*c^2*d^4 + 2 
*c*d^5 - 3*I*d^6)*e^(6*I*f*x + 6*I*e))*log(((I*c + d)*e^(2*I*f*x + 2*I*e) 
+ I*c - d)/(I*c + d)))/((a^3*c^8 + 2*I*a^3*c^7*d + 2*a^3*c^6*d^2 + 6*I*a^3 
*c^5*d^3 + 6*I*a^3*c^3*d^5 - 2*a^3*c^2*d^6 + 2*I*a^3*c*d^7 - a^3*d^8)*f*e^ 
(8*I*f*x + 8*I*e) + (a^3*c^8 + 4*I*a^3*c^7*d - 4*a^3*c^6*d^2 + 4*I*a^3*c^5 
*d^3 - 10*a^3*c^4*d^4 - 4*I*a^3*c^3*d^5 - 4*a^3*c^2*d^6 - 4*I*a^3*c*d^7 + 
a^3*d^8)*f*e^(6*I*f*x + 6*I*e))
 

Sympy [A] (verification not implemented)

Time = 65.27 (sec) , antiderivative size = 1792, normalized size of antiderivative = 5.02 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*d**5/(a**3*c**7*f + 3*I*a**3*c**6*d*f - a**3*c**5*d**2*f + 5*I*a**3*c**4 
*d**3*f - 5*a**3*c**3*d**4*f + I*a**3*c**2*d**5*f - 3*a**3*c*d**6*f - I*a* 
*3*d**7*f + (a**3*c**7*f*exp(2*I*e) + I*a**3*c**6*d*f*exp(2*I*e) + 3*a**3* 
c**5*d**2*f*exp(2*I*e) + 3*I*a**3*c**4*d**3*f*exp(2*I*e) + 3*a**3*c**3*d** 
4*f*exp(2*I*e) + 3*I*a**3*c**2*d**5*f*exp(2*I*e) + a**3*c*d**6*f*exp(2*I*e 
) + I*a**3*d**7*f*exp(2*I*e))*exp(2*I*f*x)) + x*(c**3 + 7*I*c**2*d - 23*c* 
d**2 - 49*I*d**3)/(8*a**3*c**5 + 40*I*a**3*c**4*d - 80*a**3*c**3*d**2 - 80 
*I*a**3*c**2*d**3 + 40*a**3*c*d**4 + 8*I*a**3*d**5) + Piecewise((((512*I*a 
**6*c**7*f**2*exp(6*I*e) - 3584*a**6*c**6*d*f**2*exp(6*I*e) - 10752*I*a**6 
*c**5*d**2*f**2*exp(6*I*e) + 17920*a**6*c**4*d**3*f**2*exp(6*I*e) + 17920* 
I*a**6*c**3*d**4*f**2*exp(6*I*e) - 10752*a**6*c**2*d**5*f**2*exp(6*I*e) - 
3584*I*a**6*c*d**6*f**2*exp(6*I*e) + 512*a**6*d**7*f**2*exp(6*I*e))*exp(-6 
*I*f*x) + (2304*I*a**6*c**7*f**2*exp(8*I*e) - 19200*a**6*c**6*d*f**2*exp(8 
*I*e) - 66816*I*a**6*c**5*d**2*f**2*exp(8*I*e) + 126720*a**6*c**4*d**3*f** 
2*exp(8*I*e) + 142080*I*a**6*c**3*d**4*f**2*exp(8*I*e) - 94464*a**6*c**2*d 
**5*f**2*exp(8*I*e) - 34560*I*a**6*c*d**6*f**2*exp(8*I*e) + 5376*a**6*d**7 
*f**2*exp(8*I*e))*exp(-4*I*f*x) + (4608*I*a**6*c**7*f**2*exp(10*I*e) - 445 
44*a**6*c**6*d*f**2*exp(10*I*e) - 188928*I*a**6*c**5*d**2*f**2*exp(10*I*e) 
 + 437760*a**6*c**4*d**3*f**2*exp(10*I*e) + 591360*I*a**6*c**3*d**4*f**2*e 
xp(10*I*e) - 465408*a**6*c**2*d**5*f**2*exp(10*I*e) - 198144*I*a**6*c*d...
 

Maxima [F(-2)]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=-\frac {i \, {\left (c^{3} + 7 i \, c^{2} d - 23 \, c d^{2} - 49 i \, d^{3}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{16 \, {\left (a^{3} c^{5} f + 5 i \, a^{3} c^{4} d f - 10 \, a^{3} c^{3} d^{2} f - 10 i \, a^{3} c^{2} d^{3} f + 5 \, a^{3} c d^{4} f + i \, a^{3} d^{5} f\right )}} - \frac {i \, {\left (5 \, c d^{5} - 3 i \, d^{6}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{a^{3} c^{7} d f + 3 i \, a^{3} c^{6} d^{2} f - a^{3} c^{5} d^{3} f + 5 i \, a^{3} c^{4} d^{4} f - 5 \, a^{3} c^{3} d^{5} f + i \, a^{3} c^{2} d^{6} f - 3 \, a^{3} c d^{7} f - i \, a^{3} d^{8} f} + \frac {i \, \log \left (\tan \left (f x + e\right ) + i\right )}{16 \, {\left (a^{3} c^{2} f - 2 i \, a^{3} c d f - a^{3} d^{2} f\right )}} - \frac {i \, {\left (-10 i \, c^{6} + 34 \, c^{5} d + 16 i \, c^{4} d^{2} + 104 \, c^{3} d^{3} + 2 i \, c^{2} d^{4} + 70 \, c d^{5} - 24 i \, d^{6} + 3 \, {\left (i \, c^{5} d - 5 \, c^{4} d^{2} - 10 i \, c^{3} d^{3} - 30 \, c^{2} d^{4} - 11 i \, c d^{5} - 25 \, d^{6}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (i \, c^{6} - 2 \, c^{5} d + 5 i \, c^{4} d^{2} - 40 \, c^{3} d^{3} + 67 i \, c^{2} d^{4} - 38 \, c d^{5} + 63 i \, d^{6}\right )} \tan \left (f x + e\right )^{2} + {\left (9 \, c^{6} + 35 i \, c^{5} d - 20 \, c^{4} d^{2} + 178 i \, c^{3} d^{3} + 113 \, c^{2} d^{4} + 143 i \, c d^{5} + 142 \, d^{6}\right )} \tan \left (f x + e\right )\right )}}{24 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{3} {\left (c + i \, d\right )}^{5} {\left (c - i \, d\right )}^{2} f {\left (\tan \left (f x + e\right ) - i\right )}^{3}} \] Input:

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

Output:

-1/16*I*(c^3 + 7*I*c^2*d - 23*c*d^2 - 49*I*d^3)*log(tan(f*x + e) - I)/(a^3 
*c^5*f + 5*I*a^3*c^4*d*f - 10*a^3*c^3*d^2*f - 10*I*a^3*c^2*d^3*f + 5*a^3*c 
*d^4*f + I*a^3*d^5*f) - I*(5*c*d^5 - 3*I*d^6)*log(abs(d*tan(f*x + e) + c)) 
/(a^3*c^7*d*f + 3*I*a^3*c^6*d^2*f - a^3*c^5*d^3*f + 5*I*a^3*c^4*d^4*f - 5* 
a^3*c^3*d^5*f + I*a^3*c^2*d^6*f - 3*a^3*c*d^7*f - I*a^3*d^8*f) + 1/16*I*lo 
g(tan(f*x + e) + I)/(a^3*c^2*f - 2*I*a^3*c*d*f - a^3*d^2*f) - 1/24*I*(-10* 
I*c^6 + 34*c^5*d + 16*I*c^4*d^2 + 104*c^3*d^3 + 2*I*c^2*d^4 + 70*c*d^5 - 2 
4*I*d^6 + 3*(I*c^5*d - 5*c^4*d^2 - 10*I*c^3*d^3 - 30*c^2*d^4 - 11*I*c*d^5 
- 25*d^6)*tan(f*x + e)^3 + 3*(I*c^6 - 2*c^5*d + 5*I*c^4*d^2 - 40*c^3*d^3 + 
 67*I*c^2*d^4 - 38*c*d^5 + 63*I*d^6)*tan(f*x + e)^2 + (9*c^6 + 35*I*c^5*d 
- 20*c^4*d^2 + 178*I*c^3*d^3 + 113*c^2*d^4 + 143*I*c*d^5 + 142*d^6)*tan(f* 
x + e))/((d*tan(f*x + e) + c)*a^3*(c + I*d)^5*(c - I*d)^2*f*(tan(f*x + e) 
- I)^3)
 

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 2653, normalized size of antiderivative = 7.43 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(tan(e + f*x)*(c*d^7*550i + 625*d^8 + 129*c^2*d^6 + c^3*d^5*60i 
+ 47*c^4*d^4 - c^5*d^3*10i - c^6*d^2)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2 
*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2) - 
root(a^9*c^7*d^7*e^3*18432i + 9984*a^9*c^10*d^4*e^3 - 9984*a^9*c^4*d^10*e^ 
3 + a^9*c^9*d^5*e^3*9728i + a^9*c^5*d^9*e^3*9728i + 6912*a^9*c^8*d^6*e^3 - 
 6912*a^9*c^6*d^8*e^3 + 2816*a^9*c^12*d^2*e^3 - 2816*a^9*c^2*d^12*e^3 - a^ 
9*c^11*d^3*e^3*1024i - a^9*c^3*d^11*e^3*1024i - a^9*c^13*d*e^3*1536i - a^9 
*c*d^13*e^3*1536i + 256*a^9*d^14*e^3 - 256*a^9*c^14*e^3 + a^3*c*d^9*e*7510 
i - a^3*c^9*d*e*10i - 6525*a^3*c^2*d^8*e - 350*a^3*c^4*d^6*e - a^3*c^3*d^7 
*e*200i - 130*a^3*c^6*d^4*e + a^3*c^7*d^3*e*120i + a^3*c^5*d^5*e*100i + 45 
*a^3*c^8*d^2*e + 2353*a^3*d^10*e - a^3*c^10*e + c^2*d^6*94i + 32*c^3*d^5 - 
 c^4*d^4*5i - 176*c*d^7 + d^8*147i, e, k)*((a^3*d^8 + a^3*c*d^7*6i - 15*a^ 
3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^ 
2)*(8*a^3*c^10 - 200*a^3*d^10 + a^3*c*d^9*704i + a^3*c^9*d*64i + 552*a^3*c 
^2*d^8 + a^3*c^3*d^7*768i + 1456*a^3*c^4*d^6 - a^3*c^5*d^5*512i + 464*a^3* 
c^6*d^4 - a^3*c^7*d^3*512i - 232*a^3*c^8*d^2) - root(a^9*c^7*d^7*e^3*18432 
i + 9984*a^9*c^10*d^4*e^3 - 9984*a^9*c^4*d^10*e^3 + a^9*c^9*d^5*e^3*9728i 
+ a^9*c^5*d^9*e^3*9728i + 6912*a^9*c^8*d^6*e^3 - 6912*a^9*c^6*d^8*e^3 + 28 
16*a^9*c^12*d^2*e^3 - 2816*a^9*c^2*d^12*e^3 - a^9*c^11*d^3*e^3*1024i - a^9 
*c^3*d^11*e^3*1024i - a^9*c^13*d*e^3*1536i - a^9*c*d^13*e^3*1536i + 256...
 

Reduce [F]

\[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx=-\frac {\int \frac {1}{\tan \left (f x +e \right )^{5} d^{2} i +2 \tan \left (f x +e \right )^{4} c d i +3 \tan \left (f x +e \right )^{4} d^{2}+\tan \left (f x +e \right )^{3} c^{2} i +6 \tan \left (f x +e \right )^{3} c d -3 \tan \left (f x +e \right )^{3} d^{2} i +3 \tan \left (f x +e \right )^{2} c^{2}-6 \tan \left (f x +e \right )^{2} c d i -\tan \left (f x +e \right )^{2} d^{2}-3 \tan \left (f x +e \right ) c^{2} i -2 \tan \left (f x +e \right ) c d -c^{2}}d x}{a^{3}} \] Input:

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

Output:

( - int(1/(tan(e + f*x)**5*d**2*i + 2*tan(e + f*x)**4*c*d*i + 3*tan(e + f* 
x)**4*d**2 + tan(e + f*x)**3*c**2*i + 6*tan(e + f*x)**3*c*d - 3*tan(e + f* 
x)**3*d**2*i + 3*tan(e + f*x)**2*c**2 - 6*tan(e + f*x)**2*c*d*i - tan(e + 
f*x)**2*d**2 - 3*tan(e + f*x)*c**2*i - 2*tan(e + f*x)*c*d - c**2),x))/a**3