\(\int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx\) [1093]

Optimal result

Integrand size = 28, antiderivative size = 271 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {\left (c^4+4 i c^3 d-6 c^2 d^2+12 i c d^3+9 d^4\right ) x}{4 a^2 (c-i d)^2 (c+i d)^4}-\frac {2 (2 c-i d) d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 (c-i d)^2 (c+i d)^4 f}+\frac {d \left (c^2+4 i c d+9 d^2\right )}{4 a^2 (c-i d) (c+i d)^3 f (c+d \tan (e+f x))}+\frac {i c-4 d}{4 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \] Output:

1/4*(c^4+4*I*c^3*d-6*c^2*d^2+12*I*c*d^3+9*d^4)*x/a^2/(c-I*d)^2/(c+I*d)^4-2 
*(2*c-I*d)*d^3*ln(c*cos(f*x+e)+d*sin(f*x+e))/a^2/(c-I*d)^2/(c+I*d)^4/f+1/4 
*d*(c^2+4*I*c*d+9*d^2)/a^2/(c-I*d)/(c+I*d)^3/f/(c+d*tan(f*x+e))+1/4*(I*c-4 
*d)/a^2/(c+I*d)^2/f/(1+I*tan(f*x+e))/(c+d*tan(f*x+e))-1/4/(I*c-d)/f/(a+I*a 
*tan(f*x+e))^2/(c+d*tan(f*x+e))
 

Mathematica [A] (verified)

Time = 2.14 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=-\frac {\frac {2 (c+4 i d) ((i c+d) \log (i-\tan (e+f x))+(-i c+d) \log (i+\tan (e+f x))-2 d \log (c+d \tan (e+f x)))}{c^2+d^2}+\frac {2 i (c+i d)}{(-i+\tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {2 (c+4 i d)}{(-i+\tan (e+f x)) (c+d \tan (e+f x))}-\left (c^2+4 i c d+9 d^2\right ) \left (\frac {i \log (i-\tan (e+f x))}{(c+i d)^2}-\frac {i \log (i+\tan (e+f x))}{(c-i d)^2}+\frac {2 d \left (-2 c \log (c+d \tan (e+f x))+\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}\right )}{8 a^2 (c+i d)^2 f} \] Input:

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

Output:

-1/8*((2*(c + (4*I)*d)*((I*c + d)*Log[I - Tan[e + f*x]] + ((-I)*c + d)*Log 
[I + Tan[e + f*x]] - 2*d*Log[c + d*Tan[e + f*x]]))/(c^2 + d^2) + ((2*I)*(c 
 + I*d))/((-I + Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (2*(c + (4*I)*d))/ 
((-I + Tan[e + f*x])*(c + d*Tan[e + f*x])) - (c^2 + (4*I)*c*d + 9*d^2)*((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*Tan[e + f*x]] + (c^2 + d^2)/(c + d*Tan[e + f*x])) 
)/(c^2 + d^2)^2))/(a^2*(c + I*d)^2*f)
 

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4042, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4079

\(\displaystyle \frac {-\frac {\int \frac {2 \left (\left (c^2+4 i d c-9 d^2\right ) a^2+2 (c+4 i d) d \tan (e+f x) a^2\right )}{(c+d \tan (e+f x))^2}dx}{2 a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\left (c^2+4 i d c-9 d^2\right ) a^2+2 (c+4 i d) d \tan (e+f x) a^2}{(c+d \tan (e+f x))^2}dx}{a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {\left (c^2+4 i d c-9 d^2\right ) a^2+2 (c+4 i d) d \tan (e+f x) a^2}{(c+d \tan (e+f x))^2}dx}{a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (c^3+4 i d c^2-7 d^2 c+8 i d^3\right ) a^2+d \left (c^2+4 i d c+9 d^2\right ) \tan (e+f x) a^2}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^2 d \left (c^2+4 i c d+9 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (c^3+4 i d c^2-7 d^2 c+8 i d^3\right ) a^2+d \left (c^2+4 i d c+9 d^2\right ) \tan (e+f x) a^2}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^2 d \left (c^2+4 i c d+9 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4014

\(\displaystyle \frac {-\frac {\frac {\frac {a^2 x \left (c^4+4 i c^3 d-6 c^2 d^2+12 i c d^3+9 d^4\right )}{c^2+d^2}-\frac {8 a^2 d^3 (2 c-i d) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}}{c^2+d^2}+\frac {a^2 d \left (c^2+4 i c d+9 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\frac {a^2 x \left (c^4+4 i c^3 d-6 c^2 d^2+12 i c d^3+9 d^4\right )}{c^2+d^2}-\frac {8 a^2 d^3 (2 c-i d) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}}{c^2+d^2}+\frac {a^2 d \left (c^2+4 i c d+9 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

\(\Big \downarrow \) 4013

\(\displaystyle \frac {-\frac {\frac {a^2 d \left (c^2+4 i c d+9 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}+\frac {\frac {a^2 x \left (c^4+4 i c^3 d-6 c^2 d^2+12 i c d^3+9 d^4\right )}{c^2+d^2}-\frac {8 a^2 d^3 (2 c-i d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )}}{c^2+d^2}}{a^2 (-d+i c)}-\frac {c+4 i d}{f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))}}{4 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}\)

Input:

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

Output:

-1/4*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) + (-((c 
 + (4*I)*d)/((c + I*d)*f*(1 + I*Tan[e + f*x])*(c + d*Tan[e + f*x]))) - ((( 
a^2*(c^4 + (4*I)*c^3*d - 6*c^2*d^2 + (12*I)*c*d^3 + 9*d^4)*x)/(c^2 + d^2) 
- (8*a^2*(2*c - I*d)*d^3*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2 
)*f))/(c^2 + d^2) + (a^2*d*(c^2 + (4*I)*c*d + 9*d^2))/((c^2 + d^2)*f*(c + 
d*Tan[e + f*x])))/(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 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. 571 vs. \(2 (248 ) = 496\).

Time = 0.52 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.11

method	result	size
derivativedivides	\(\frac {2 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4}}+\frac {3 i c d}{2 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )}+\frac {3 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c d}{8 f \,a^{2} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{2}}{8 f \,a^{2} \left (i d +c \right )^{4}}-\frac {17 \arctan \left (\tan \left (f x +e \right )\right ) d^{2}}{8 f \,a^{2} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{2}}{16 f \,a^{2} \left (i d +c \right )^{4}}+\frac {17 i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d^{2}}{16 f \,a^{2} \left (i d +c \right )^{4}}+\frac {c^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )}-\frac {5 d^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )}+\frac {3 i \arctan \left (\tan \left (f x +e \right )\right ) c d}{4 f \,a^{2} \left (i d +c \right )^{4}}+\frac {i d^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {c d}{2 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {d^{3} c^{2}}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{5}}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{16 f \,a^{2} \left (i d -c \right )^{2}}-\frac {4 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4}}-\frac {i c^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{2} \left (i d -c \right )^{2}}\)	\(572\)
default	\(\frac {2 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4}}+\frac {3 i c d}{2 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )}+\frac {3 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c d}{8 f \,a^{2} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{2}}{8 f \,a^{2} \left (i d +c \right )^{4}}-\frac {17 \arctan \left (\tan \left (f x +e \right )\right ) d^{2}}{8 f \,a^{2} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{2}}{16 f \,a^{2} \left (i d +c \right )^{4}}+\frac {17 i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d^{2}}{16 f \,a^{2} \left (i d +c \right )^{4}}+\frac {c^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )}-\frac {5 d^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )}+\frac {3 i \arctan \left (\tan \left (f x +e \right )\right ) c d}{4 f \,a^{2} \left (i d +c \right )^{4}}+\frac {i d^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {c d}{2 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {d^{3} c^{2}}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{5}}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{16 f \,a^{2} \left (i d -c \right )^{2}}-\frac {4 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f \,a^{2} \left (i d -c \right )^{2} \left (i d +c \right )^{4}}-\frac {i c^{2}}{4 f \,a^{2} \left (i d +c \right )^{4} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{2} \left (i d -c \right )^{2}}\)	\(572\)
risch	\(-\frac {x}{4 a^{2} \left (2 i c d -c^{2}+d^{2}\right )}-\frac {3 \,{\mathrm e}^{-2 i \left (f x +e \right )} d}{4 a^{2} \left (2 i c d +c^{2}-d^{2}\right ) \left (i d +c \right ) f}+\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} c}{4 a^{2} \left (2 i c d +c^{2}-d^{2}\right ) \left (i d +c \right ) f}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )}}{16 a^{2} \left (2 i c d +c^{2}-d^{2}\right ) f}+\frac {4 d^{4} x}{a^{2} \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}+\frac {4 d^{4} e}{a^{2} f \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}+\frac {8 i d^{3} c x}{a^{2} \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}+\frac {8 i d^{3} c e}{a^{2} f \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}-\frac {2 i d^{4}}{\left (-i c +d \right )^{3} f \,a^{2} \left (i c +d \right )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )} d -d +i {\mathrm e}^{2 i \left (f x +e \right )} c +i c \right )}+\frac {2 i d^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{a^{2} f \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}-\frac {4 d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) c}{a^{2} f \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}\)	\(627\)
norman	\(\frac {\frac {4 i c -7 d}{8 a f \left (2 i c d +c^{2}-d^{2}\right )}+\frac {3 d \tan \left (f x +e \right )^{4}}{8 a f \left (2 i c d +c^{2}-d^{2}\right )}+\frac {\left (4 i c^{4} d +12 i c^{2} d^{3}+c^{5}-6 c^{3} d^{2}+9 c \,d^{4}\right ) x}{4 \left (-i d +c \right )^{2} \left (i d +c \right )^{4} a}+\frac {\left (4 i c^{4} d +12 i c^{2} d^{3}+c^{5}-6 c^{3} d^{2}+9 c \,d^{4}\right ) x \tan \left (f x +e \right )^{2}}{2 \left (-i d +c \right )^{2} \left (i d +c \right )^{4} a}+\frac {\left (4 i c^{4} d +12 i c^{2} d^{3}+c^{5}-6 c^{3} d^{2}+9 c \,d^{4}\right ) x \tan \left (f x +e \right )^{4}}{4 \left (-i d +c \right )^{2} \left (i d +c \right )^{4} a}-\frac {d \left (4 i c^{3} d +12 i c \,d^{3}+c^{4}-6 c^{2} d^{2}+9 d^{4}\right ) x \tan \left (f x +e \right )}{4 \left (i c +d \right )^{2} \left (-i c +d \right )^{4} a}-\frac {d \left (4 i c^{3} d +12 i c \,d^{3}+c^{4}-6 c^{2} d^{2}+9 d^{4}\right ) x \tan \left (f x +e \right )^{3}}{2 \left (i c +d \right )^{2} \left (-i c +d \right )^{4} a}-\frac {d \left (4 i c^{3} d +12 i c \,d^{3}+c^{4}-6 c^{2} d^{2}+9 d^{4}\right ) x \tan \left (f x +e \right )^{5}}{4 \left (i c +d \right )^{2} \left (-i c +d \right )^{4} a}+\frac {\left (-12 i c^{3} d -4 i c \,d^{3}-6 c^{4}-7 c^{2} d^{2}+15 d^{4}\right ) \tan \left (f x +e \right )}{8 a f \left (i c +d \right ) \left (-i c +d \right )^{3} c}+\frac {\left (-4 i c^{3} d +4 i c \,d^{3}-c^{4}-2 c^{2} d^{2}+15 d^{4}\right ) \tan \left (f x +e \right )^{3}}{4 a f \left (i c +d \right ) \left (-i c +d \right )^{3} c}+\frac {d \left (8 i c \,d^{2}-c^{2} d +15 d^{3}\right ) \tan \left (f x +e \right )^{5}}{8 a f \left (i c +d \right ) \left (-i c +d \right )^{3} c}}{a \left (c +d \tan \left (f x +e \right )\right ) \left (1+\tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (-i d^{4}+2 c \,d^{3}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{a^{2} f \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}-\frac {2 \left (-i d^{4}+2 c \,d^{3}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{a^{2} f \left (2 i c^{5} d +4 i c^{3} d^{3}+2 i c \,d^{5}+c^{6}+c^{4} d^{2}-c^{2} d^{4}-d^{6}\right )}\)	\(814\)

Input:

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

Output:

2*I/f/a^2*d^4/(I*d-c)^2/(c+I*d)^4*ln(c+d*tan(f*x+e))+3/2*I/f/a^2/(c+I*d)^4 
/(-I+tan(f*x+e))*c*d+3/8/f/a^2/(c+I*d)^4*ln(1+tan(f*x+e)^2)*c*d+1/8/f/a^2/ 
(c+I*d)^4*arctan(tan(f*x+e))*c^2-17/8/f/a^2/(c+I*d)^4*arctan(tan(f*x+e))*d 
^2-1/16*I/f/a^2/(c+I*d)^4*ln(1+tan(f*x+e)^2)*c^2+17/16*I/f/a^2/(c+I*d)^4*l 
n(1+tan(f*x+e)^2)*d^2+1/4/f/a^2/(c+I*d)^4/(-I+tan(f*x+e))*c^2-5/4/f/a^2/(c 
+I*d)^4/(-I+tan(f*x+e))*d^2+3/4*I/f/a^2/(c+I*d)^4*arctan(tan(f*x+e))*c*d+1 
/4*I/f/a^2/(c+I*d)^4/(-I+tan(f*x+e))^2*d^2+1/2/f/a^2/(c+I*d)^4/(-I+tan(f*x 
+e))^2*c*d+1/f/a^2*d^3/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e))*c^2+1/f/a^2*d^ 
5/(I*d-c)^2/(c+I*d)^4/(c+d*tan(f*x+e))+1/16*I/f/a^2/(I*d-c)^2*ln(1+tan(f*x 
+e)^2)-4/f/a^2*d^3/(I*d-c)^2/(c+I*d)^4*ln(c+d*tan(f*x+e))*c-1/4*I/f/a^2/(c 
+I*d)^4/(-I+tan(f*x+e))^2*c^2+1/8/f/a^2/(I*d-c)^2*arctan(tan(f*x+e))
                                                                                    
                                                                                    
 

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. 505 vs. \(2 (233) = 466\).

Time = 0.10 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=-\frac {c^{5} + i \, c^{4} d + 2 \, c^{3} d^{2} + 2 i \, c^{2} d^{3} + c d^{4} + i \, d^{5} - 4 \, {\left (i \, c^{5} - 3 \, c^{4} d - 2 i \, c^{3} d^{2} - 34 \, c^{2} d^{3} + 45 i \, c d^{4} + 17 \, d^{5}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 4 \, {\left (c^{5} + i \, c^{4} d + 6 \, c^{3} d^{2} - 2 i \, c^{2} d^{3} - 3 \, c d^{4} - 11 i \, d^{5} - {\left (i \, c^{5} - 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 22 \, c^{2} d^{3} - 11 i \, c d^{4} - 17 \, d^{5}\right )} f x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (5 \, c^{5} + 11 i \, c^{4} d + 10 \, c^{3} d^{2} + 22 i \, c^{2} d^{3} + 5 \, c d^{4} + 11 i \, d^{5}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 32 \, {\left ({\left (-2 i \, c^{2} d^{3} - 3 \, c d^{4} + i \, d^{5}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-2 i \, c^{2} d^{3} + c d^{4} - i \, d^{5}\right )} e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right )}{16 \, {\left ({\left (i \, a^{2} c^{7} - a^{2} c^{6} d + 3 i \, a^{2} c^{5} d^{2} - 3 \, a^{2} c^{4} d^{3} + 3 i \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} + i \, a^{2} c d^{6} - a^{2} d^{7}\right )} f e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (i \, a^{2} c^{7} - 3 \, a^{2} c^{6} d - i \, a^{2} c^{5} d^{2} - 5 \, a^{2} c^{4} d^{3} - 5 i \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 3 i \, a^{2} c d^{6} + a^{2} d^{7}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )}} \] Input:

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

Output:

-1/16*(c^5 + I*c^4*d + 2*c^3*d^2 + 2*I*c^2*d^3 + c*d^4 + I*d^5 - 4*(I*c^5 
- 3*c^4*d - 2*I*c^3*d^2 - 34*c^2*d^3 + 45*I*c*d^4 + 17*d^5)*f*x*e^(6*I*f*x 
 + 6*I*e) + 4*(c^5 + I*c^4*d + 6*c^3*d^2 - 2*I*c^2*d^3 - 3*c*d^4 - 11*I*d^ 
5 - (I*c^5 - 5*c^4*d - 10*I*c^3*d^2 - 22*c^2*d^3 - 11*I*c*d^4 - 17*d^5)*f* 
x)*e^(4*I*f*x + 4*I*e) + (5*c^5 + 11*I*c^4*d + 10*c^3*d^2 + 22*I*c^2*d^3 + 
 5*c*d^4 + 11*I*d^5)*e^(2*I*f*x + 2*I*e) - 32*((-2*I*c^2*d^3 - 3*c*d^4 + I 
*d^5)*e^(6*I*f*x + 6*I*e) + (-2*I*c^2*d^3 + c*d^4 - I*d^5)*e^(4*I*f*x + 4* 
I*e))*log(((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c - d)/(I*c + d)))/((I*a^2*c^ 
7 - a^2*c^6*d + 3*I*a^2*c^5*d^2 - 3*a^2*c^4*d^3 + 3*I*a^2*c^3*d^4 - 3*a^2* 
c^2*d^5 + I*a^2*c*d^6 - a^2*d^7)*f*e^(6*I*f*x + 6*I*e) + (I*a^2*c^7 - 3*a^ 
2*c^6*d - I*a^2*c^5*d^2 - 5*a^2*c^4*d^3 - 5*I*a^2*c^3*d^4 - a^2*c^2*d^5 - 
3*I*a^2*c*d^6 + a^2*d^7)*f*e^(4*I*f*x + 4*I*e))
 

Sympy [A] (verification not implemented)

Time = 29.11 (sec) , antiderivative size = 966, normalized size of antiderivative = 3.56 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=- \frac {2 i d^{4}}{a^{2} c^{6} f + 2 i a^{2} c^{5} d f + a^{2} c^{4} d^{2} f + 4 i a^{2} c^{3} d^{3} f - a^{2} c^{2} d^{4} f + 2 i a^{2} c d^{5} f - a^{2} d^{6} f + \left (a^{2} c^{6} f e^{2 i e} + 3 a^{2} c^{4} d^{2} f e^{2 i e} + 3 a^{2} c^{2} d^{4} f e^{2 i e} + a^{2} d^{6} f e^{2 i e}\right ) e^{2 i f x}} + \frac {x \left (c^{2} + 6 i c d - 17 d^{2}\right )}{4 a^{2} c^{4} + 16 i a^{2} c^{3} d - 24 a^{2} c^{2} d^{2} - 16 i a^{2} c d^{3} + 4 a^{2} d^{4}} + \begin {cases} \frac {\left (4 i a^{2} c^{3} f e^{2 i e} - 12 a^{2} c^{2} d f e^{2 i e} - 12 i a^{2} c d^{2} f e^{2 i e} + 4 a^{2} d^{3} f e^{2 i e}\right ) e^{- 4 i f x} + \left (16 i a^{2} c^{3} f e^{4 i e} - 80 a^{2} c^{2} d f e^{4 i e} - 112 i a^{2} c d^{2} f e^{4 i e} + 48 a^{2} d^{3} f e^{4 i e}\right ) e^{- 2 i f x}}{64 a^{4} c^{5} f^{2} e^{6 i e} + 320 i a^{4} c^{4} d f^{2} e^{6 i e} - 640 a^{4} c^{3} d^{2} f^{2} e^{6 i e} - 640 i a^{4} c^{2} d^{3} f^{2} e^{6 i e} + 320 a^{4} c d^{4} f^{2} e^{6 i e} + 64 i a^{4} d^{5} f^{2} e^{6 i e}} & \text {for}\: 64 a^{4} c^{5} f^{2} e^{6 i e} + 320 i a^{4} c^{4} d f^{2} e^{6 i e} - 640 a^{4} c^{3} d^{2} f^{2} e^{6 i e} - 640 i a^{4} c^{2} d^{3} f^{2} e^{6 i e} + 320 a^{4} c d^{4} f^{2} e^{6 i e} + 64 i a^{4} d^{5} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac {c^{2} + 6 i c d - 17 d^{2}}{4 a^{2} c^{4} + 16 i a^{2} c^{3} d - 24 a^{2} c^{2} d^{2} - 16 i a^{2} c d^{3} + 4 a^{2} d^{4}} + \frac {c^{2} e^{4 i e} + 2 c^{2} e^{2 i e} + c^{2} + 6 i c d e^{4 i e} + 8 i c d e^{2 i e} + 2 i c d - 17 d^{2} e^{4 i e} - 6 d^{2} e^{2 i e} - d^{2}}{4 a^{2} c^{4} e^{4 i e} + 16 i a^{2} c^{3} d e^{4 i e} - 24 a^{2} c^{2} d^{2} e^{4 i e} - 16 i a^{2} c d^{3} e^{4 i e} + 4 a^{2} d^{4} e^{4 i e}}\right ) & \text {otherwise} \end {cases} - \frac {2 d^{3} \cdot \left (2 c - i d\right ) \log {\left (\frac {c + i d}{c e^{2 i e} - i d e^{2 i e}} + e^{2 i f x} \right )}}{a^{2} f \left (c - i d\right )^{2} \left (c + i d\right )^{4}} \] Input:

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

Output:

-2*I*d**4/(a**2*c**6*f + 2*I*a**2*c**5*d*f + a**2*c**4*d**2*f + 4*I*a**2*c 
**3*d**3*f - a**2*c**2*d**4*f + 2*I*a**2*c*d**5*f - a**2*d**6*f + (a**2*c* 
*6*f*exp(2*I*e) + 3*a**2*c**4*d**2*f*exp(2*I*e) + 3*a**2*c**2*d**4*f*exp(2 
*I*e) + a**2*d**6*f*exp(2*I*e))*exp(2*I*f*x)) + x*(c**2 + 6*I*c*d - 17*d** 
2)/(4*a**2*c**4 + 16*I*a**2*c**3*d - 24*a**2*c**2*d**2 - 16*I*a**2*c*d**3 
+ 4*a**2*d**4) + Piecewise((((4*I*a**2*c**3*f*exp(2*I*e) - 12*a**2*c**2*d* 
f*exp(2*I*e) - 12*I*a**2*c*d**2*f*exp(2*I*e) + 4*a**2*d**3*f*exp(2*I*e))*e 
xp(-4*I*f*x) + (16*I*a**2*c**3*f*exp(4*I*e) - 80*a**2*c**2*d*f*exp(4*I*e) 
- 112*I*a**2*c*d**2*f*exp(4*I*e) + 48*a**2*d**3*f*exp(4*I*e))*exp(-2*I*f*x 
))/(64*a**4*c**5*f**2*exp(6*I*e) + 320*I*a**4*c**4*d*f**2*exp(6*I*e) - 640 
*a**4*c**3*d**2*f**2*exp(6*I*e) - 640*I*a**4*c**2*d**3*f**2*exp(6*I*e) + 3 
20*a**4*c*d**4*f**2*exp(6*I*e) + 64*I*a**4*d**5*f**2*exp(6*I*e)), Ne(64*a* 
*4*c**5*f**2*exp(6*I*e) + 320*I*a**4*c**4*d*f**2*exp(6*I*e) - 640*a**4*c** 
3*d**2*f**2*exp(6*I*e) - 640*I*a**4*c**2*d**3*f**2*exp(6*I*e) + 320*a**4*c 
*d**4*f**2*exp(6*I*e) + 64*I*a**4*d**5*f**2*exp(6*I*e), 0)), (x*(-(c**2 + 
6*I*c*d - 17*d**2)/(4*a**2*c**4 + 16*I*a**2*c**3*d - 24*a**2*c**2*d**2 - 1 
6*I*a**2*c*d**3 + 4*a**2*d**4) + (c**2*exp(4*I*e) + 2*c**2*exp(2*I*e) + c* 
*2 + 6*I*c*d*exp(4*I*e) + 8*I*c*d*exp(2*I*e) + 2*I*c*d - 17*d**2*exp(4*I*e 
) - 6*d**2*exp(2*I*e) - d**2)/(4*a**2*c**4*exp(4*I*e) + 16*I*a**2*c**3*d*e 
xp(4*I*e) - 24*a**2*c**2*d**2*exp(4*I*e) - 16*I*a**2*c*d**3*exp(4*I*e) ...
                                                                                    
                                                                                    
 

Maxima [F(-2)]

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

integrate(1/(a+I*a*tan(f*x+e))^2/(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 = 0.55 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {{\left (-i \, c^{2} + 6 \, c d + 17 i \, d^{2}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{8 \, {\left (a^{2} c^{4} f + 4 i \, a^{2} c^{3} d f - 6 \, a^{2} c^{2} d^{2} f - 4 i \, a^{2} c d^{3} f + a^{2} d^{4} f\right )}} - \frac {2 \, {\left (2 \, c d^{4} - i \, d^{5}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{a^{2} c^{6} d f + 2 i \, a^{2} c^{5} d^{2} f + a^{2} c^{4} d^{3} f + 4 i \, a^{2} c^{3} d^{4} f - a^{2} c^{2} d^{5} f + 2 i \, a^{2} c d^{6} f - a^{2} d^{7} f} - \frac {\log \left (\tan \left (f x + e\right ) + i\right )}{8 i \, a^{2} c^{2} f + 16 \, a^{2} c d f - 8 i \, a^{2} d^{2} f} + \frac {-2 i \, c^{5} + 4 \, c^{4} d - 8 i \, c^{3} d^{2} - 6 i \, c d^{4} - 4 \, d^{5} + {\left (c^{4} d + 4 i \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 4 i \, c d^{4} + 9 \, d^{5}\right )} \tan \left (f x + e\right )^{2} + {\left (c^{5} + 2 i \, c^{4} d + 10 \, c^{3} d^{2} - 12 i \, c^{2} d^{3} + 9 \, c d^{4} - 14 i \, d^{5}\right )} \tan \left (f x + e\right )}{4 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{2} {\left (c + i \, d\right )}^{4} {\left (c - i \, d\right )}^{2} f {\left (\tan \left (f x + e\right ) - i\right )}^{2}} \] Input:

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

Output:

1/8*(-I*c^2 + 6*c*d + 17*I*d^2)*log(tan(f*x + e) - I)/(a^2*c^4*f + 4*I*a^2 
*c^3*d*f - 6*a^2*c^2*d^2*f - 4*I*a^2*c*d^3*f + a^2*d^4*f) - 2*(2*c*d^4 - I 
*d^5)*log(abs(d*tan(f*x + e) + c))/(a^2*c^6*d*f + 2*I*a^2*c^5*d^2*f + a^2* 
c^4*d^3*f + 4*I*a^2*c^3*d^4*f - a^2*c^2*d^5*f + 2*I*a^2*c*d^6*f - a^2*d^7* 
f) - log(tan(f*x + e) + I)/(8*I*a^2*c^2*f + 16*a^2*c*d*f - 8*I*a^2*d^2*f) 
+ 1/4*(-2*I*c^5 + 4*c^4*d - 8*I*c^3*d^2 - 6*I*c*d^4 - 4*d^5 + (c^4*d + 4*I 
*c^3*d^2 + 10*c^2*d^3 + 4*I*c*d^4 + 9*d^5)*tan(f*x + e)^2 + (c^5 + 2*I*c^4 
*d + 10*c^3*d^2 - 12*I*c^2*d^3 + 9*c*d^4 - 14*I*d^5)*tan(f*x + e))/((d*tan 
(f*x + e) + c)*a^2*(c + I*d)^4*(c - I*d)^2*f*(tan(f*x + e) - I)^2)
 

Mupad [B] (verification not implemented)

Time = 7.59 (sec) , antiderivative size = 1984, normalized size of antiderivative = 7.32 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((a^2*d^6 + a^2*c*d^5*4i - 6*a^2*c^2*d^4 - a^2*c^3*d^3*4i + a^2* 
c^4*d^2)*(c^5*d - 95*c*d^5 + d^6*72i + c^2*d^4*16i - 14*c^3*d^3 + c^4*d^2* 
8i) - root(1792*a^6*c^6*d^6*e^3 + 1088*a^6*c^8*d^4*e^3 + 1088*a^6*c^4*d^8* 
e^3 - a^6*c^9*d^3*e^3*768i + a^6*c^3*d^9*e^3*768i - a^6*c^7*d^5*e^3*512i + 
 a^6*c^5*d^7*e^3*512i + 128*a^6*c^10*d^2*e^3 + 128*a^6*c^2*d^10*e^3 - a^6* 
c^11*d*e^3*256i + a^6*c*d^11*e^3*256i - 64*a^6*d^12*e^3 - 64*a^6*c^12*e^3 
- a^2*c*d^7*e*984i - a^2*c^7*d*e*8i + 1020*a^2*c^2*d^6*e + a^2*c^3*d^5*e*7 
2i + 42*a^2*c^4*d^4*e + a^2*c^5*d^3*e*24i + 28*a^2*c^6*d^2*e - 273*a^2*d^8 
*e - a^2*c^8*e - c^2*d^4*22i - 4*c^3*d^3 + 56*c*d^5 - d^6*34i, e, k)*((a^2 
*d^6 + a^2*c*d^5*4i - 6*a^2*c^2*d^4 - a^2*c^3*d^3*4i + a^2*c^4*d^2)*(a^2*c 
*d^7*88i - 36*a^2*d^8 - 4*a^2*c^8 - a^2*c^7*d*24i + a^2*c^3*d^5*152i + 104 
*a^2*c^4*d^4 + a^2*c^5*d^3*40i + 64*a^2*c^6*d^2) + root(1792*a^6*c^6*d^6*e 
^3 + 1088*a^6*c^8*d^4*e^3 + 1088*a^6*c^4*d^8*e^3 - a^6*c^9*d^3*e^3*768i + 
a^6*c^3*d^9*e^3*768i - a^6*c^7*d^5*e^3*512i + a^6*c^5*d^7*e^3*512i + 128*a 
^6*c^10*d^2*e^3 + 128*a^6*c^2*d^10*e^3 - a^6*c^11*d*e^3*256i + a^6*c*d^11* 
e^3*256i - 64*a^6*d^12*e^3 - 64*a^6*c^12*e^3 - a^2*c*d^7*e*984i - a^2*c^7* 
d*e*8i + 1020*a^2*c^2*d^6*e + a^2*c^3*d^5*e*72i + 42*a^2*c^4*d^4*e + a^2*c 
^5*d^3*e*24i + 28*a^2*c^6*d^2*e - 273*a^2*d^8*e - a^2*c^8*e - c^2*d^4*22i 
- 4*c^3*d^3 + 56*c*d^5 - d^6*34i, e, k)*((a^2*d^6 + a^2*c*d^5*4i - 6*a^2*c 
^2*d^4 - a^2*c^3*d^3*4i + a^2*c^4*d^2)*(a^4*c^2*d^8*512i - 128*a^4*c^9*...
 

Reduce [F]

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

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

Output:

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