\(\int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx\) [1098]

Optimal result

Integrand size = 28, antiderivative size = 273 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx=\frac {\left (c^4+4 i c^3 d+6 c^2 d^2-12 i c d^3-3 d^4\right ) x}{2 a (c-i d)^3 (c+i d)^4}+\frac {2 d^2 \left (3 c^2-2 i c d-d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{a (c+i d)^4 (i c+d)^3 f}+\frac {(c-2 i d) d}{2 a (c-i d) (c+i d)^2 f (c+d \tan (e+f x))^2}-\frac {1}{2 (i c-d) f (a+i a \tan (e+f x)) (c+d \tan (e+f x))^2}+\frac {d \left (c^2-8 i c d-3 d^2\right )}{2 a (c-i d)^2 (c+i d)^3 f (c+d \tan (e+f x))} \] Output:

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

Mathematica [A] (verified)

Time = 3.33 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx=-\frac {i \left (\frac {3 \log (i-\tan (e+f x))}{(c+i d)^2}-\frac {3 \log (i+\tan (e+f x))}{(c-i d)^2}+\frac {2 i}{(-i+\tan (e+f x)) (c+d \tan (e+f x))^2}+\frac {6 i d \left (2 c \log (c+d \tan (e+f x))-\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}+2 i (c-2 i d) \left (\frac {i \log (i-\tan (e+f x))}{(c+i d)^3}-\frac {\log (i+\tan (e+f x))}{(i c+d)^3}+\frac {d \left (\left (-6 c^2+2 d^2\right ) \log (c+d \tan (e+f x))+\frac {\left (c^2+d^2\right ) \left (5 c^2+d^2+4 c d \tan (e+f x)\right )}{(c+d \tan (e+f x))^2}\right )}{\left (c^2+d^2\right )^3}\right )\right )}{4 a (c+i d) f} \] Input:

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

Output:

((-1/4*I)*((3*Log[I - Tan[e + f*x]])/(c + I*d)^2 - (3*Log[I + Tan[e + f*x] 
])/(c - I*d)^2 + (2*I)/((-I + Tan[e + f*x])*(c + d*Tan[e + f*x])^2) + ((6* 
I)*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 + (2*I)*(c - (2*I)*d)*((I*Log[I - Tan[e + f*x]])/(c + I*d)^3 - 
Log[I + Tan[e + f*x]]/(I*c + d)^3 + (d*((-6*c^2 + 2*d^2)*Log[c + d*Tan[e + 
 f*x]] + ((c^2 + d^2)*(5*c^2 + d^2 + 4*c*d*Tan[e + f*x]))/(c + d*Tan[e + f 
*x])^2))/(c^2 + d^2)^3)))/(a*(c + I*d)*f)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4035, 3042, 4012, 25, 3042, 4012, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4035

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4014

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4013

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (250 ) = 500\).

Time = 0.55 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.27

method	result	size
derivativedivides	\(\frac {3 i d^{4} c}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c}{8 f a \left (i d +c \right )^{4}}+\frac {d^{3} c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{5}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {3 i d^{2} c^{3}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {7 i \arctan \left (\tan \left (f x +e \right )\right ) d}{4 f a \left (i d +c \right )^{4}}-\frac {4 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}-\frac {6 i d^{2} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}+\frac {i d^{6}}{2 f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {2 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}+\frac {i d^{2} c^{4}}{2 f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{4 f a \left (i d -c \right )^{3}}+\frac {1}{2 f a \left (i d +c \right )^{3} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{8 f a \left (i d -c \right )^{3}}+\frac {7 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d}{8 f a \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c}{4 f a \left (i d +c \right )^{4}}+\frac {i d^{4} c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}\)	\(619\)
default	\(\frac {3 i d^{4} c}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c}{8 f a \left (i d +c \right )^{4}}+\frac {d^{3} c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{5}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {3 i d^{2} c^{3}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {7 i \arctan \left (\tan \left (f x +e \right )\right ) d}{4 f a \left (i d +c \right )^{4}}-\frac {4 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}-\frac {6 i d^{2} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}+\frac {i d^{6}}{2 f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {2 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}+\frac {i d^{2} c^{4}}{2 f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{4 f a \left (i d -c \right )^{3}}+\frac {1}{2 f a \left (i d +c \right )^{3} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{8 f a \left (i d -c \right )^{3}}+\frac {7 \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d}{8 f a \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c}{4 f a \left (i d +c \right )^{4}}+\frac {i d^{4} c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}\)	\(619\)
risch	\(-\frac {x}{2 a \left (3 i c^{2} d -i d^{3}-c^{3}+3 c \,d^{2}\right )}+\frac {i {\mathrm e}^{-2 i \left (f x +e \right )}}{4 a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) f}+\frac {12 d^{2} c^{2} x}{a \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}-\frac {4 d^{4} x}{a \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}+\frac {12 d^{2} c^{2} e}{a f \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}-\frac {4 d^{4} e}{a f \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}-\frac {8 i d^{3} c x}{a \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}-\frac {8 i d^{3} c e}{a f \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}+\frac {2 d^{3} \left (-4 i c d \,{\mathrm e}^{2 i \left (f x +e \right )}+4 c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 i c d +4 c^{2}+d^{2}\right )}{\left (-i c +d \right )^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )} d -d +i {\mathrm e}^{2 i \left (f x +e \right )} c +i c \right )^{2} f a \left (i c +d \right )^{3}}+\frac {6 i d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) c^{2}}{a f \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}-\frac {2 i d^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{a f \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}+\frac {4 d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) c}{a f \left (i c^{6} d +3 i c^{4} d^{3}+3 i c^{2} d^{5}+i d^{7}+c^{7}+3 c^{5} d^{2}+3 c^{3} d^{4}+c \,d^{6}\right )}\)	\(902\)
norman	\(\frac {\frac {d c \left (4 i c^{3} d -12 i c \,d^{3}+c^{4}+6 c^{2} d^{2}-3 d^{4}\right ) x \tan \left (f x +e \right )}{\left (c^{2}+d^{2}\right ) a \left (i c^{4} d +2 i c^{2} d^{3}+i d^{5}+c^{5}+2 c^{3} d^{2}+c \,d^{4}\right )}+\frac {d c \left (4 i c^{3} d -12 i c \,d^{3}+c^{4}+6 c^{2} d^{2}-3 d^{4}\right ) x \tan \left (f x +e \right )^{3}}{\left (c^{2}+d^{2}\right ) a \left (i c^{4} d +2 i c^{2} d^{3}+i d^{5}+c^{5}+2 c^{3} d^{2}+c \,d^{4}\right )}+\frac {\tan \left (f x +e \right )}{2 a f \left (i d +c \right )}+\frac {\left (4 i c^{3} d -12 i c \,d^{3}+c^{4}+6 c^{2} d^{2}-3 d^{4}\right ) x \tan \left (f x +e \right )^{2}}{2 a \left (i c^{4} d +2 i c^{2} d^{3}+i d^{5}+c^{5}+2 c^{3} d^{2}+c \,d^{4}\right )}+\frac {2 i c^{4} d -8 i c^{2} d^{3}-2 i d^{5}+3 c^{3} d^{2}-c \,d^{4}}{4 a f \left (i c^{4} d +2 i c^{2} d^{3}+i d^{5}+c^{5}+2 c^{3} d^{2}+c \,d^{4}\right ) d}+\frac {c^{2} \left (4 i c^{3} d -12 i c \,d^{3}+c^{4}+6 c^{2} d^{2}-3 d^{4}\right ) x}{2 \left (c^{2}+d^{2}\right ) a \left (i c^{4} d +2 i c^{2} d^{3}+i d^{5}+c^{5}+2 c^{3} d^{2}+c \,d^{4}\right )}+\frac {d \left (i c^{2} d^{2}-3 i d^{4}+8 c \,d^{3}\right ) \tan \left (f x +e \right )^{4}}{4 \left (-i c^{5}-2 i c^{3} d^{2}-i c \,d^{4}+c^{4} d +2 c^{2} d^{3}+d^{5}\right ) a f c}+\frac {\left (-12 i c^{3} d^{3}+4 i c \,d^{5}+3 c^{4} d^{2}-2 c^{2} d^{4}+3 d^{6}\right ) \tan \left (f x +e \right )^{2}}{4 c a f \left (i c^{4} d +2 i c^{2} d^{3}+i d^{5}+c^{5}+2 c^{3} d^{2}+c \,d^{4}\right ) d}+\frac {i d^{2} \left (-4 i c^{3} d +12 i c \,d^{3}-c^{4}-6 c^{2} d^{2}+3 d^{4}\right ) x \tan \left (f x +e \right )^{4}}{2 \left (c^{2}+d^{2}\right ) a \left (-i c^{5}-2 i c^{3} d^{2}-i c \,d^{4}+c^{4} d +2 c^{2} d^{3}+d^{5}\right )}}{\left (c +d \tan \left (f x +e \right )\right )^{2} \left (1+\tan \left (f x +e \right )^{2}\right )}+\frac {d^{2} \left (2 i c d -3 c^{2}+d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{a f \left (-i c^{7}-3 i c^{5} d^{2}-3 i c^{3} d^{4}-i c \,d^{6}+c^{6} d +3 c^{4} d^{3}+3 c^{2} d^{5}+d^{7}\right )}-\frac {2 d^{2} \left (2 i c d -3 c^{2}+d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{a f \left (-i c^{7}-3 i c^{5} d^{2}-3 i c^{3} d^{4}-i c \,d^{6}+c^{6} d +3 c^{4} d^{3}+3 c^{2} d^{5}+d^{7}\right )}\)	\(977\)

Input:

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

Output:

3*I/f/a*d^4/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))*c-1/8*I/f/a/(c+I*d)^4*ln( 
1+tan(f*x+e)^2)*c+1/f/a*d^3/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))*c^2+1/f/a 
*d^5/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))+3*I/f/a*d^2/(I*d-c)^3/(c+I*d)^4/ 
(c+d*tan(f*x+e))*c^3+7/4*I/f/a/(c+I*d)^4*arctan(tan(f*x+e))*d-4/f/a*d^3/(I 
*d-c)^3/(c+I*d)^4*ln(c+d*tan(f*x+e))*c-6*I/f/a*d^2/(I*d-c)^3/(c+I*d)^4*ln( 
c+d*tan(f*x+e))*c^2+1/2*I/f/a*d^6/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))^2+2 
*I/f/a*d^4/(I*d-c)^3/(c+I*d)^4*ln(c+d*tan(f*x+e))+1/2*I/f/a*d^2/(I*d-c)^3/ 
(c+I*d)^4/(c+d*tan(f*x+e))^2*c^4-1/4/f/a/(I*d-c)^3*arctan(tan(f*x+e))+1/2/ 
f/a/(c+I*d)^3/(-I+tan(f*x+e))-1/8*I/f/a/(I*d-c)^3*ln(1+tan(f*x+e)^2)+7/8/f 
/a/(c+I*d)^4*ln(1+tan(f*x+e)^2)*d+1/4/f/a/(c+I*d)^4*arctan(tan(f*x+e))*c+I 
/f/a*d^4/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))^2*c^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. 709 vs. \(2 (235) = 470\).

Time = 0.11 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.60 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6 + 2*(-I*c^6 + 2*c^5*d - 25*I*c^4*d^ 
2 - 60*c^3*d^3 + 65*I*c^2*d^4 + 34*c*d^5 - 7*I*d^6)*f*x*e^(6*I*f*x + 6*I*e 
) + (c^6 - 4*I*c^5*d - 5*c^4*d^2 + 32*I*c^3*d^3 - 5*c^2*d^4 + 36*I*c*d^5 + 
 d^6 + 4*(-I*c^6 + 4*c^5*d - 19*I*c^4*d^2 - 16*c^3*d^3 - 11*I*c^2*d^4 - 20 
*c*d^5 + 7*I*d^6)*f*x)*e^(4*I*f*x + 4*I*e) + 2*(c^6 - 2*I*c^5*d + c^4*d^2 
+ 12*I*c^3*d^3 - 29*c^2*d^4 - 10*I*c*d^5 - 5*d^6 + (-I*c^6 + 6*c^5*d - 9*I 
*c^4*d^2 + 12*c^3*d^3 - 15*I*c^2*d^4 + 6*c*d^5 - 7*I*d^6)*f*x)*e^(2*I*f*x 
+ 2*I*e) + 8*((3*c^4*d^2 - 8*I*c^3*d^3 - 8*c^2*d^4 + 4*I*c*d^5 + d^6)*e^(6 
*I*f*x + 6*I*e) + 2*(3*c^4*d^2 - 2*I*c^3*d^3 + 2*c^2*d^4 - 2*I*c*d^5 - d^6 
)*e^(4*I*f*x + 4*I*e) + (3*c^4*d^2 + 4*I*c^3*d^3 + d^6)*e^(2*I*f*x + 2*I*e 
))*log(((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c - d)/(I*c + d)))/((-I*a*c^9 - 
a*c^8*d - 4*I*a*c^7*d^2 - 4*a*c^6*d^3 - 6*I*a*c^5*d^4 - 6*a*c^4*d^5 - 4*I* 
a*c^3*d^6 - 4*a*c^2*d^7 - I*a*c*d^8 - a*d^9)*f*e^(6*I*f*x + 6*I*e) + 2*(-I 
*a*c^9 + a*c^8*d - 4*I*a*c^7*d^2 + 4*a*c^6*d^3 - 6*I*a*c^5*d^4 + 6*a*c^4*d 
^5 - 4*I*a*c^3*d^6 + 4*a*c^2*d^7 - I*a*c*d^8 + a*d^9)*f*e^(4*I*f*x + 4*I*e 
) + (-I*a*c^9 + 3*a*c^8*d + 8*a*c^6*d^3 + 6*I*a*c^5*d^4 + 6*a*c^4*d^5 + 8* 
I*a*c^3*d^6 + 3*I*a*c*d^8 - a*d^9)*f*e^(2*I*f*x + 2*I*e))
 

Sympy [A] (verification not implemented)

Time = 49.55 (sec) , antiderivative size = 821, normalized size of antiderivative = 3.01 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx=\frac {x \left (c + 7 i d\right )}{2 a c^{4} + 8 i a c^{3} d - 12 a c^{2} d^{2} - 8 i a c d^{3} + 2 a d^{4}} + \frac {- 8 c^{2} d^{3} - 6 i c d^{4} - 2 d^{5} + \left (- 8 c^{2} d^{3} e^{2 i e} + 8 i c d^{4} e^{2 i e}\right ) e^{2 i f x}}{a c^{8} f + 2 i a c^{7} d f + 2 a c^{6} d^{2} f + 6 i a c^{5} d^{3} f + 6 i a c^{3} d^{5} f - 2 a c^{2} d^{6} f + 2 i a c d^{7} f - a d^{8} f + \left (2 a c^{8} f e^{2 i e} + 8 a c^{6} d^{2} f e^{2 i e} + 12 a c^{4} d^{4} f e^{2 i e} + 8 a c^{2} d^{6} f e^{2 i e} + 2 a d^{8} f e^{2 i e}\right ) e^{2 i f x} + \left (a c^{8} f e^{4 i e} - 2 i a c^{7} d f e^{4 i e} + 2 a c^{6} d^{2} f e^{4 i e} - 6 i a c^{5} d^{3} f e^{4 i e} - 6 i a c^{3} d^{5} f e^{4 i e} - 2 a c^{2} d^{6} f e^{4 i e} - 2 i a c d^{7} f e^{4 i e} - a d^{8} f e^{4 i e}\right ) e^{4 i f x}} + \begin {cases} \frac {i e^{- 2 i f x}}{4 a c^{3} f e^{2 i e} + 12 i a c^{2} d f e^{2 i e} - 12 a c d^{2} f e^{2 i e} - 4 i a d^{3} f e^{2 i e}} & \text {for}\: 4 a c^{3} f e^{2 i e} + 12 i a c^{2} d f e^{2 i e} - 12 a c d^{2} f e^{2 i e} - 4 i a d^{3} f e^{2 i e} \neq 0 \\x \left (- \frac {c + 7 i d}{2 a c^{4} + 8 i a c^{3} d - 12 a c^{2} d^{2} - 8 i a c d^{3} + 2 a d^{4}} + \frac {c e^{2 i e} + c + 7 i d e^{2 i e} + i d}{2 a c^{4} e^{2 i e} + 8 i a c^{3} d e^{2 i e} - 12 a c^{2} d^{2} e^{2 i e} - 8 i a c d^{3} e^{2 i e} + 2 a d^{4} e^{2 i e}}\right ) & \text {otherwise} \end {cases} + \frac {2 i d^{2} \cdot \left (3 c^{2} - 2 i c d - d^{2}\right ) \log {\left (\frac {c + i d}{c e^{2 i e} - i d e^{2 i e}} + e^{2 i f x} \right )}}{a f \left (c - i d\right )^{3} \left (c + i d\right )^{4}} \] Input:

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

Output:

x*(c + 7*I*d)/(2*a*c**4 + 8*I*a*c**3*d - 12*a*c**2*d**2 - 8*I*a*c*d**3 + 2 
*a*d**4) + (-8*c**2*d**3 - 6*I*c*d**4 - 2*d**5 + (-8*c**2*d**3*exp(2*I*e) 
+ 8*I*c*d**4*exp(2*I*e))*exp(2*I*f*x))/(a*c**8*f + 2*I*a*c**7*d*f + 2*a*c* 
*6*d**2*f + 6*I*a*c**5*d**3*f + 6*I*a*c**3*d**5*f - 2*a*c**2*d**6*f + 2*I* 
a*c*d**7*f - a*d**8*f + (2*a*c**8*f*exp(2*I*e) + 8*a*c**6*d**2*f*exp(2*I*e 
) + 12*a*c**4*d**4*f*exp(2*I*e) + 8*a*c**2*d**6*f*exp(2*I*e) + 2*a*d**8*f* 
exp(2*I*e))*exp(2*I*f*x) + (a*c**8*f*exp(4*I*e) - 2*I*a*c**7*d*f*exp(4*I*e 
) + 2*a*c**6*d**2*f*exp(4*I*e) - 6*I*a*c**5*d**3*f*exp(4*I*e) - 6*I*a*c**3 
*d**5*f*exp(4*I*e) - 2*a*c**2*d**6*f*exp(4*I*e) - 2*I*a*c*d**7*f*exp(4*I*e 
) - a*d**8*f*exp(4*I*e))*exp(4*I*f*x)) + Piecewise((I*exp(-2*I*f*x)/(4*a*c 
**3*f*exp(2*I*e) + 12*I*a*c**2*d*f*exp(2*I*e) - 12*a*c*d**2*f*exp(2*I*e) - 
 4*I*a*d**3*f*exp(2*I*e)), Ne(4*a*c**3*f*exp(2*I*e) + 12*I*a*c**2*d*f*exp( 
2*I*e) - 12*a*c*d**2*f*exp(2*I*e) - 4*I*a*d**3*f*exp(2*I*e), 0)), (x*(-(c 
+ 7*I*d)/(2*a*c**4 + 8*I*a*c**3*d - 12*a*c**2*d**2 - 8*I*a*c*d**3 + 2*a*d* 
*4) + (c*exp(2*I*e) + c + 7*I*d*exp(2*I*e) + I*d)/(2*a*c**4*exp(2*I*e) + 8 
*I*a*c**3*d*exp(2*I*e) - 12*a*c**2*d**2*exp(2*I*e) - 8*I*a*c*d**3*exp(2*I* 
e) + 2*a*d**4*exp(2*I*e))), True)) + 2*I*d**2*(3*c**2 - 2*I*c*d - d**2)*lo 
g((c + I*d)/(c*exp(2*I*e) - I*d*exp(2*I*e)) + exp(2*I*f*x))/(a*f*(c - I*d) 
**3*(c + I*d)**4)
 

Maxima [F(-2)]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx=-\frac {i \, {\left (c + 7 i \, d\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{4 \, {\left (a c^{4} f + 4 i \, a c^{3} d f - 6 \, a c^{2} d^{2} f - 4 i \, a c d^{3} f + a d^{4} f\right )}} + \frac {2 i \, {\left (3 \, c^{2} d^{3} - 2 i \, c d^{4} - d^{5}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{a c^{7} d f + i \, a c^{6} d^{2} f + 3 \, a c^{5} d^{3} f + 3 i \, a c^{4} d^{4} f + 3 \, a c^{3} d^{5} f + 3 i \, a c^{2} d^{6} f + a c d^{7} f + i \, a d^{8} f} + \frac {i \, \log \left (\tan \left (f x + e\right ) + i\right )}{4 \, {\left (a c^{3} f - 3 i \, a c^{2} d f - 3 \, a c d^{2} f + i \, a d^{3} f\right )}} - \frac {i \, {\left (c^{6} - 2 i \, c^{5} d - 7 \, c^{4} d^{2} - 9 \, c^{2} d^{4} + 2 i \, c d^{5} - d^{6} + i \, {\left (-i \, c^{4} d^{2} - 8 \, c^{3} d^{3} + 2 i \, c^{2} d^{4} - 8 \, c d^{5} + 3 i \, d^{6}\right )} \tan \left (f x + e\right )^{2} + i \, {\left (-2 i \, c^{5} d - 11 \, c^{4} d^{2} + 8 i \, c^{3} d^{3} - 10 \, c^{2} d^{4} + 10 i \, c d^{5} + d^{6}\right )} \tan \left (f x + e\right )\right )}}{2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} a {\left (i \, c + d\right )}^{3} {\left (-i \, c + d\right )}^{4} f {\left (\tan \left (f x + e\right ) - i\right )}} \] Input:

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

Output:

-1/4*I*(c + 7*I*d)*log(tan(f*x + e) - I)/(a*c^4*f + 4*I*a*c^3*d*f - 6*a*c^ 
2*d^2*f - 4*I*a*c*d^3*f + a*d^4*f) + 2*I*(3*c^2*d^3 - 2*I*c*d^4 - d^5)*log 
(abs(d*tan(f*x + e) + c))/(a*c^7*d*f + I*a*c^6*d^2*f + 3*a*c^5*d^3*f + 3*I 
*a*c^4*d^4*f + 3*a*c^3*d^5*f + 3*I*a*c^2*d^6*f + a*c*d^7*f + I*a*d^8*f) + 
1/4*I*log(tan(f*x + e) + I)/(a*c^3*f - 3*I*a*c^2*d*f - 3*a*c*d^2*f + I*a*d 
^3*f) - 1/2*I*(c^6 - 2*I*c^5*d - 7*c^4*d^2 - 9*c^2*d^4 + 2*I*c*d^5 - d^6 + 
 I*(-I*c^4*d^2 - 8*c^3*d^3 + 2*I*c^2*d^4 - 8*c*d^5 + 3*I*d^6)*tan(f*x + e) 
^2 + I*(-2*I*c^5*d - 11*c^4*d^2 + 8*I*c^3*d^3 - 10*c^2*d^4 + 10*I*c*d^5 + 
d^6)*tan(f*x + e))/((d*tan(f*x + e) + c)^2*a*(I*c + d)^3*(-I*c + d)^4*f*(t 
an(f*x + e) - I))
 

Mupad [B] (verification not implemented)

Time = 7.18 (sec) , antiderivative size = 1910, normalized size of antiderivative = 7.00 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((a*d^4 - a*c^2*d^2 + a*c*d^3*2i)*(47*c*d^5 - c^5*d - d^6*12i + 
c^2*d^4*56i - 34*c^3*d^3 + c^4*d^2*4i) - root(a^3*c^7*d^7*e^3*640i + a^3*c 
^9*d^5*e^3*480i + a^3*c^5*d^9*e^3*480i + a^3*c^11*d^3*e^3*192i + a^3*c^3*d 
^11*e^3*192i + 144*a^3*c^10*d^4*e^3 - 144*a^3*c^4*d^10*e^3 + 80*a^3*c^12*d 
^2*e^3 + 80*a^3*c^8*d^6*e^3 - 80*a^3*c^6*d^8*e^3 - 80*a^3*c^2*d^12*e^3 + a 
^3*c^13*d*e^3*32i + a^3*c*d^13*e^3*32i - 16*a^3*d^14*e^3 + 16*a^3*c^14*e^3 
 - a*c^3*d^5*e*744i - 660*a*c^2*d^6*e + 558*a*c^4*d^4*e + a*c^5*d^3*e*24i 
- 4*a*c^6*d^2*e + a*c*d^7*e*264i + a*c^7*d*e*8i + 57*a*d^8*e + a*c^8*e + 3 
8*c^2*d^3 - c^3*d^2*6i - c*d^4*26i - 14*d^5, e, k)*((a*d^4 - a*c^2*d^2 + a 
*c*d^3*2i)*(2*a*c^9 + a*d^9*6i + a*c^2*d^7*28i - 4*a*c^3*d^6 + a*c^4*d^5*4 
8i + a*c^6*d^3*36i + 4*a*c^7*d^2 - 2*a*c*d^8 + a*c^8*d*10i) + root(a^3*c^7 
*d^7*e^3*640i + a^3*c^9*d^5*e^3*480i + a^3*c^5*d^9*e^3*480i + a^3*c^11*d^3 
*e^3*192i + a^3*c^3*d^11*e^3*192i + 144*a^3*c^10*d^4*e^3 - 144*a^3*c^4*d^1 
0*e^3 + 80*a^3*c^12*d^2*e^3 + 80*a^3*c^8*d^6*e^3 - 80*a^3*c^6*d^8*e^3 - 80 
*a^3*c^2*d^12*e^3 + a^3*c^13*d*e^3*32i + a^3*c*d^13*e^3*32i - 16*a^3*d^14* 
e^3 + 16*a^3*c^14*e^3 - a*c^3*d^5*e*744i - 660*a*c^2*d^6*e + 558*a*c^4*d^4 
*e + a*c^5*d^3*e*24i - 4*a*c^6*d^2*e + a*c*d^7*e*264i + a*c^7*d*e*8i + 57* 
a*d^8*e + a*c^8*e + 38*c^2*d^3 - c^3*d^2*6i - c*d^4*26i - 14*d^5, e, k)*(( 
a*d^4 - a*c^2*d^2 + a*c*d^3*2i)*(32*a^2*c^11*d - 32*a^2*c*d^11 + a^2*c^2*d 
^10*64i - 96*a^2*c^3*d^9 + a^2*c^4*d^8*256i - 64*a^2*c^5*d^7 + a^2*c^6*...
 

Reduce [F]

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

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

Output:

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