\(\int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx\) [1099]

Optimal result

Integrand size = 28, antiderivative size = 354 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=\frac {\left (c^5+5 i c^4 d-10 c^3 d^2+30 i c^2 d^3+45 c d^4-15 i d^5\right ) x}{4 a^2 (c-i d)^3 (c+i d)^5}-\frac {2 d^3 \left (5 c^2-5 i c d-2 d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 (i c-d)^5 (i c+d)^3 f}+\frac {d \left (c^2+5 i c d+8 d^2\right )}{4 a^2 (c-i d) (c+i d)^3 f (c+d \tan (e+f x))^2}+\frac {i c-5 d}{4 a^2 (c+i d)^2 f (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}+\frac {(c-3 i d) d \left (c^2+8 i c d+5 d^2\right )}{4 a^2 (c-i d)^2 (c+i d)^4 f (c+d \tan (e+f x))} \] Output:

1/4*(c^5+5*I*c^4*d-10*c^3*d^2+30*I*c^2*d^3+45*c*d^4-15*I*d^5)*x/a^2/(c-I*d 
)^3/(c+I*d)^5-2*d^3*(5*c^2-5*I*c*d-2*d^2)*ln(c*cos(f*x+e)+d*sin(f*x+e))/a^ 
2/(I*c-d)^5/(I*c+d)^3/f+1/4*d*(c^2+5*I*c*d+8*d^2)/a^2/(c-I*d)/(c+I*d)^3/f/ 
(c+d*tan(f*x+e))^2+1/4*(I*c-5*d)/a^2/(c+I*d)^2/f/(1+I*tan(f*x+e))/(c+d*tan 
(f*x+e))^2-1/4/(I*c-d)/f/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^2+1/4*(c-3* 
I*d)*d*(c^2+8*I*c*d+5*d^2)/a^2/(c-I*d)^2/(c+I*d)^4/f/(c+d*tan(f*x+e))
 

Mathematica [A] (verified)

Time = 4.62 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=-\frac {\frac {2 i (c+i d)}{(-i+\tan (e+f x))^2 (c+d \tan (e+f x))^2}-\frac {2 (c+5 i d)}{(-i+\tan (e+f x)) (c+d \tan (e+f x))^2}-6 (c+5 i d) \left (-\frac {i \log (i-\tan (e+f x))}{2 (c+i d)^2}+\frac {i \log (i+\tan (e+f x))}{2 (c-i d)^2}+\frac {d \left (2 c \log (c+d \tan (e+f x))-\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}\right )+2 \left (c^2+5 i c d+8 d^2\right ) \left (\frac {\log (i-\tan (e+f x))}{(-i c+d)^3}+\frac {\log (i+\tan (e+f x))}{(i c+d)^3}+\frac {d \left (\left (6 c^2-2 d^2\right ) \log (c+d \tan (e+f x))-\frac {\left (c^2+d^2\right ) \left (5 c^2+d^2+4 c d \tan (e+f x)\right )}{(c+d \tan (e+f x))^2}\right )}{\left (c^2+d^2\right )^3}\right )}{8 a^2 (c+i d)^2 f} \] Input:

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

Output:

-1/8*(((2*I)*(c + I*d))/((-I + Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2) - ( 
2*(c + (5*I)*d))/((-I + Tan[e + f*x])*(c + d*Tan[e + f*x])^2) - 6*(c + (5* 
I)*d)*(((-1/2*I)*Log[I - Tan[e + f*x]])/(c + I*d)^2 + ((I/2)*Log[I + Tan[e 
 + f*x]])/(c - I*d)^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) + 2*(c^2 + (5*I)*c*d + 8*d^2)*(Log[I - T 
an[e + f*x]]/((-I)*c + 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*Ta 
n[e + f*x]))/(c + d*Tan[e + f*x])^2))/(c^2 + d^2)^3))/(a^2*(c + I*d)^2*f)
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {3042, 4042, 27, 3042, 4079, 3042, 4012, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4042

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4079

\(\displaystyle \frac {-\frac {\int \frac {\left (c^2+5 i d c-16 d^2\right ) a^2+3 (c+5 i d) d \tan (e+f x) a^2}{(c+d \tan (e+f x))^3}dx}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {\left (c^2+5 i d c-16 d^2\right ) a^2+3 (c+5 i d) d \tan (e+f x) a^2}{(c+d \tan (e+f x))^3}dx}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (c^3+5 i d c^2-13 d^2 c+15 i d^3\right ) a^2+2 d \left (c^2+5 i d c+8 d^2\right ) \tan (e+f x) a^2}{(c+d \tan (e+f x))^2}dx}{c^2+d^2}+\frac {a^2 d \left (c^2+5 i c d+8 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\int \frac {\left (c^3+5 i d c^2-13 d^2 c+15 i d^3\right ) a^2+2 d \left (c^2+5 i d c+8 d^2\right ) \tan (e+f x) a^2}{(c+d \tan (e+f x))^2}dx}{c^2+d^2}+\frac {a^2 d \left (c^2+5 i c d+8 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (c^4+5 i d c^3-11 d^2 c^2+25 i d^3 c+16 d^4\right ) a^2+d \left (c^3+5 i d c^2+29 d^2 c-15 i d^3\right ) \tan (e+f x) a^2}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^2 d \left (c^3+5 i c^2 d+29 c d^2-15 i d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{c^2+d^2}+\frac {a^2 d \left (c^2+5 i c d+8 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {\left (c^4+5 i d c^3-11 d^2 c^2+25 i d^3 c+16 d^4\right ) a^2+d \left (c^3+5 i d c^2+29 d^2 c-15 i d^3\right ) \tan (e+f x) a^2}{c+d \tan (e+f x)}dx}{c^2+d^2}+\frac {a^2 d \left (c^3+5 i c^2 d+29 c d^2-15 i d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{c^2+d^2}+\frac {a^2 d \left (c^2+5 i c d+8 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 4014

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {a^2 x \left (c^5+5 i c^4 d-10 c^3 d^2+30 i c^2 d^3+45 c d^4-15 i d^5\right )}{c^2+d^2}-\frac {8 a^2 d^3 \left (5 c^2-5 i c d-2 d^2\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}}{c^2+d^2}+\frac {a^2 d \left (c^3+5 i c^2 d+29 c d^2-15 i d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{c^2+d^2}+\frac {a^2 d \left (c^2+5 i c d+8 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {a^2 x \left (c^5+5 i c^4 d-10 c^3 d^2+30 i c^2 d^3+45 c d^4-15 i d^5\right )}{c^2+d^2}-\frac {8 a^2 d^3 \left (5 c^2-5 i c d-2 d^2\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)}dx}{c^2+d^2}}{c^2+d^2}+\frac {a^2 d \left (c^3+5 i c^2 d+29 c d^2-15 i d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{c^2+d^2}+\frac {a^2 d \left (c^2+5 i c d+8 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

\(\Big \downarrow \) 4013

\(\displaystyle \frac {-\frac {\frac {a^2 d \left (c^2+5 i c d+8 d^2\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}+\frac {\frac {a^2 d \left (c^3+5 i c^2 d+29 c d^2-15 i d^3\right )}{f \left (c^2+d^2\right ) (c+d \tan (e+f x))}+\frac {\frac {a^2 x \left (c^5+5 i c^4 d-10 c^3 d^2+30 i c^2 d^3+45 c d^4-15 i d^5\right )}{c^2+d^2}-\frac {8 a^2 d^3 \left (5 c^2-5 i c d-2 d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )}}{c^2+d^2}}{c^2+d^2}}{2 a^2 (-d+i c)}-\frac {c+5 i d}{2 f (c+i d) (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}}{2 a^2 (-d+i c)}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}\)

Input:

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

Output:

-1/4*1/((I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2) + (-1 
/2*(c + (5*I)*d)/((c + I*d)*f*(1 + I*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) 
 - ((a^2*d*(c^2 + (5*I)*c*d + 8*d^2))/((c^2 + d^2)*f*(c + d*Tan[e + f*x])^ 
2) + (((a^2*(c^5 + (5*I)*c^4*d - 10*c^3*d^2 + (30*I)*c^2*d^3 + 45*c*d^4 - 
(15*I)*d^5)*x)/(c^2 + d^2) - (8*a^2*d^3*(5*c^2 - (5*I)*c*d - 2*d^2)*Log[c* 
Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)*f))/(c^2 + d^2) + (a^2*d*(c^3 
 + (5*I)*c^2*d + 29*c*d^2 - (15*I)*d^3))/((c^2 + d^2)*f*(c + d*Tan[e + f*x 
])))/(c^2 + d^2))/(2*a^2*(I*c - d)))/(2*a^2*(I*c - d))
 

Defintions of rubi rules used

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. 833 vs. \(2 (324 ) = 648\).

Time = 0.86 (sec) , antiderivative size = 834, normalized size of antiderivative = 2.36

method	result	size
derivativedivides	\(\frac {2 i d^{4} c^{2}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}+\frac {10 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5}}-\frac {4 d^{5} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5}}+\frac {31 i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d^{2}}{16 f \,a^{2} \left (i d +c \right )^{5}}+\frac {2 i d^{6}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {4 d^{3} c^{3}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {4 d^{5} c}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {d^{3} c^{4}}{2 f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {d^{5} c^{2}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {d^{7}}{2 f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{2}}{16 f \,a^{2} \left (i d +c \right )^{5}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{2} \left (i d -c \right )^{3}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{16 f \,a^{2} \left (i d -c \right )^{3}}-\frac {i c^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {c d}{2 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {i \arctan \left (\tan \left (f x +e \right )\right ) c d}{f \,a^{2} \left (i d +c \right )^{5}}+\frac {c^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {7 d^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {2 i c d}{f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {i d^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right ) c d}{2 f \,a^{2} \left (i d +c \right )^{5}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{2}}{8 f \,a^{2} \left (i d +c \right )^{5}}-\frac {31 \arctan \left (\tan \left (f x +e \right )\right ) d^{2}}{8 f \,a^{2} \left (i d +c \right )^{5}}-\frac {10 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5}}\)	\(834\)
default	\(\frac {2 i d^{4} c^{2}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}+\frac {10 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5}}-\frac {4 d^{5} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5}}+\frac {31 i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) d^{2}}{16 f \,a^{2} \left (i d +c \right )^{5}}+\frac {2 i d^{6}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {4 d^{3} c^{3}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {4 d^{5} c}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )}-\frac {d^{3} c^{4}}{2 f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {d^{5} c^{2}}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {d^{7}}{2 f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5} \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right ) c^{2}}{16 f \,a^{2} \left (i d +c \right )^{5}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{2} \left (i d -c \right )^{3}}-\frac {i \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{16 f \,a^{2} \left (i d -c \right )^{3}}-\frac {i c^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {c d}{2 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {i \arctan \left (\tan \left (f x +e \right )\right ) c d}{f \,a^{2} \left (i d +c \right )^{5}}+\frac {c^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {7 d^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {2 i c d}{f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {i d^{2}}{4 f \,a^{2} \left (i d +c \right )^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right ) c d}{2 f \,a^{2} \left (i d +c \right )^{5}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{2}}{8 f \,a^{2} \left (i d +c \right )^{5}}-\frac {31 \arctan \left (\tan \left (f x +e \right )\right ) d^{2}}{8 f \,a^{2} \left (i d +c \right )^{5}}-\frac {10 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f \,a^{2} \left (i d -c \right )^{3} \left (i d +c \right )^{5}}\)	\(834\)
risch	\(-\frac {x}{4 a^{2} \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 )} c}{4 a^{2} \left (-i c +d \right )^{2} \left (-2 i c d -c^{2}+d^{2}\right ) f}-\frac {{\mathrm e}^{-2 i \left (f x +e \right )} d}{a^{2} \left (-i c +d \right )^{2} \left (-2 i c d -c^{2}+d^{2}\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 ) \left (i d +c \right ) f}+\frac {20 d^{4} c x}{a^{2} \left (2 i c^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}+\frac {20 d^{4} c e}{a^{2} f \left (2 i c^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}+\frac {20 i d^{3} c^{2} x}{a^{2} \left (2 i c^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}-\frac {8 i d^{5} x}{a^{2} \left (2 i c^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}+\frac {20 i d^{3} c^{2} e}{a^{2} f \left (2 i c^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}-\frac {8 i d^{5} e}{a^{2} f \left (2 i c^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}-\frac {2 d^{4} \left (6 i c d \,{\mathrm e}^{2 i \left (f x +e \right )}-5 c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-3 i c d -5 c^{2}-2 d^{2}\right )}{\left (-i c +d \right )^{4} \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^{2} \left (i c +d \right )^{3}}+\frac {10 i d^{4} \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^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}-\frac {10 d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right ) c^{2}}{a^{2} f \left (2 i c^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}+\frac {4 d^{5} \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^{7} d +6 i c^{5} d^{3}+6 i c^{3} d^{5}+2 i c \,d^{7}+c^{8}+2 c^{6} d^{2}-2 c^{2} d^{6}-d^{8}\right )}\)	\(1011\)
norman	\(\text {Expression too large to display}\)	\(1498\)

Input:

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

Output:

2*I/f/a^2*d^4/(I*d-c)^3/(c+I*d)^5/(c+d*tan(f*x+e))*c^2+10/f/a^2*d^3/(I*d-c 
)^3/(c+I*d)^5*ln(c+d*tan(f*x+e))*c^2-4/f/a^2*d^5/(I*d-c)^3/(c+I*d)^5*ln(c+ 
d*tan(f*x+e))+31/16*I/f/a^2/(c+I*d)^5*ln(1+tan(f*x+e)^2)*d^2+2*I/f/a^2*d^6 
/(I*d-c)^3/(c+I*d)^5/(c+d*tan(f*x+e))-4/f/a^2*d^3/(I*d-c)^3/(c+I*d)^5/(c+d 
*tan(f*x+e))*c^3-4/f/a^2*d^5/(I*d-c)^3/(c+I*d)^5/(c+d*tan(f*x+e))*c-1/2/f/ 
a^2*d^3/(I*d-c)^3/(c+I*d)^5/(c+d*tan(f*x+e))^2*c^4-1/f/a^2*d^5/(I*d-c)^3/( 
c+I*d)^5/(c+d*tan(f*x+e))^2*c^2-1/2/f/a^2*d^7/(I*d-c)^3/(c+I*d)^5/(c+d*tan 
(f*x+e))^2-1/16*I/f/a^2/(c+I*d)^5*ln(1+tan(f*x+e)^2)*c^2-1/8/f/a^2/(I*d-c) 
^3*arctan(tan(f*x+e))-1/16*I/f/a^2/(I*d-c)^3*ln(1+tan(f*x+e)^2)-1/4*I/f/a^ 
2/(c+I*d)^5/(-I+tan(f*x+e))^2*c^2+1/2/f/a^2/(c+I*d)^5/(-I+tan(f*x+e))^2*c* 
d+I/f/a^2/(c+I*d)^5*arctan(tan(f*x+e))*c*d+1/4/f/a^2/(c+I*d)^5/(-I+tan(f*x 
+e))*c^2-7/4/f/a^2/(c+I*d)^5/(-I+tan(f*x+e))*d^2+2*I/f/a^2/(c+I*d)^5/(-I+t 
an(f*x+e))*c*d+1/4*I/f/a^2/(c+I*d)^5/(-I+tan(f*x+e))^2*d^2+1/2/f/a^2/(c+I* 
d)^5*ln(1+tan(f*x+e)^2)*c*d+1/8/f/a^2/(c+I*d)^5*arctan(tan(f*x+e))*c^2-31/ 
8/f/a^2/(c+I*d)^5*arctan(tan(f*x+e))*d^2-10*I/f/a^2*d^4/(I*d-c)^3/(c+I*d)^ 
5*ln(c+d*tan(f*x+e))*c
 

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. 907 vs. \(2 (304) = 608\).

Time = 0.11 (sec) , antiderivative size = 907, normalized size of antiderivative = 2.56 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/16*(I*c^7 - c^6*d + 3*I*c^5*d^2 - 3*c^4*d^3 + 3*I*c^3*d^4 - 3*c^2*d^5 + 
I*c*d^6 - d^7 + 4*(c^7 + 3*I*c^6*d - c^5*d^2 + 85*I*c^4*d^3 + 235*c^3*d^4 
- 271*I*c^2*d^5 - 147*c*d^6 + 31*I*d^7)*f*x*e^(8*I*f*x + 8*I*e) - 4*(-I*c^ 
7 - 11*I*c^5*d^2 - 20*c^4*d^3 + 45*I*c^3*d^4 - 8*c^2*d^5 + 55*I*c*d^6 + 12 
*d^7 - 2*(c^7 + 5*I*c^6*d - 9*c^5*d^2 + 75*I*c^4*d^3 + 75*c^3*d^4 + 39*I*c 
^2*d^5 + 85*c*d^6 - 31*I*d^7)*f*x)*e^(6*I*f*x + 6*I*e) + (9*I*c^7 - 13*c^6 
*d + 71*I*c^5*d^2 + 5*c^4*d^3 - 45*I*c^3*d^4 + 305*c^2*d^5 + 85*I*c*d^6 + 
95*d^7 + 4*(c^7 + 7*I*c^6*d - 21*c^5*d^2 + 45*I*c^4*d^3 - 45*c^3*d^4 + 69* 
I*c^2*d^5 - 23*c*d^6 + 31*I*d^7)*f*x)*e^(4*I*f*x + 4*I*e) - 2*(-3*I*c^7 + 
7*c^6*d - 9*I*c^5*d^2 + 21*c^4*d^3 - 9*I*c^3*d^4 + 21*c^2*d^5 - 3*I*c*d^6 
+ 7*d^7)*e^(2*I*f*x + 2*I*e) - 32*((5*c^4*d^3 - 15*I*c^3*d^4 - 17*c^2*d^5 
+ 9*I*c*d^6 + 2*d^7)*e^(8*I*f*x + 8*I*e) + 2*(5*c^4*d^3 - 5*I*c^3*d^4 + 3* 
c^2*d^5 - 5*I*c*d^6 - 2*d^7)*e^(6*I*f*x + 6*I*e) + (5*c^4*d^3 + 5*I*c^3*d^ 
4 + 3*c^2*d^5 + I*c*d^6 + 2*d^7)*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)))/((a^2*c^10 + 5*a^2*c^8*d^2 + 10*a^2* 
c^6*d^4 + 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 + a^2*d^10)*f*e^(8*I*f*x + 8*I*e) 
 + 2*(a^2*c^10 + 2*I*a^2*c^9*d + 3*a^2*c^8*d^2 + 8*I*a^2*c^7*d^3 + 2*a^2*c 
^6*d^4 + 12*I*a^2*c^5*d^5 - 2*a^2*c^4*d^6 + 8*I*a^2*c^3*d^7 - 3*a^2*c^2*d^ 
8 + 2*I*a^2*c*d^9 - a^2*d^10)*f*e^(6*I*f*x + 6*I*e) + (a^2*c^10 + 4*I*a^2* 
c^9*d - 3*a^2*c^8*d^2 + 8*I*a^2*c^7*d^3 - 14*a^2*c^6*d^4 - 14*a^2*c^4*d...
 

Sympy [A] (verification not implemented)

Time = 91.62 (sec) , antiderivative size = 1583, normalized size of antiderivative = 4.47 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

x*(c**2 + 8*I*c*d - 31*d**2)/(4*a**2*c**5 + 20*I*a**2*c**4*d - 40*a**2*c** 
3*d**2 - 40*I*a**2*c**2*d**3 + 20*a**2*c*d**4 + 4*I*a**2*d**5) + (-10*I*c* 
*2*d**4 + 6*c*d**5 - 4*I*d**6 + (-10*I*c**2*d**4*exp(2*I*e) - 12*c*d**5*ex 
p(2*I*e) + 2*I*d**6*exp(2*I*e))*exp(2*I*f*x))/(a**2*c**9*f + 3*I*a**2*c**8 
*d*f + 8*I*a**2*c**6*d**3*f - 6*a**2*c**5*d**4*f + 6*I*a**2*c**4*d**5*f - 
8*a**2*c**3*d**6*f - 3*a**2*c*d**8*f - I*a**2*d**9*f + (2*a**2*c**9*f*exp( 
2*I*e) + 2*I*a**2*c**8*d*f*exp(2*I*e) + 8*a**2*c**7*d**2*f*exp(2*I*e) + 8* 
I*a**2*c**6*d**3*f*exp(2*I*e) + 12*a**2*c**5*d**4*f*exp(2*I*e) + 12*I*a**2 
*c**4*d**5*f*exp(2*I*e) + 8*a**2*c**3*d**6*f*exp(2*I*e) + 8*I*a**2*c**2*d* 
*7*f*exp(2*I*e) + 2*a**2*c*d**8*f*exp(2*I*e) + 2*I*a**2*d**9*f*exp(2*I*e)) 
*exp(2*I*f*x) + (a**2*c**9*f*exp(4*I*e) - I*a**2*c**8*d*f*exp(4*I*e) + 4*a 
**2*c**7*d**2*f*exp(4*I*e) - 4*I*a**2*c**6*d**3*f*exp(4*I*e) + 6*a**2*c**5 
*d**4*f*exp(4*I*e) - 6*I*a**2*c**4*d**5*f*exp(4*I*e) + 4*a**2*c**3*d**6*f* 
exp(4*I*e) - 4*I*a**2*c**2*d**7*f*exp(4*I*e) + a**2*c*d**8*f*exp(4*I*e) - 
I*a**2*d**9*f*exp(4*I*e))*exp(4*I*f*x)) + Piecewise((((4*I*a**2*c**4*f*exp 
(2*I*e) - 16*a**2*c**3*d*f*exp(2*I*e) - 24*I*a**2*c**2*d**2*f*exp(2*I*e) + 
 16*a**2*c*d**3*f*exp(2*I*e) + 4*I*a**2*d**4*f*exp(2*I*e))*exp(-4*I*f*x) + 
 (16*I*a**2*c**4*f*exp(4*I*e) - 112*a**2*c**3*d*f*exp(4*I*e) - 240*I*a**2* 
c**2*d**2*f*exp(4*I*e) + 208*a**2*c*d**3*f*exp(4*I*e) + 64*I*a**2*d**4*f*e 
xp(4*I*e))*exp(-2*I*f*x))/(64*a**4*c**7*f**2*exp(6*I*e) + 448*I*a**4*c*...
 

Maxima [F(-2)]

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

integrate(1/(a+I*a*tan(f*x+e))^2/(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 = 505, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=\frac {{\left (c^{2} + 8 i \, c d - 31 \, d^{2}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{8 i \, a^{2} c^{5} f - 40 \, a^{2} c^{4} d f - 80 i \, a^{2} c^{3} d^{2} f + 80 \, a^{2} c^{2} d^{3} f + 40 i \, a^{2} c d^{4} f - 8 \, a^{2} d^{5} f} + \frac {2 \, {\left (5 i \, c^{2} d^{4} + 5 \, c d^{5} - 2 i \, d^{6}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{-i \, a^{2} c^{8} d f + 2 \, a^{2} c^{7} d^{2} f - 2 i \, a^{2} c^{6} d^{3} f + 6 \, a^{2} c^{5} d^{4} f + 6 \, a^{2} c^{3} d^{6} f + 2 i \, a^{2} c^{2} d^{7} f + 2 \, a^{2} c d^{8} f + i \, a^{2} d^{9} f} - \frac {\log \left (\tan \left (f x + e\right ) + i\right )}{8 i \, a^{2} c^{3} f + 24 \, a^{2} c^{2} d f - 24 i \, a^{2} c d^{2} f - 8 \, a^{2} d^{3} f} + \frac {-2 i \, c^{6} + 2 \, c^{5} d - 18 i \, c^{4} d^{2} - 40 \, c^{3} d^{3} + 34 i \, c^{2} d^{4} + 6 \, c d^{5} + 2 i \, d^{6} - i \, {\left (i \, c^{4} d^{2} - 4 \, c^{3} d^{3} + 34 i \, c^{2} d^{4} + 44 \, c d^{5} - 15 i \, d^{6}\right )} \tan \left (f x + e\right )^{3} - 2 i \, {\left (i \, c^{5} d - 3 \, c^{4} d^{2} + 28 i \, c^{3} d^{3} + 58 \, c^{2} d^{4} - 45 i \, c d^{5} - 11 \, d^{6}\right )} \tan \left (f x + e\right )^{2} - i \, {\left (i \, c^{6} + 22 i \, c^{4} d^{2} + 92 \, c^{3} d^{3} - 119 i \, c^{2} d^{4} - 52 \, c d^{5} + 4 i \, d^{6}\right )} \tan \left (f x + e\right )}{4 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} a^{2} {\left (c + i \, d\right )}^{4} {\left (c - i \, d\right )}^{3} f {\left (\tan \left (f x + e\right ) - i\right )}^{2}} \] Input:

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

Output:

(c^2 + 8*I*c*d - 31*d^2)*log(tan(f*x + e) - I)/(8*I*a^2*c^5*f - 40*a^2*c^4 
*d*f - 80*I*a^2*c^3*d^2*f + 80*a^2*c^2*d^3*f + 40*I*a^2*c*d^4*f - 8*a^2*d^ 
5*f) + 2*(5*I*c^2*d^4 + 5*c*d^5 - 2*I*d^6)*log(abs(d*tan(f*x + e) + c))/(- 
I*a^2*c^8*d*f + 2*a^2*c^7*d^2*f - 2*I*a^2*c^6*d^3*f + 6*a^2*c^5*d^4*f + 6* 
a^2*c^3*d^6*f + 2*I*a^2*c^2*d^7*f + 2*a^2*c*d^8*f + I*a^2*d^9*f) - log(tan 
(f*x + e) + I)/(8*I*a^2*c^3*f + 24*a^2*c^2*d*f - 24*I*a^2*c*d^2*f - 8*a^2* 
d^3*f) + 1/4*(-2*I*c^6 + 2*c^5*d - 18*I*c^4*d^2 - 40*c^3*d^3 + 34*I*c^2*d^ 
4 + 6*c*d^5 + 2*I*d^6 - I*(I*c^4*d^2 - 4*c^3*d^3 + 34*I*c^2*d^4 + 44*c*d^5 
 - 15*I*d^6)*tan(f*x + e)^3 - 2*I*(I*c^5*d - 3*c^4*d^2 + 28*I*c^3*d^3 + 58 
*c^2*d^4 - 45*I*c*d^5 - 11*d^6)*tan(f*x + e)^2 - I*(I*c^6 + 22*I*c^4*d^2 + 
 92*c^3*d^3 - 119*I*c^2*d^4 - 52*c*d^5 + 4*I*d^6)*tan(f*x + e))/((d*tan(f* 
x + e) + c)^2*a^2*(c + I*d)^4*(c - I*d)^3*f*(tan(f*x + e) - I)^2)
 

Mupad [B] (verification not implemented)

Time = 8.84 (sec) , antiderivative size = 2640, normalized size of antiderivative = 7.46 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx=\text {Too large to display} \] Input:

int(1/((a + a*tan(e + f*x)*1i)^2*(c + d*tan(e + f*x))^3),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)*(839*c*d^7 + c^7*d - d^8*240i + c^2*d^6*970i - 353*c^3*d^5 + c^4* 
d^4*100i - 7*c^5*d^3 + c^6*d^2*10i) - root(5760*a^6*c^8*d^8*e^3 + 4096*a^6 
*c^10*d^6*e^3 + 4096*a^6*c^6*d^10*e^3 - a^6*c^11*d^5*e^3*2304i + a^6*c^5*d 
^11*e^3*2304i - a^6*c^13*d^3*e^3*1280i - a^6*c^9*d^7*e^3*1280i + a^6*c^7*d 
^9*e^3*1280i + a^6*c^3*d^13*e^3*1280i + 1280*a^6*c^12*d^4*e^3 + 1280*a^6*c 
^4*d^12*e^3 - a^6*c^15*d*e^3*256i + a^6*c*d^15*e^3*256i - 64*a^6*d^16*e^3 
- 64*a^6*c^16*e^3 + a^2*c*d^9*e*5190i - a^2*c^9*d*e*10i - a^2*c^3*d^7*e*12 
600i - 11565*a^2*c^2*d^8*e + 6450*a^2*c^4*d^6*e + a^2*c^5*d^5*e*180i + 110 
*a^2*c^6*d^4*e + a^2*c^7*d^3*e*40i + 45*a^2*c^8*d^2*e + 993*a^2*d^10*e - a 
^2*c^10*e + 234*c^2*d^5 - c^3*d^4*70i - 10*c^4*d^3 - c*d^6*278i - 124*d^7, 
 e, k)*(root(5760*a^6*c^8*d^8*e^3 + 4096*a^6*c^10*d^6*e^3 + 4096*a^6*c^6*d 
^10*e^3 - a^6*c^11*d^5*e^3*2304i + a^6*c^5*d^11*e^3*2304i - a^6*c^13*d^3*e 
^3*1280i - a^6*c^9*d^7*e^3*1280i + a^6*c^7*d^9*e^3*1280i + a^6*c^3*d^13*e^ 
3*1280i + 1280*a^6*c^12*d^4*e^3 + 1280*a^6*c^4*d^12*e^3 - a^6*c^15*d*e^3*2 
56i + a^6*c*d^15*e^3*256i - 64*a^6*d^16*e^3 - 64*a^6*c^16*e^3 + a^2*c*d^9* 
e*5190i - a^2*c^9*d*e*10i - a^2*c^3*d^7*e*12600i - 11565*a^2*c^2*d^8*e + 6 
450*a^2*c^4*d^6*e + a^2*c^5*d^5*e*180i + 110*a^2*c^6*d^4*e + a^2*c^7*d^3*e 
*40i + 45*a^2*c^8*d^2*e + 993*a^2*d^10*e - a^2*c^10*e + 234*c^2*d^5 - c^3* 
d^4*70i - 10*c^4*d^3 - c*d^6*278i - 124*d^7, e, k)*((a^2*d^6 + a^2*c*d^...
 

Reduce [F]

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

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

Output:

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