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

Optimal result

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

[Out]

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))

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3640, 3677, 3610, 3612, 3611} \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\frac {d \left (c^2+4 i c d+9 d^2\right )}{4 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))}+\frac {x \left (c^4+4 i c^3 d-6 c^2 d^2+12 i c d^3+9 d^4\right )}{4 a^2 (c-i d)^2 (c+i d)^4}-\frac {2 d^3 (2 c-i d) \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 f (c-i d)^2 (c+i d)^4}+\frac {-4 d+i c}{4 a^2 f (c+i d)^2 (1+i \tan (e+f x)) (c+d \tan (e+f x))}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \]

[In]

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

[Out]

((c^4 + (4*I)*c^3*d - 6*c^2*d^2 + (12*I)*c*d^3 + 9*d^4)*x)/(4*a^2*(c - I*d)^2*(c + I*d)^4) - (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) + (d*(c^2 + (4*I)*c*d + 9*d^2))/(4*a^2
*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])) + (I*c - 4*d)/(4*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])*(c + d*
Tan[e + f*x])) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x]))

Rule 3610

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] + Dist[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 3611

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*Sin[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 3612

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] + Dist[(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] && NeQ[a*c + b*d, 0]

Rule 3640

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Dist[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[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 3677

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[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
] /; FreeQ[{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]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {\int \frac {-a (2 i c-5 d)-3 i a d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx}{4 a^2 (i c-d)} \\ & = \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))}-\frac {\int \frac {-2 a^2 \left (c^2+4 i c d-9 d^2\right )-4 a^2 (c+4 i d) d \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx}{8 a^4 (c+i d)^2} \\ & = \frac {d \left (c^2+4 i c d+9 d^2\right )}{4 a^2 (c+i d)^2 \left (c^2+d^2\right ) 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))}-\frac {\int \frac {-2 a^2 \left (c^3+4 i c^2 d-7 c d^2+8 i d^3\right )-2 a^2 d \left (c^2+4 i c d+9 d^2\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{8 a^4 (c+i d)^2 \left (c^2+d^2\right )} \\ & = \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 \left (c^2+d^2\right )^2}+\frac {d \left (c^2+4 i c d+9 d^2\right )}{4 a^2 (c+i d)^2 \left (c^2+d^2\right ) 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))}-\frac {\left (2 (2 c-i d) d^3\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^2 (c+i d)^2 \left (c^2+d^2\right )^2} \\ & = \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 \left (c^2+d^2\right )^2}-\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 \left (c^2+d^2\right )^2 f}+\frac {d \left (c^2+4 i c d+9 d^2\right )}{4 a^2 (c+i d)^2 \left (c^2+d^2\right ) 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))} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.23 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=-\frac {\frac {2 (c+4 i d) ((i c+d) \log (i-\tan (e+f x))+(-i c+d) \log (i+\tan (e+f x))-2 d \log (c+d \tan (e+f x)))}{c^2+d^2}+\frac {2 i (c+i d)}{(-i+\tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {2 (c+4 i d)}{(-i+\tan (e+f x)) (c+d \tan (e+f x))}-\left (c^2+4 i c d+9 d^2\right ) \left (\frac {i \log (i-\tan (e+f x))}{(c+i d)^2}-\frac {i \log (i+\tan (e+f x))}{(c-i d)^2}+\frac {2 d \left (-2 c \log (c+d \tan (e+f x))+\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}\right )}{8 a^2 (c+i d)^2 f} \]

[In]

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

[Out]

-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*T
an[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)

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 = 0.77 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.11

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

[In]

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

[Out]

3/2*I/f/a^2/(c+I*d)^4/(tan(f*x+e)-I)*c*d+1/8/f/a^2/(I*d-c)^2*arctan(tan(f*x+e))+2*I/f/a^2*d^4/(I*d-c)^2/(c+I*d
)^4*ln(c+d*tan(f*x+e))+3/4*I/f/a^2/(c+I*d)^4*arctan(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/(I*d-c)^2
*ln(1+tan(f*x+e)^2)+17/16*I/f/a^2/(c+I*d)^4*ln(1+tan(f*x+e)^2)*d^2+1/4/f/a^2/(c+I*d)^4/(tan(f*x+e)-I)*c^2-5/4/
f/a^2/(c+I*d)^4/(tan(f*x+e)-I)*d^2+1/4*I/f/a^2/(c+I*d)^4/(tan(f*x+e)-I)^2*d^2-1/16*I/f/a^2/(c+I*d)^4*ln(1+tan(
f*x+e)^2)*c^2+1/2/f/a^2/(c+I*d)^4/(tan(f*x+e)-I)^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))-1/4*I/f/a^2/(c+I*d)^4/(tan(f*x+e)-I)^2*c^2-4/f/a^2*d^3/(I*d-c)^
2/(c+I*d)^4*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. 505 vs. \(2 (233) = 466\).

Time = 0.25 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=-\frac {c^{5} + i \, c^{4} d + 2 \, c^{3} d^{2} + 2 i \, c^{2} d^{3} + c d^{4} + i \, d^{5} - 4 \, {\left (i \, c^{5} - 3 \, c^{4} d - 2 i \, c^{3} d^{2} - 34 \, c^{2} d^{3} + 45 i \, c d^{4} + 17 \, d^{5}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 4 \, {\left (c^{5} + i \, c^{4} d + 6 \, c^{3} d^{2} - 2 i \, c^{2} d^{3} - 3 \, c d^{4} - 11 i \, d^{5} - {\left (i \, c^{5} - 5 \, c^{4} d - 10 i \, c^{3} d^{2} - 22 \, c^{2} d^{3} - 11 i \, c d^{4} - 17 \, d^{5}\right )} f x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (5 \, c^{5} + 11 i \, c^{4} d + 10 \, c^{3} d^{2} + 22 i \, c^{2} d^{3} + 5 \, c d^{4} + 11 i \, d^{5}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 32 \, {\left ({\left (-2 i \, c^{2} d^{3} - 3 \, c d^{4} + i \, d^{5}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-2 i \, c^{2} d^{3} + c d^{4} - i \, d^{5}\right )} e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right )}{16 \, {\left ({\left (i \, a^{2} c^{7} - a^{2} c^{6} d + 3 i \, a^{2} c^{5} d^{2} - 3 \, a^{2} c^{4} d^{3} + 3 i \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} + i \, a^{2} c d^{6} - a^{2} d^{7}\right )} f e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (i \, a^{2} c^{7} - 3 \, a^{2} c^{6} d - i \, a^{2} c^{5} d^{2} - 5 \, a^{2} c^{4} d^{3} - 5 i \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 3 i \, a^{2} c d^{6} + a^{2} d^{7}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )}} \]

[In]

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

[Out]

-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 = 26.28 (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}} \]

[In]

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

[Out]

-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))*exp(-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) + 320*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 - 16*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*exp(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) + 4*a**2*d**4*exp(4*I*e))), True)) - 2*d**3*(2*c - I*d)*log((c
+ I*d)/(c*exp(2*I*e) - I*d*exp(2*I*e)) + exp(2*I*f*x))/(a**2*f*(c - I*d)**2*(c + I*d)**4)

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (233) = 466\).

Time = 0.57 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=-\frac {\frac {2 \, {\left (2 \, c d^{4} - i \, d^{5}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{a^{2} c^{6} d + 2 i \, a^{2} c^{5} d^{2} + a^{2} c^{4} d^{3} + 4 i \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} + 2 i \, a^{2} c d^{6} - a^{2} d^{7}} - \frac {2 \, {\left (c^{2} + 6 i \, c d - 17 \, d^{2}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{16 i \, a^{2} c^{4} - 64 \, a^{2} c^{3} d - 96 i \, a^{2} c^{2} d^{2} + 64 \, a^{2} c d^{3} + 16 i \, a^{2} d^{4}} + \frac {2 \, \log \left (\tan \left (f x + e\right ) + i\right )}{16 i \, a^{2} c^{2} + 32 \, a^{2} c d - 16 i \, a^{2} d^{2}} - \frac {4 \, c d^{4} \tan \left (f x + e\right ) - 2 i \, d^{5} \tan \left (f x + e\right ) + 5 \, c^{2} d^{3} - 2 i \, c d^{4} + d^{5}}{{\left (a^{2} c^{6} + 2 i \, a^{2} c^{5} d + a^{2} c^{4} d^{2} + 4 i \, a^{2} c^{3} d^{3} - a^{2} c^{2} d^{4} + 2 i \, a^{2} c d^{5} - a^{2} d^{6}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}} + \frac {2 \, {\left (3 \, c^{2} \tan \left (f x + e\right )^{2} + 18 i \, c d \tan \left (f x + e\right )^{2} - 51 \, d^{2} \tan \left (f x + e\right )^{2} - 10 i \, c^{2} \tan \left (f x + e\right ) + 60 \, c d \tan \left (f x + e\right ) + 122 i \, d^{2} \tan \left (f x + e\right ) - 11 \, c^{2} - 50 i \, c d + 75 \, d^{2}\right )}}{-32 \, {\left (-i \, a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 i \, a^{2} c^{2} d^{2} - 4 \, a^{2} c d^{3} - i \, a^{2} d^{4}\right )} {\left (\tan \left (f x + e\right ) - i\right )}^{2}}}{f} \]

[In]

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

[Out]

-(2*(2*c*d^4 - I*d^5)*log(d*tan(f*x + e) + c)/(a^2*c^6*d + 2*I*a^2*c^5*d^2 + a^2*c^4*d^3 + 4*I*a^2*c^3*d^4 - a
^2*c^2*d^5 + 2*I*a^2*c*d^6 - a^2*d^7) - 2*(c^2 + 6*I*c*d - 17*d^2)*log(tan(f*x + e) - I)/(16*I*a^2*c^4 - 64*a^
2*c^3*d - 96*I*a^2*c^2*d^2 + 64*a^2*c*d^3 + 16*I*a^2*d^4) + 2*log(tan(f*x + e) + I)/(16*I*a^2*c^2 + 32*a^2*c*d
- 16*I*a^2*d^2) - (4*c*d^4*tan(f*x + e) - 2*I*d^5*tan(f*x + e) + 5*c^2*d^3 - 2*I*c*d^4 + d^5)/((a^2*c^6 + 2*I
*a^2*c^5*d + a^2*c^4*d^2 + 4*I*a^2*c^3*d^3 - a^2*c^2*d^4 + 2*I*a^2*c*d^5 - a^2*d^6)*(d*tan(f*x + e) + c)) + 2*
(3*c^2*tan(f*x + e)^2 + 18*I*c*d*tan(f*x + e)^2 - 51*d^2*tan(f*x + e)^2 - 10*I*c^2*tan(f*x + e) + 60*c*d*tan(f
*x + e) + 122*I*d^2*tan(f*x + e) - 11*c^2 - 50*I*c*d + 75*d^2)/((32*I*a^2*c^4 - 128*a^2*c^3*d - 192*I*a^2*c^2*
d^2 + 128*a^2*c*d^3 + 32*I*a^2*d^4)*(tan(f*x + e) - I)^2))/f

Mupad [B] (verification not implemented)

Time = 11.94 (sec) , antiderivative size = 1984, normalized size of antiderivative = 7.32 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*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^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*d - 128*a^4*c*d^9 + 512*a^4*c^3*d^7 + a^4*c^4*d^6*512i + 1280*a
^4*c^5*d^5 - a^4*c^6*d^4*512i + 512*a^4*c^7*d^3 - a^4*c^8*d^2*512i) + tan(e + f*x)*(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)*(32*a^4*c^10 - 96*a^4*d^10 + a^4*c*d^9*384i + a^4*c^9*d*128i + 41
6*a^4*c^2*d^8 + a^4*c^3*d^7*256i + 832*a^4*c^4*d^6 - a^4*c^5*d^5*512i + 64*a^4*c^6*d^4 - a^4*c^7*d^3*256i - 22
4*a^4*c^8*d^2)) + tan(e + f*x)*(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*d^
8*96i + 72*a^2*c*d^7 - 8*a^2*c^7*d + a^2*c^2*d^6*272i + 264*a^2*c^3*d^5 + a^2*c^4*d^4*128i + 184*a^2*c^5*d^3 -
a^2*c^6*d^2*48i)) + tan(e + f*x)*(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*d
^5*72i + 81*d^6 + 2*c^2*d^4 + c^3*d^3*8i + c^4*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 + 1
28*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*3
4i, e, k), k, 1, 3)/f + ((tan(e + f*x)^2*(c*d*4i + c^2 + 9*d^2))/(4*a^2*(c*d^3*2i + c^3*d*2i + c^4 - d^4)) - (
(9*c*d^2 + c^2*d*6i + 3*c^3 - d^3*6i)*1i)/(6*a^2*d*(c^2 + d^2)*(c*d*2i + c^2 - d^2)) + (tan(e + f*x)*(18*c*d^2
+ c^2*d*4i + 2*c^3 - d^3*28i))/(8*a^2*d*(c*d^3*2i + c^3*d*2i + c^4 - d^4)))/(f*(tan(e + f*x)^2*(c/d - 2i) - c
/d - tan(e + f*x)*((c*2i)/d + 1) + tan(e + f*x)^3))