\(\int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx\) [738]

Optimal result

Integrand size = 41, antiderivative size = 287 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {7 (4 A+i B) x}{128 a^3 c^5}+\frac {A+i B}{192 a^3 c^5 f (i-\tan (e+f x))^3}-\frac {3 i A-2 B}{128 a^3 c^5 f (i-\tan (e+f x))^2}-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (i-\tan (e+f x))}+\frac {A-i B}{80 a^3 c^5 f (i+\tan (e+f x))^5}-\frac {2 i A+B}{64 a^3 c^5 f (i+\tan (e+f x))^4}-\frac {5 A-i B}{96 a^3 c^5 f (i+\tan (e+f x))^3}+\frac {5 i A}{64 a^3 c^5 f (i+\tan (e+f x))^2}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (i+\tan (e+f x))} \]

[Out]

7/128*(4*A+I*B)*x/a^3/c^5+1/192*(A+I*B)/a^3/c^5/f/(I-tan(f*x+e))^3+1/128*(-3*I*A+2*B)/a^3/c^5/f/(I-tan(f*x+e))
^2-3/256*(7*A+3*I*B)/a^3/c^5/f/(I-tan(f*x+e))+1/80*(A-I*B)/a^3/c^5/f/(I+tan(f*x+e))^5+1/64*(-2*I*A-B)/a^3/c^5/
f/(I+tan(f*x+e))^4+1/96*(-5*A+I*B)/a^3/c^5/f/(I+tan(f*x+e))^3+5/64*I*A/a^3/c^5/f/(I+tan(f*x+e))^2+5/256*(7*A+I
*B)/a^3/c^5/f/(I+tan(f*x+e))

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3669, 78, 209} \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (-\tan (e+f x)+i)}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (\tan (e+f x)+i)}-\frac {-2 B+3 i A}{128 a^3 c^5 f (-\tan (e+f x)+i)^2}+\frac {A+i B}{192 a^3 c^5 f (-\tan (e+f x)+i)^3}-\frac {5 A-i B}{96 a^3 c^5 f (\tan (e+f x)+i)^3}-\frac {B+2 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^4}+\frac {A-i B}{80 a^3 c^5 f (\tan (e+f x)+i)^5}+\frac {7 x (4 A+i B)}{128 a^3 c^5}+\frac {5 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^2} \]

[In]

Int[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^5),x]

[Out]

(7*(4*A + I*B)*x)/(128*a^3*c^5) + (A + I*B)/(192*a^3*c^5*f*(I - Tan[e + f*x])^3) - ((3*I)*A - 2*B)/(128*a^3*c^
5*f*(I - Tan[e + f*x])^2) - (3*(7*A + (3*I)*B))/(256*a^3*c^5*f*(I - Tan[e + f*x])) + (A - I*B)/(80*a^3*c^5*f*(
I + Tan[e + f*x])^5) - ((2*I)*A + B)/(64*a^3*c^5*f*(I + Tan[e + f*x])^4) - (5*A - I*B)/(96*a^3*c^5*f*(I + Tan[
e + f*x])^3) + (((5*I)/64)*A)/(a^3*c^5*f*(I + Tan[e + f*x])^2) + (5*(7*A + I*B))/(256*a^3*c^5*f*(I + Tan[e + f
*x]))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3669

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a*(c/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x
], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^4 (c-i c x)^6} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (\frac {A+i B}{64 a^4 c^6 (-i+x)^4}+\frac {i (3 A+2 i B)}{64 a^4 c^6 (-i+x)^3}-\frac {3 (7 A+3 i B)}{256 a^4 c^6 (-i+x)^2}+\frac {-A+i B}{16 a^4 c^6 (i+x)^6}+\frac {2 i A+B}{16 a^4 c^6 (i+x)^5}+\frac {5 A-i B}{32 a^4 c^6 (i+x)^4}-\frac {5 i A}{32 a^4 c^6 (i+x)^3}-\frac {5 (7 A+i B)}{256 a^4 c^6 (i+x)^2}+\frac {7 (4 A+i B)}{128 a^4 c^6 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {A+i B}{192 a^3 c^5 f (i-\tan (e+f x))^3}-\frac {3 i A-2 B}{128 a^3 c^5 f (i-\tan (e+f x))^2}-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (i-\tan (e+f x))}+\frac {A-i B}{80 a^3 c^5 f (i+\tan (e+f x))^5}-\frac {2 i A+B}{64 a^3 c^5 f (i+\tan (e+f x))^4}-\frac {5 A-i B}{96 a^3 c^5 f (i+\tan (e+f x))^3}+\frac {5 i A}{64 a^3 c^5 f (i+\tan (e+f x))^2}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (i+\tan (e+f x))}+\frac {(7 (4 A+i B)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 a^3 c^5 f} \\ & = \frac {7 (4 A+i B) x}{128 a^3 c^5}+\frac {A+i B}{192 a^3 c^5 f (i-\tan (e+f x))^3}-\frac {3 i A-2 B}{128 a^3 c^5 f (i-\tan (e+f x))^2}-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (i-\tan (e+f x))}+\frac {A-i B}{80 a^3 c^5 f (i+\tan (e+f x))^5}-\frac {2 i A+B}{64 a^3 c^5 f (i+\tan (e+f x))^4}-\frac {5 A-i B}{96 a^3 c^5 f (i+\tan (e+f x))^3}+\frac {5 i A}{64 a^3 c^5 f (i+\tan (e+f x))^2}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (i+\tan (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.79 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {\sec ^7(e+f x) (2100 i A \cos (e+f x)+63 (-8 i A+7 B) \cos (3 (e+f x))-56 i A \cos (5 (e+f x))+119 B \cos (5 (e+f x))-4 i A \cos (7 (e+f x))+16 B \cos (7 (e+f x))+872 A \sin (e+f x)+218 i B \sin (e+f x)-420 (4 A+i B) \arctan (\tan (e+f x)) \sec (e+f x) (\cos (2 (e+f x))-i \sin (2 (e+f x)))-956 A \sin (3 (e+f x))-239 i B \sin (3 (e+f x))-164 A \sin (5 (e+f x))-41 i B \sin (5 (e+f x))-16 A \sin (7 (e+f x))-4 i B \sin (7 (e+f x)))}{7680 a^3 c^5 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^5} \]

[In]

Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^5),x]

[Out]

(Sec[e + f*x]^7*((2100*I)*A*Cos[e + f*x] + 63*((-8*I)*A + 7*B)*Cos[3*(e + f*x)] - (56*I)*A*Cos[5*(e + f*x)] +
119*B*Cos[5*(e + f*x)] - (4*I)*A*Cos[7*(e + f*x)] + 16*B*Cos[7*(e + f*x)] + 872*A*Sin[e + f*x] + (218*I)*B*Sin
[e + f*x] - 420*(4*A + I*B)*ArcTan[Tan[e + f*x]]*Sec[e + f*x]*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]) - 956*A*
Sin[3*(e + f*x)] - (239*I)*B*Sin[3*(e + f*x)] - 164*A*Sin[5*(e + f*x)] - (41*I)*B*Sin[5*(e + f*x)] - 16*A*Sin[
7*(e + f*x)] - (4*I)*B*Sin[7*(e + f*x)]))/(7680*a^3*c^5*f*(-I + Tan[e + f*x])^3*(I + Tan[e + f*x])^5)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.17

method	result	size
norman	\(\frac {\frac {7 \left (i B +4 A \right ) x}{128 a c}-\frac {8 i A +3 B}{40 a c f}+\frac {B \tan \left (f x +e \right )^{2}}{8 a c f}+\frac {79 \left (i B +4 A \right ) \tan \left (f x +e \right )^{3}}{192 a c f}+\frac {7 \left (i B +4 A \right ) \tan \left (f x +e \right )^{5}}{15 a c f}+\frac {49 \left (i B +4 A \right ) \tan \left (f x +e \right )^{7}}{192 a c f}+\frac {7 \left (i B +4 A \right ) \tan \left (f x +e \right )^{9}}{128 a c f}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{2}}{128 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{4}}{64 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{6}}{64 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{8}}{128 a c}+\frac {7 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{10}}{128 a c}+\frac {\left (-7 i B +100 A \right ) \tan \left (f x +e \right )}{128 a c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{5} a^{2} c^{4}}\)	\(335\)
risch	\(-\frac {i \sin \left (4 f x +4 e \right ) B}{128 a^{3} c^{5} f}+\frac {7 x A}{32 a^{3} c^{5}}-\frac {{\mathrm e}^{10 i \left (f x +e \right )} B}{2560 a^{3} c^{5} f}+\frac {7 i x B}{128 a^{3} c^{5}}-\frac {3 \,{\mathrm e}^{8 i \left (f x +e \right )} B}{1024 a^{3} c^{5} f}-\frac {13 i \sin \left (6 f x +6 e \right ) B}{1536 a^{3} c^{5} f}-\frac {5 \cos \left (6 f x +6 e \right ) B}{512 a^{3} c^{5} f}-\frac {21 i \cos \left (2 f x +2 e \right ) A}{256 a^{3} c^{5} f}-\frac {i {\mathrm e}^{8 i \left (f x +e \right )} A}{256 a^{3} c^{5} f}+\frac {29 \sin \left (6 f x +6 e \right ) A}{1536 a^{3} c^{5} f}-\frac {5 \cos \left (4 f x +4 e \right ) B}{256 a^{3} c^{5} f}+\frac {7 i \sin \left (2 f x +2 e \right ) B}{256 a^{3} c^{5} f}-\frac {3 i \cos \left (4 f x +4 e \right ) A}{64 a^{3} c^{5} f}+\frac {\sin \left (4 f x +4 e \right ) A}{16 a^{3} c^{5} f}-\frac {7 \cos \left (2 f x +2 e \right ) B}{256 a^{3} c^{5} f}-\frac {i {\mathrm e}^{10 i \left (f x +e \right )} A}{2560 a^{3} c^{5} f}-\frac {9 i \cos \left (6 f x +6 e \right ) A}{512 a^{3} c^{5} f}+\frac {49 \sin \left (2 f x +2 e \right ) A}{256 a^{3} c^{5} f}\)	\(367\)
derivativedivides	\(-\frac {i A}{32 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {A}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {9 i B}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {5 A}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i B}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}+\frac {35 A}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}-\frac {i B}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {B}{64 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{5}}-\frac {3 i A}{128 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {7 A \arctan \left (\tan \left (f x +e \right )\right )}{32 f \,a^{3} c^{5}}-\frac {A}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i A}{64 a^{3} c^{5} f \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {i B}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {21 A}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{64 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}\)	\(394\)
default	\(-\frac {i A}{32 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {A}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {9 i B}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}-\frac {5 A}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i B}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}+\frac {35 A}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}-\frac {i B}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {B}{64 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{5}}-\frac {3 i A}{128 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {7 A \arctan \left (\tan \left (f x +e \right )\right )}{32 f \,a^{3} c^{5}}-\frac {A}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {5 i A}{64 a^{3} c^{5} f \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {i B}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {21 A}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{64 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}\)	\(394\)

[In]

int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

(7/128*(4*A+I*B)/a/c*x-1/40*(3*B+8*I*A)/a/c/f+1/8/a/c/f*B*tan(f*x+e)^2+79/192*(4*A+I*B)/a/c/f*tan(f*x+e)^3+7/1
5*(4*A+I*B)/a/c/f*tan(f*x+e)^5+49/192*(4*A+I*B)/a/c/f*tan(f*x+e)^7+7/128*(4*A+I*B)/a/c/f*tan(f*x+e)^9+35/128*(
4*A+I*B)/a/c*x*tan(f*x+e)^2+35/64*(4*A+I*B)/a/c*x*tan(f*x+e)^4+35/64*(4*A+I*B)/a/c*x*tan(f*x+e)^6+35/128*(4*A+
I*B)/a/c*x*tan(f*x+e)^8+7/128*(4*A+I*B)/a/c*x*tan(f*x+e)^10+1/128*(100*A-7*I*B)/a/c/f*tan(f*x+e))/(1+tan(f*x+e
)^2)^5/a^2/c^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.55 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {{\left (840 \, {\left (4 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 6 \, {\left (i \, A + B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} - 15 \, {\left (4 i \, A + 3 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} - 140 \, {\left (2 i \, A + B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} - 210 \, {\left (4 i \, A + B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 2100 i \, A e^{\left (8 i \, f x + 8 i \, e\right )} - 420 \, {\left (-2 i \, A + B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 30 \, {\left (-4 i \, A + 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 10 i \, A - 10 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{15360 \, a^{3} c^{5} f} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, algorithm="fricas")

[Out]

1/15360*(840*(4*A + I*B)*f*x*e^(6*I*f*x + 6*I*e) - 6*(I*A + B)*e^(16*I*f*x + 16*I*e) - 15*(4*I*A + 3*B)*e^(14*
I*f*x + 14*I*e) - 140*(2*I*A + B)*e^(12*I*f*x + 12*I*e) - 210*(4*I*A + B)*e^(10*I*f*x + 10*I*e) - 2100*I*A*e^(
8*I*f*x + 8*I*e) - 420*(-2*I*A + B)*e^(4*I*f*x + 4*I*e) - 30*(-4*I*A + 3*B)*e^(2*I*f*x + 2*I*e) + 10*I*A - 10*
B)*e^(-6*I*f*x - 6*I*e)/(a^3*c^5*f)

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 646, normalized size of antiderivative = 2.25 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\begin {cases} \frac {\left (- 7263405479023135948800 i A a^{21} c^{35} f^{7} e^{14 i e} e^{2 i f x} + \left (34587645138205409280 i A a^{21} c^{35} f^{7} e^{6 i e} - 34587645138205409280 B a^{21} c^{35} f^{7} e^{6 i e}\right ) e^{- 6 i f x} + \left (415051741658464911360 i A a^{21} c^{35} f^{7} e^{8 i e} - 311288806243848683520 B a^{21} c^{35} f^{7} e^{8 i e}\right ) e^{- 4 i f x} + \left (2905362191609254379520 i A a^{21} c^{35} f^{7} e^{10 i e} - 1452681095804627189760 B a^{21} c^{35} f^{7} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 2905362191609254379520 i A a^{21} c^{35} f^{7} e^{16 i e} - 726340547902313594880 B a^{21} c^{35} f^{7} e^{16 i e}\right ) e^{4 i f x} + \left (- 968454063869751459840 i A a^{21} c^{35} f^{7} e^{18 i e} - 484227031934875729920 B a^{21} c^{35} f^{7} e^{18 i e}\right ) e^{6 i f x} + \left (- 207525870829232455680 i A a^{21} c^{35} f^{7} e^{20 i e} - 155644403121924341760 B a^{21} c^{35} f^{7} e^{20 i e}\right ) e^{8 i f x} + \left (- 20752587082923245568 i A a^{21} c^{35} f^{7} e^{22 i e} - 20752587082923245568 B a^{21} c^{35} f^{7} e^{22 i e}\right ) e^{10 i f x}\right ) e^{- 12 i e}}{53126622932283508654080 a^{24} c^{40} f^{8}} & \text {for}\: a^{24} c^{40} f^{8} e^{12 i e} \neq 0 \\x \left (- \frac {28 A + 7 i B}{128 a^{3} c^{5}} + \frac {\left (A e^{16 i e} + 8 A e^{14 i e} + 28 A e^{12 i e} + 56 A e^{10 i e} + 70 A e^{8 i e} + 56 A e^{6 i e} + 28 A e^{4 i e} + 8 A e^{2 i e} + A - i B e^{16 i e} - 6 i B e^{14 i e} - 14 i B e^{12 i e} - 14 i B e^{10 i e} + 14 i B e^{6 i e} + 14 i B e^{4 i e} + 6 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{256 a^{3} c^{5}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (28 A + 7 i B\right )}{128 a^{3} c^{5}} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**5,x)

[Out]

Piecewise(((-7263405479023135948800*I*A*a**21*c**35*f**7*exp(14*I*e)*exp(2*I*f*x) + (34587645138205409280*I*A*
a**21*c**35*f**7*exp(6*I*e) - 34587645138205409280*B*a**21*c**35*f**7*exp(6*I*e))*exp(-6*I*f*x) + (41505174165
8464911360*I*A*a**21*c**35*f**7*exp(8*I*e) - 311288806243848683520*B*a**21*c**35*f**7*exp(8*I*e))*exp(-4*I*f*x
) + (2905362191609254379520*I*A*a**21*c**35*f**7*exp(10*I*e) - 1452681095804627189760*B*a**21*c**35*f**7*exp(1
0*I*e))*exp(-2*I*f*x) + (-2905362191609254379520*I*A*a**21*c**35*f**7*exp(16*I*e) - 726340547902313594880*B*a*
*21*c**35*f**7*exp(16*I*e))*exp(4*I*f*x) + (-968454063869751459840*I*A*a**21*c**35*f**7*exp(18*I*e) - 48422703
1934875729920*B*a**21*c**35*f**7*exp(18*I*e))*exp(6*I*f*x) + (-207525870829232455680*I*A*a**21*c**35*f**7*exp(
20*I*e) - 155644403121924341760*B*a**21*c**35*f**7*exp(20*I*e))*exp(8*I*f*x) + (-20752587082923245568*I*A*a**2
1*c**35*f**7*exp(22*I*e) - 20752587082923245568*B*a**21*c**35*f**7*exp(22*I*e))*exp(10*I*f*x))*exp(-12*I*e)/(5
3126622932283508654080*a**24*c**40*f**8), Ne(a**24*c**40*f**8*exp(12*I*e), 0)), (x*(-(28*A + 7*I*B)/(128*a**3*
c**5) + (A*exp(16*I*e) + 8*A*exp(14*I*e) + 28*A*exp(12*I*e) + 56*A*exp(10*I*e) + 70*A*exp(8*I*e) + 56*A*exp(6*
I*e) + 28*A*exp(4*I*e) + 8*A*exp(2*I*e) + A - I*B*exp(16*I*e) - 6*I*B*exp(14*I*e) - 14*I*B*exp(12*I*e) - 14*I*
B*exp(10*I*e) + 14*I*B*exp(6*I*e) + 14*I*B*exp(4*I*e) + 6*I*B*exp(2*I*e) + I*B)*exp(-6*I*e)/(256*a**3*c**5)),
True)) + x*(28*A + 7*I*B)/(128*a**3*c**5)

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.89 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=-\frac {\frac {420 \, {\left (-4 i \, A + B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{5}} + \frac {420 \, {\left (4 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{5}} + \frac {10 \, {\left (-308 i \, A \tan \left (f x + e\right )^{3} + 77 \, B \tan \left (f x + e\right )^{3} - 1050 \, A \tan \left (f x + e\right )^{2} - 285 i \, B \tan \left (f x + e\right )^{2} + 1212 i \, A \tan \left (f x + e\right ) - 363 \, B \tan \left (f x + e\right ) + 478 \, A + 163 i \, B\right )}}{a^{3} c^{5} {\left (\tan \left (f x + e\right ) - i\right )}^{3}} + \frac {3836 i \, A \tan \left (f x + e\right )^{5} - 959 \, B \tan \left (f x + e\right )^{5} - 21280 \, A \tan \left (f x + e\right )^{4} - 5095 i \, B \tan \left (f x + e\right )^{4} - 47960 i \, A \tan \left (f x + e\right )^{3} + 10790 \, B \tan \left (f x + e\right )^{3} + 55360 \, A \tan \left (f x + e\right )^{2} + 11230 i \, B \tan \left (f x + e\right )^{2} + 33260 i \, A \tan \left (f x + e\right ) - 5435 \, B \tan \left (f x + e\right ) - 8608 \, A - 667 i \, B}{a^{3} c^{5} {\left (\tan \left (f x + e\right ) + i\right )}^{5}}}{15360 \, f} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, algorithm="giac")

[Out]

-1/15360*(420*(-4*I*A + B)*log(tan(f*x + e) + I)/(a^3*c^5) + 420*(4*I*A - B)*log(tan(f*x + e) - I)/(a^3*c^5) +
10*(-308*I*A*tan(f*x + e)^3 + 77*B*tan(f*x + e)^3 - 1050*A*tan(f*x + e)^2 - 285*I*B*tan(f*x + e)^2 + 1212*I*A
*tan(f*x + e) - 363*B*tan(f*x + e) + 478*A + 163*I*B)/(a^3*c^5*(tan(f*x + e) - I)^3) + (3836*I*A*tan(f*x + e)^
5 - 959*B*tan(f*x + e)^5 - 21280*A*tan(f*x + e)^4 - 5095*I*B*tan(f*x + e)^4 - 47960*I*A*tan(f*x + e)^3 + 10790
*B*tan(f*x + e)^3 + 55360*A*tan(f*x + e)^2 + 11230*I*B*tan(f*x + e)^2 + 33260*I*A*tan(f*x + e) - 5435*B*tan(f*
x + e) - 8608*A - 667*I*B)/(a^3*c^5*(tan(f*x + e) + I)^5))/f

Mupad [B] (verification not implemented)

Time = 10.41 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {\frac {3\,B}{40\,a^3\,c^5}+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (-\frac {7\,B}{24\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{6\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (-\frac {7\,B}{64\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{16\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {7\,A}{32\,a^3\,c^5}+\frac {B\,7{}\mathrm {i}}{128\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {35\,A}{96\,a^3\,c^5}+\frac {B\,35{}\mathrm {i}}{384\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-\frac {77\,B}{320\,a^3\,c^5}+\frac {A\,77{}\mathrm {i}}{80\,a^3\,c^5}\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {49\,A}{480\,a^3\,c^5}+\frac {B\,49{}\mathrm {i}}{1920\,a^3\,c^5}\right )-\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {61\,A}{160\,a^3\,c^5}+\frac {B\,61{}\mathrm {i}}{640\,a^3\,c^5}\right )+\frac {A\,1{}\mathrm {i}}{5\,a^3\,c^5}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,2{}\mathrm {i}+2\,{\mathrm {tan}\left (e+f\,x\right )}^6+{\mathrm {tan}\left (e+f\,x\right )}^5\,6{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^3\,6{}\mathrm {i}-2\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )}+\frac {7\,x\,\left (4\,A+B\,1{}\mathrm {i}\right )}{128\,a^3\,c^5} \]

[In]

int((A + B*tan(e + f*x))/((a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i)^5),x)

[Out]

(tan(e + f*x)^4*((A*7i)/(6*a^3*c^5) - (7*B)/(24*a^3*c^5)) - tan(e + f*x)*((61*A)/(160*a^3*c^5) + (B*61i)/(640*
a^3*c^5)) + tan(e + f*x)^6*((A*7i)/(16*a^3*c^5) - (7*B)/(64*a^3*c^5)) + tan(e + f*x)^7*((7*A)/(32*a^3*c^5) + (
B*7i)/(128*a^3*c^5)) + tan(e + f*x)^5*((35*A)/(96*a^3*c^5) + (B*35i)/(384*a^3*c^5)) + tan(e + f*x)^2*((A*77i)/
(80*a^3*c^5) - (77*B)/(320*a^3*c^5)) - tan(e + f*x)^3*((49*A)/(480*a^3*c^5) + (B*49i)/(1920*a^3*c^5)) + (A*1i)
/(5*a^3*c^5) + (3*B)/(40*a^3*c^5))/(f*(tan(e + f*x)*2i - 2*tan(e + f*x)^2 + tan(e + f*x)^3*6i + tan(e + f*x)^5
*6i + 2*tan(e + f*x)^6 + tan(e + f*x)^7*2i + tan(e + f*x)^8 - 1)) + (7*x*(4*A + B*1i))/(128*a^3*c^5)