\(\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx\) [691]

Optimal result

Integrand size = 41, antiderivative size = 135 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=\frac {4 a^3 (i A+B) c^5 (1-i \tan (e+f x))^5}{5 f}-\frac {2 a^3 (i A+2 B) c^5 (1-i \tan (e+f x))^6}{3 f}+\frac {a^3 (i A+5 B) c^5 (1-i \tan (e+f x))^7}{7 f}-\frac {a^3 B c^5 (1-i \tan (e+f x))^8}{8 f} \]

[Out]

4/5*a^3*(I*A+B)*c^5*(1-I*tan(f*x+e))^5/f-2/3*a^3*(I*A+2*B)*c^5*(1-I*tan(f*x+e))^6/f+1/7*a^3*(I*A+5*B)*c^5*(1-I
*tan(f*x+e))^7/f-1/8*a^3*B*c^5*(1-I*tan(f*x+e))^8/f

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 78} \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=\frac {a^3 c^5 (5 B+i A) (1-i \tan (e+f x))^7}{7 f}-\frac {2 a^3 c^5 (2 B+i A) (1-i \tan (e+f x))^6}{3 f}+\frac {4 a^3 c^5 (B+i A) (1-i \tan (e+f x))^5}{5 f}-\frac {a^3 B c^5 (1-i \tan (e+f x))^8}{8 f} \]

[In]

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

[Out]

(4*a^3*(I*A + B)*c^5*(1 - I*Tan[e + f*x])^5)/(5*f) - (2*a^3*(I*A + 2*B)*c^5*(1 - I*Tan[e + f*x])^6)/(3*f) + (a
^3*(I*A + 5*B)*c^5*(1 - I*Tan[e + f*x])^7)/(7*f) - (a^3*B*c^5*(1 - I*Tan[e + f*x])^8)/(8*f)

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 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 (a+i a x)^2 (A+B x) (c-i c x)^4 \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^4-\frac {4 a^2 (A-2 i B) (c-i c x)^5}{c}+\frac {a^2 (A-5 i B) (c-i c x)^6}{c^2}+\frac {i a^2 B (c-i c x)^7}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {4 a^3 (i A+B) c^5 (1-i \tan (e+f x))^5}{5 f}-\frac {2 a^3 (i A+2 B) c^5 (1-i \tan (e+f x))^6}{3 f}+\frac {a^3 (i A+5 B) c^5 (1-i \tan (e+f x))^7}{7 f}-\frac {a^3 B c^5 (1-i \tan (e+f x))^8}{8 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.69 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.73 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=\frac {a^3 c^5 \sec ^8(e+f x) (140 (-i A+B) \cos (2 (e+f x))+84 (A-i B) \sin (2 (e+f x))+(4 A+i B) (-35 i+28 \sin (4 (e+f x))+8 \sin (6 (e+f x))+\sin (8 (e+f x))))}{840 f} \]

[In]

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

[Out]

(a^3*c^5*Sec[e + f*x]^8*(140*((-I)*A + B)*Cos[2*(e + f*x)] + 84*(A - I*B)*Sin[2*(e + f*x)] + (4*A + I*B)*(-35*
I + 28*Sin[4*(e + f*x)] + 8*Sin[6*(e + f*x)] + Sin[8*(e + f*x)])))/(840*f)

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {32 c^{5} a^{3} \left (84 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+84 B \,{\mathrm e}^{6 i \left (f x +e \right )}+112 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-28 B \,{\mathrm e}^{4 i \left (f x +e \right )}+32 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-8 B \,{\mathrm e}^{2 i \left (f x +e \right )}+4 i A -B \right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{8}}\)	\(108\)
derivativedivides	\(\frac {c^{5} a^{3} \left (-\frac {B \tan \left (f x +e \right )^{8}}{8}-\frac {\left (2 i B +A \right ) \tan \left (f x +e \right )^{7}}{7}-\frac {\left (-7 B -2 i A +4 i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{6}}{6}-\frac {\left (-7 A +4 i \left (-2 i A -B \right )+8 i B \right ) \tan \left (f x +e \right )^{5}}{5}-\frac {\left (8 i A +7 B -4 i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{4}}{4}-\frac {\left (7 A -4 i \left (-2 i A -B \right )-2 i B \right ) \tan \left (f x +e \right )^{3}}{3}-\frac {\left (2 i A -B \right ) \tan \left (f x +e \right )^{2}}{2}+A \tan \left (f x +e \right )\right )}{f}\)	\(177\)
default	\(\frac {c^{5} a^{3} \left (-\frac {B \tan \left (f x +e \right )^{8}}{8}-\frac {\left (2 i B +A \right ) \tan \left (f x +e \right )^{7}}{7}-\frac {\left (-7 B -2 i A +4 i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{6}}{6}-\frac {\left (-7 A +4 i \left (-2 i A -B \right )+8 i B \right ) \tan \left (f x +e \right )^{5}}{5}-\frac {\left (8 i A +7 B -4 i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{4}}{4}-\frac {\left (7 A -4 i \left (-2 i A -B \right )-2 i B \right ) \tan \left (f x +e \right )^{3}}{3}-\frac {\left (2 i A -B \right ) \tan \left (f x +e \right )^{2}}{2}+A \tan \left (f x +e \right )\right )}{f}\)	\(177\)
norman	\(\frac {A \,a^{3} c^{5} \tan \left (f x +e \right )}{f}-\frac {\left (2 i A \,a^{3} c^{5}+B \,a^{3} c^{5}\right ) \tan \left (f x +e \right )^{6}}{6 f}-\frac {\left (2 i B \,a^{3} c^{5}+A \,a^{3} c^{5}\right ) \tan \left (f x +e \right )^{7}}{7 f}-\frac {\left (4 i B \,a^{3} c^{5}+A \,a^{3} c^{5}\right ) \tan \left (f x +e \right )^{5}}{5 f}+\frac {\left (-4 i A \,a^{3} c^{5}+B \,a^{3} c^{5}\right ) \tan \left (f x +e \right )^{4}}{4 f}+\frac {\left (-2 i A \,a^{3} c^{5}+B \,a^{3} c^{5}\right ) \tan \left (f x +e \right )^{2}}{2 f}+\frac {\left (-2 i B \,a^{3} c^{5}+A \,a^{3} c^{5}\right ) \tan \left (f x +e \right )^{3}}{3 f}-\frac {B \,a^{3} c^{5} \tan \left (f x +e \right )^{8}}{8 f}\)	\(231\)
parallelrisch	\(-\frac {240 i B \tan \left (f x +e \right )^{7} a^{3} c^{5}+105 B \tan \left (f x +e \right )^{8} a^{3} c^{5}+280 i A \tan \left (f x +e \right )^{6} a^{3} c^{5}+120 A \tan \left (f x +e \right )^{7} a^{3} c^{5}+672 i B \tan \left (f x +e \right )^{5} a^{3} c^{5}+140 B \tan \left (f x +e \right )^{6} a^{3} c^{5}+840 i A \tan \left (f x +e \right )^{4} a^{3} c^{5}+168 A \tan \left (f x +e \right )^{5} a^{3} c^{5}+560 i B \tan \left (f x +e \right )^{3} a^{3} c^{5}-210 B \tan \left (f x +e \right )^{4} a^{3} c^{5}+840 i A \tan \left (f x +e \right )^{2} a^{3} c^{5}-280 A \tan \left (f x +e \right )^{3} a^{3} c^{5}-420 B \tan \left (f x +e \right )^{2} a^{3} c^{5}-840 A \tan \left (f x +e \right ) a^{3} c^{5}}{840 f}\)	\(249\)
parts	\(\frac {\left (-6 i A \,a^{3} c^{5}+2 B \,a^{3} c^{5}\right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (-6 i B \,a^{3} c^{5}-2 A \,a^{3} c^{5}\right ) \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (-2 i A \,a^{3} c^{5}-2 B \,a^{3} c^{5}\right ) \left (\frac {\tan \left (f x +e \right )^{6}}{6}-\frac {\tan \left (f x +e \right )^{4}}{4}+\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (-2 i A \,a^{3} c^{5}+B \,a^{3} c^{5}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (-2 i B \,a^{3} c^{5}-A \,a^{3} c^{5}\right ) \left (\frac {\tan \left (f x +e \right )^{7}}{7}-\frac {\tan \left (f x +e \right )^{5}}{5}+\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (-2 i B \,a^{3} c^{5}+2 A \,a^{3} c^{5}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+A \,a^{3} c^{5} x -\frac {6 i A \,a^{3} c^{5} \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}-\frac {6 i B \,a^{3} c^{5} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}-\frac {B \,a^{3} c^{5} \left (\frac {\tan \left (f x +e \right )^{8}}{8}-\frac {\tan \left (f x +e \right )^{6}}{6}+\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}\)	\(484\)

[In]

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

[Out]

32/105*c^5*a^3*(84*I*A*exp(6*I*(f*x+e))+84*B*exp(6*I*(f*x+e))+112*I*A*exp(4*I*(f*x+e))-28*B*exp(4*I*(f*x+e))+3
2*I*A*exp(2*I*(f*x+e))-8*B*exp(2*I*(f*x+e))+4*I*A-B)/f/(exp(2*I*(f*x+e))+1)^8

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.35 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=-\frac {32 \, {\left (84 \, {\left (-i \, A - B\right )} a^{3} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, {\left (-4 i \, A + B\right )} a^{3} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, {\left (-4 i \, A + B\right )} a^{3} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-4 i \, A + B\right )} a^{3} c^{5}\right )}}{105 \, {\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

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

[Out]

-32/105*(84*(-I*A - B)*a^3*c^5*e^(6*I*f*x + 6*I*e) + 28*(-4*I*A + B)*a^3*c^5*e^(4*I*f*x + 4*I*e) + 8*(-4*I*A +
B)*a^3*c^5*e^(2*I*f*x + 2*I*e) + (-4*I*A + B)*a^3*c^5)/(f*e^(16*I*f*x + 16*I*e) + 8*f*e^(14*I*f*x + 14*I*e) +
28*f*e^(12*I*f*x + 12*I*e) + 56*f*e^(10*I*f*x + 10*I*e) + 70*f*e^(8*I*f*x + 8*I*e) + 56*f*e^(6*I*f*x + 6*I*e)
+ 28*f*e^(4*I*f*x + 4*I*e) + 8*f*e^(2*I*f*x + 2*I*e) + f)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (110) = 220\).

Time = 0.83 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.27 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=\frac {128 i A a^{3} c^{5} - 32 B a^{3} c^{5} + \left (1024 i A a^{3} c^{5} e^{2 i e} - 256 B a^{3} c^{5} e^{2 i e}\right ) e^{2 i f x} + \left (3584 i A a^{3} c^{5} e^{4 i e} - 896 B a^{3} c^{5} e^{4 i e}\right ) e^{4 i f x} + \left (2688 i A a^{3} c^{5} e^{6 i e} + 2688 B a^{3} c^{5} e^{6 i e}\right ) e^{6 i f x}}{105 f e^{16 i e} e^{16 i f x} + 840 f e^{14 i e} e^{14 i f x} + 2940 f e^{12 i e} e^{12 i f x} + 5880 f e^{10 i e} e^{10 i f x} + 7350 f e^{8 i e} e^{8 i f x} + 5880 f e^{6 i e} e^{6 i f x} + 2940 f e^{4 i e} e^{4 i f x} + 840 f e^{2 i e} e^{2 i f x} + 105 f} \]

[In]

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

[Out]

(128*I*A*a**3*c**5 - 32*B*a**3*c**5 + (1024*I*A*a**3*c**5*exp(2*I*e) - 256*B*a**3*c**5*exp(2*I*e))*exp(2*I*f*x
) + (3584*I*A*a**3*c**5*exp(4*I*e) - 896*B*a**3*c**5*exp(4*I*e))*exp(4*I*f*x) + (2688*I*A*a**3*c**5*exp(6*I*e)
+ 2688*B*a**3*c**5*exp(6*I*e))*exp(6*I*f*x))/(105*f*exp(16*I*e)*exp(16*I*f*x) + 840*f*exp(14*I*e)*exp(14*I*f*
x) + 2940*f*exp(12*I*e)*exp(12*I*f*x) + 5880*f*exp(10*I*e)*exp(10*I*f*x) + 7350*f*exp(8*I*e)*exp(8*I*f*x) + 58
80*f*exp(6*I*e)*exp(6*I*f*x) + 2940*f*exp(4*I*e)*exp(4*I*f*x) + 840*f*exp(2*I*e)*exp(2*I*f*x) + 105*f)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.23 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=-\frac {105 \, B a^{3} c^{5} \tan \left (f x + e\right )^{8} + 120 \, {\left (A + 2 i \, B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{7} - 140 \, {\left (-2 i \, A - B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{6} + 168 \, {\left (A + 4 i \, B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{5} - 210 \, {\left (-4 i \, A + B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{4} - 280 \, {\left (A - 2 i \, B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{3} - 420 \, {\left (-2 i \, A + B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{2} - 840 \, A a^{3} c^{5} \tan \left (f x + e\right )}{840 \, f} \]

[In]

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

[Out]

-1/840*(105*B*a^3*c^5*tan(f*x + e)^8 + 120*(A + 2*I*B)*a^3*c^5*tan(f*x + e)^7 - 140*(-2*I*A - B)*a^3*c^5*tan(f
*x + e)^6 + 168*(A + 4*I*B)*a^3*c^5*tan(f*x + e)^5 - 210*(-4*I*A + B)*a^3*c^5*tan(f*x + e)^4 - 280*(A - 2*I*B)
*a^3*c^5*tan(f*x + e)^3 - 420*(-2*I*A + B)*a^3*c^5*tan(f*x + e)^2 - 840*A*a^3*c^5*tan(f*x + e))/f

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. 227 vs. \(2 (113) = 226\).

Time = 1.20 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.68 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=-\frac {32 \, {\left (-84 i \, A a^{3} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )} - 84 \, B a^{3} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )} - 112 i \, A a^{3} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} + 28 \, B a^{3} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} - 32 i \, A a^{3} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, B a^{3} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 i \, A a^{3} c^{5} + B a^{3} c^{5}\right )}}{105 \, {\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

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

[Out]

-32/105*(-84*I*A*a^3*c^5*e^(6*I*f*x + 6*I*e) - 84*B*a^3*c^5*e^(6*I*f*x + 6*I*e) - 112*I*A*a^3*c^5*e^(4*I*f*x +
4*I*e) + 28*B*a^3*c^5*e^(4*I*f*x + 4*I*e) - 32*I*A*a^3*c^5*e^(2*I*f*x + 2*I*e) + 8*B*a^3*c^5*e^(2*I*f*x + 2*I
*e) - 4*I*A*a^3*c^5 + B*a^3*c^5)/(f*e^(16*I*f*x + 16*I*e) + 8*f*e^(14*I*f*x + 14*I*e) + 28*f*e^(12*I*f*x + 12*
I*e) + 56*f*e^(10*I*f*x + 10*I*e) + 70*f*e^(8*I*f*x + 8*I*e) + 56*f*e^(6*I*f*x + 6*I*e) + 28*f*e^(4*I*f*x + 4*
I*e) + 8*f*e^(2*I*f*x + 2*I*e) + f)

Mupad [B] (verification not implemented)

Time = 8.60 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.29 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx=-\frac {\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-B+A\,2{}\mathrm {i}\right )}{2}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (-B+A\,4{}\mathrm {i}\right )}{4}-A\,a^3\,c^5\,\mathrm {tan}\left (e+f\,x\right )-\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (A-B\,2{}\mathrm {i}\right )}{3}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (B+A\,2{}\mathrm {i}\right )}{6}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (A+B\,4{}\mathrm {i}\right )}{5}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (A+B\,2{}\mathrm {i}\right )}{7}+\frac {B\,a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^8}{8}}{f} \]

[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]

-((a^3*c^5*tan(e + f*x)^2*(A*2i - B))/2 + (a^3*c^5*tan(e + f*x)^4*(A*4i - B))/4 - A*a^3*c^5*tan(e + f*x) - (a^
3*c^5*tan(e + f*x)^3*(A - B*2i))/3 + (a^3*c^5*tan(e + f*x)^6*(A*2i + B))/6 + (a^3*c^5*tan(e + f*x)^5*(A + B*4i
))/5 + (a^3*c^5*tan(e + f*x)^7*(A + B*2i))/7 + (B*a^3*c^5*tan(e + f*x)^8)/8)/f