\(\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx\) [678]

Optimal result

Integrand size = 41, antiderivative size = 99 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {a^2 (i A+B) c^4 (1-i \tan (e+f x))^4}{2 f}-\frac {a^2 (i A+3 B) c^4 (1-i \tan (e+f x))^5}{5 f}+\frac {a^2 B c^4 (1-i \tan (e+f x))^6}{6 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 99, 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} \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {a^2 c^4 (3 B+i A) (1-i \tan (e+f x))^5}{5 f}+\frac {a^2 c^4 (B+i A) (1-i \tan (e+f x))^4}{2 f}+\frac {a^2 B c^4 (1-i \tan (e+f x))^6}{6 f} \]

[In]

Int[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 2.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {a^2 c^4 \left (6 i A-7 B-30 A \tan (e+f x)+(30 i A-15 B) \tan ^2(e+f x)+20 i B \tan ^3(e+f x)+15 i A \tan ^4(e+f x)+6 (A+2 i B) \tan ^5(e+f x)+5 B \tan ^6(e+f x)\right )}{30 f} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]

[Out]

-1/30*(a^2*c^4*((6*I)*A - 7*B - 30*A*Tan[e + f*x] + ((30*I)*A - 15*B)*Tan[e + f*x]^2 + (20*I)*B*Tan[e + f*x]^3
+ (15*I)*A*Tan[e + f*x]^4 + 6*(A + (2*I)*B)*Tan[e + f*x]^5 + 5*B*Tan[e + f*x]^6))/f

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {8 c^{4} a^{2} \left (15 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+15 B \,{\mathrm e}^{4 i \left (f x +e \right )}+18 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-6 B \,{\mathrm e}^{2 i \left (f x +e \right )}+3 i A -B \right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6}}\)	\(83\)
derivativedivides	\(\frac {c^{4} a^{2} \left (-\frac {B \tan \left (f x +e \right )^{6}}{6}+\frac {\left (-2 i B -A \right ) \tan \left (f x +e \right )^{5}}{5}+\frac {\left (i A +3 i \left (i B -A \right )+3 B \right ) \tan \left (f x +e \right )^{4}}{4}-\frac {2 i B \tan \left (f x +e \right )^{3}}{3}+\frac {\left (-3 i A -i \left (i B -A \right )\right ) \tan \left (f x +e \right )^{2}}{2}+A \tan \left (f x +e \right )\right )}{f}\)	\(116\)
default	\(\frac {c^{4} a^{2} \left (-\frac {B \tan \left (f x +e \right )^{6}}{6}+\frac {\left (-2 i B -A \right ) \tan \left (f x +e \right )^{5}}{5}+\frac {\left (i A +3 i \left (i B -A \right )+3 B \right ) \tan \left (f x +e \right )^{4}}{4}-\frac {2 i B \tan \left (f x +e \right )^{3}}{3}+\frac {\left (-3 i A -i \left (i B -A \right )\right ) \tan \left (f x +e \right )^{2}}{2}+A \tan \left (f x +e \right )\right )}{f}\)	\(116\)
norman	\(\frac {A \,a^{2} c^{4} \tan \left (f x +e \right )}{f}-\frac {\left (2 i B \,a^{2} c^{4}+A \,a^{2} c^{4}\right ) \tan \left (f x +e \right )^{5}}{5 f}+\frac {\left (-2 i A \,a^{2} c^{4}+B \,a^{2} c^{4}\right ) \tan \left (f x +e \right )^{2}}{2 f}-\frac {B \,a^{2} c^{4} \tan \left (f x +e \right )^{6}}{6 f}-\frac {i A \,a^{2} c^{4} \tan \left (f x +e \right )^{4}}{2 f}-\frac {2 i B \,a^{2} c^{4} \tan \left (f x +e \right )^{3}}{3 f}\)	\(145\)
parallelrisch	\(-\frac {12 i B \tan \left (f x +e \right )^{5} a^{2} c^{4}+5 B \tan \left (f x +e \right )^{6} a^{2} c^{4}+15 i A \,a^{2} c^{4} \tan \left (f x +e \right )^{4}+6 A \tan \left (f x +e \right )^{5} a^{2} c^{4}+20 i B \,a^{2} c^{4} \tan \left (f x +e \right )^{3}+30 i A \tan \left (f x +e \right )^{2} a^{2} c^{4}-15 B \tan \left (f x +e \right )^{2} a^{2} c^{4}-30 A \tan \left (f x +e \right ) a^{2} c^{4}}{30 f}\)	\(145\)
parts	\(\frac {\left (-4 i A \,a^{2} c^{4}+B \,a^{2} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (-4 i B \,a^{2} c^{4}-A \,a^{2} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (-2 i A \,a^{2} c^{4}-B \,a^{2} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (-2 i A \,a^{2} c^{4}+B \,a^{2} c^{4}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (-2 i B \,a^{2} c^{4}-A \,a^{2} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (-2 i B \,a^{2} c^{4}+A \,a^{2} c^{4}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+A \,a^{2} c^{4} x -\frac {B \,a^{2} c^{4} \left (\frac {\tan \left (f x +e \right )^{6}}{6}-\frac {\tan \left (f x +e \right )^{4}}{4}+\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}\)	\(356\)

[In]

int((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

8/15*c^4*a^2*(15*I*A*exp(4*I*(f*x+e))+15*B*exp(4*I*(f*x+e))+18*I*A*exp(2*I*(f*x+e))-6*B*exp(2*I*(f*x+e))+3*I*A
-B)/f/(exp(2*I*(f*x+e))+1)^6

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.37 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {8 \, {\left (15 \, {\left (-i \, A - B\right )} a^{2} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, {\left (-3 i \, A + B\right )} a^{2} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-3 i \, A + B\right )} a^{2} c^{4}\right )}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

-8/15*(15*(-I*A - B)*a^2*c^4*e^(4*I*f*x + 4*I*e) + 6*(-3*I*A + B)*a^2*c^4*e^(2*I*f*x + 2*I*e) + (-3*I*A + B)*a
^2*c^4)/(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I*f*x + 8*I*e) + 20*f*e^(6*I*f*x + 6*
I*e) + 15*f*e^(4*I*f*x + 4*I*e) + 6*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. 224 vs. \(2 (78) = 156\).

Time = 0.49 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.26 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {24 i A a^{2} c^{4} - 8 B a^{2} c^{4} + \left (144 i A a^{2} c^{4} e^{2 i e} - 48 B a^{2} c^{4} e^{2 i e}\right ) e^{2 i f x} + \left (120 i A a^{2} c^{4} e^{4 i e} + 120 B a^{2} c^{4} e^{4 i e}\right ) e^{4 i f x}}{15 f e^{12 i e} e^{12 i f x} + 90 f e^{10 i e} e^{10 i f x} + 225 f e^{8 i e} e^{8 i f x} + 300 f e^{6 i e} e^{6 i f x} + 225 f e^{4 i e} e^{4 i f x} + 90 f e^{2 i e} e^{2 i f x} + 15 f} \]

[In]

integrate((a+I*a*tan(f*x+e))**2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**4,x)

[Out]

(24*I*A*a**2*c**4 - 8*B*a**2*c**4 + (144*I*A*a**2*c**4*exp(2*I*e) - 48*B*a**2*c**4*exp(2*I*e))*exp(2*I*f*x) +
(120*I*A*a**2*c**4*exp(4*I*e) + 120*B*a**2*c**4*exp(4*I*e))*exp(4*I*f*x))/(15*f*exp(12*I*e)*exp(12*I*f*x) + 90
*f*exp(10*I*e)*exp(10*I*f*x) + 225*f*exp(8*I*e)*exp(8*I*f*x) + 300*f*exp(6*I*e)*exp(6*I*f*x) + 225*f*exp(4*I*e
)*exp(4*I*f*x) + 90*f*exp(2*I*e)*exp(2*I*f*x) + 15*f)

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {5 \, B a^{2} c^{4} \tan \left (f x + e\right )^{6} + 6 \, {\left (A + 2 i \, B\right )} a^{2} c^{4} \tan \left (f x + e\right )^{5} + 15 i \, A a^{2} c^{4} \tan \left (f x + e\right )^{4} + 20 i \, B a^{2} c^{4} \tan \left (f x + e\right )^{3} + 15 \, {\left (2 i \, A - B\right )} a^{2} c^{4} \tan \left (f x + e\right )^{2} - 30 \, A a^{2} c^{4} \tan \left (f x + e\right )}{30 \, f} \]

[In]

integrate((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

-1/30*(5*B*a^2*c^4*tan(f*x + e)^6 + 6*(A + 2*I*B)*a^2*c^4*tan(f*x + e)^5 + 15*I*A*a^2*c^4*tan(f*x + e)^4 + 20*
I*B*a^2*c^4*tan(f*x + e)^3 + 15*(2*I*A - B)*a^2*c^4*tan(f*x + e)^2 - 30*A*a^2*c^4*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. 167 vs. \(2 (83) = 166\).

Time = 0.89 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.69 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {8 \, {\left (-15 i \, A a^{2} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 15 \, B a^{2} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 18 i \, A a^{2} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 \, B a^{2} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, A a^{2} c^{4} + B a^{2} c^{4}\right )}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

-8/15*(-15*I*A*a^2*c^4*e^(4*I*f*x + 4*I*e) - 15*B*a^2*c^4*e^(4*I*f*x + 4*I*e) - 18*I*A*a^2*c^4*e^(2*I*f*x + 2*
I*e) + 6*B*a^2*c^4*e^(2*I*f*x + 2*I*e) - 3*I*A*a^2*c^4 + B*a^2*c^4)/(f*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x
+ 10*I*e) + 15*f*e^(8*I*f*x + 8*I*e) + 20*f*e^(6*I*f*x + 6*I*e) + 15*f*e^(4*I*f*x + 4*I*e) + 6*f*e^(2*I*f*x +
2*I*e) + f)

Mupad [B] (verification not implemented)

Time = 9.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.21 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {\frac {a^2\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-B+A\,2{}\mathrm {i}\right )}{2}-A\,a^2\,c^4\,\mathrm {tan}\left (e+f\,x\right )+\frac {a^2\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (A+B\,2{}\mathrm {i}\right )}{5}+\frac {B\,a^2\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^6}{6}+\frac {A\,a^2\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^4\,1{}\mathrm {i}}{2}+\frac {B\,a^2\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^3\,2{}\mathrm {i}}{3}}{f} \]

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^2*(c - c*tan(e + f*x)*1i)^4,x)

[Out]

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