3.690 \(\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^6 \, dx\)

Optimal. Leaf size=135 \[ \frac {a^3 c^6 (5 B+i A) (1-i \tan (e+f x))^8}{8 f}-\frac {4 a^3 c^6 (2 B+i A) (1-i \tan (e+f x))^7}{7 f}+\frac {2 a^3 c^6 (B+i A) (1-i \tan (e+f x))^6}{3 f}-\frac {a^3 B c^6 (1-i \tan (e+f x))^9}{9 f} \]

[Out]

2/3*a^3*(I*A+B)*c^6*(1-I*tan(f*x+e))^6/f-4/7*a^3*(I*A+2*B)*c^6*(1-I*tan(f*x+e))^7/f+1/8*a^3*(I*A+5*B)*c^6*(1-I
*tan(f*x+e))^8/f-1/9*a^3*B*c^6*(1-I*tan(f*x+e))^9/f

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Rubi [A]  time = 0.20, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3588, 77} \[ \frac {a^3 c^6 (5 B+i A) (1-i \tan (e+f x))^8}{8 f}-\frac {4 a^3 c^6 (2 B+i A) (1-i \tan (e+f x))^7}{7 f}+\frac {2 a^3 c^6 (B+i A) (1-i \tan (e+f x))^6}{3 f}-\frac {a^3 B c^6 (1-i \tan (e+f x))^9}{9 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^6,x]

[Out]

(2*a^3*(I*A + B)*c^6*(1 - I*Tan[e + f*x])^6)/(3*f) - (4*a^3*(I*A + 2*B)*c^6*(1 - I*Tan[e + f*x])^7)/(7*f) + (a
^3*(I*A + 5*B)*c^6*(1 - I*Tan[e + f*x])^8)/(8*f) - (a^3*B*c^6*(1 - I*Tan[e + f*x])^9)/(9*f)

Rule 77

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 3588

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*} \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^6 \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^5 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^5-\frac {4 a^2 (A-2 i B) (c-i c x)^6}{c}+\frac {a^2 (A-5 i B) (c-i c x)^7}{c^2}+\frac {i a^2 B (c-i c x)^8}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 a^3 (i A+B) c^6 (1-i \tan (e+f x))^6}{3 f}-\frac {4 a^3 (i A+2 B) c^6 (1-i \tan (e+f x))^7}{7 f}+\frac {a^3 (i A+5 B) c^6 (1-i \tan (e+f x))^8}{8 f}-\frac {a^3 B c^6 (1-i \tan (e+f x))^9}{9 f}\\ \end {align*}

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Mathematica [A]  time = 11.39, size = 262, normalized size = 1.94 \[ \frac {a^3 c^6 \sec (e) \sec ^9(e+f x) (63 (B-3 i A) \cos (2 e+f x)+63 (B-3 i A) \cos (f x)-189 A \sin (2 e+f x)+168 A \sin (2 e+3 f x)-84 A \sin (4 e+3 f x)+108 A \sin (4 e+5 f x)+27 A \sin (6 e+7 f x)+3 A \sin (8 e+9 f x)-84 i A \cos (2 e+3 f x)-84 i A \cos (4 e+3 f x)+189 A \sin (f x)-63 i B \sin (2 e+f x)-84 i B \sin (4 e+3 f x)+36 i B \sin (4 e+5 f x)+9 i B \sin (6 e+7 f x)+i B \sin (8 e+9 f x)+84 B \cos (2 e+3 f x)+84 B \cos (4 e+3 f x)+63 i B \sin (f x))}{1008 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^6,x]

[Out]

(a^3*c^6*Sec[e]*Sec[e + f*x]^9*(63*((-3*I)*A + B)*Cos[f*x] + 63*((-3*I)*A + B)*Cos[2*e + f*x] - (84*I)*A*Cos[2
*e + 3*f*x] + 84*B*Cos[2*e + 3*f*x] - (84*I)*A*Cos[4*e + 3*f*x] + 84*B*Cos[4*e + 3*f*x] + 189*A*Sin[f*x] + (63
*I)*B*Sin[f*x] - 189*A*Sin[2*e + f*x] - (63*I)*B*Sin[2*e + f*x] + 168*A*Sin[2*e + 3*f*x] - 84*A*Sin[4*e + 3*f*
x] - (84*I)*B*Sin[4*e + 3*f*x] + 108*A*Sin[4*e + 5*f*x] + (36*I)*B*Sin[4*e + 5*f*x] + 27*A*Sin[6*e + 7*f*x] +
(9*I)*B*Sin[6*e + 7*f*x] + 3*A*Sin[8*e + 9*f*x] + I*B*Sin[8*e + 9*f*x]))/(1008*f)

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fricas [A]  time = 0.87, size = 197, normalized size = 1.46 \[ \frac {{\left (2688 i \, A + 2688 \, B\right )} a^{3} c^{6} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (3456 i \, A - 1152 \, B\right )} a^{3} c^{6} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (864 i \, A - 288 \, B\right )} a^{3} c^{6} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (96 i \, A - 32 \, B\right )} a^{3} c^{6}}{63 \, {\left (f e^{\left (18 i \, f x + 18 i \, e\right )} + 9 \, f e^{\left (16 i \, f x + 16 i \, e\right )} + 36 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 84 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 126 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 126 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 84 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 36 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 9 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^6,x, algorithm="fricas")

[Out]

1/63*((2688*I*A + 2688*B)*a^3*c^6*e^(6*I*f*x + 6*I*e) + (3456*I*A - 1152*B)*a^3*c^6*e^(4*I*f*x + 4*I*e) + (864
*I*A - 288*B)*a^3*c^6*e^(2*I*f*x + 2*I*e) + (96*I*A - 32*B)*a^3*c^6)/(f*e^(18*I*f*x + 18*I*e) + 9*f*e^(16*I*f*
x + 16*I*e) + 36*f*e^(14*I*f*x + 14*I*e) + 84*f*e^(12*I*f*x + 12*I*e) + 126*f*e^(10*I*f*x + 10*I*e) + 126*f*e^
(8*I*f*x + 8*I*e) + 84*f*e^(6*I*f*x + 6*I*e) + 36*f*e^(4*I*f*x + 4*I*e) + 9*f*e^(2*I*f*x + 2*I*e) + f)

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giac [B]  time = 4.46, size = 255, normalized size = 1.89 \[ \frac {2688 i \, A a^{3} c^{6} e^{\left (6 i \, f x + 6 i \, e\right )} + 2688 \, B a^{3} c^{6} e^{\left (6 i \, f x + 6 i \, e\right )} + 3456 i \, A a^{3} c^{6} e^{\left (4 i \, f x + 4 i \, e\right )} - 1152 \, B a^{3} c^{6} e^{\left (4 i \, f x + 4 i \, e\right )} + 864 i \, A a^{3} c^{6} e^{\left (2 i \, f x + 2 i \, e\right )} - 288 \, B a^{3} c^{6} e^{\left (2 i \, f x + 2 i \, e\right )} + 96 i \, A a^{3} c^{6} - 32 \, B a^{3} c^{6}}{63 \, {\left (f e^{\left (18 i \, f x + 18 i \, e\right )} + 9 \, f e^{\left (16 i \, f x + 16 i \, e\right )} + 36 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 84 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 126 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 126 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 84 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 36 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 9 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^6,x, algorithm="giac")

[Out]

1/63*(2688*I*A*a^3*c^6*e^(6*I*f*x + 6*I*e) + 2688*B*a^3*c^6*e^(6*I*f*x + 6*I*e) + 3456*I*A*a^3*c^6*e^(4*I*f*x
+ 4*I*e) - 1152*B*a^3*c^6*e^(4*I*f*x + 4*I*e) + 864*I*A*a^3*c^6*e^(2*I*f*x + 2*I*e) - 288*B*a^3*c^6*e^(2*I*f*x
+ 2*I*e) + 96*I*A*a^3*c^6 - 32*B*a^3*c^6)/(f*e^(18*I*f*x + 18*I*e) + 9*f*e^(16*I*f*x + 16*I*e) + 36*f*e^(14*I
*f*x + 14*I*e) + 84*f*e^(12*I*f*x + 12*I*e) + 126*f*e^(10*I*f*x + 10*I*e) + 126*f*e^(8*I*f*x + 8*I*e) + 84*f*e
^(6*I*f*x + 6*I*e) + 36*f*e^(4*I*f*x + 4*I*e) + 9*f*e^(2*I*f*x + 2*I*e) + f)

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maple [A]  time = 0.02, size = 193, normalized size = 1.43 \[ \frac {c^{6} a^{3} \left (-i B \left (\tan ^{3}\left (f x +e \right )\right )-\frac {i B \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {i A \left (\tan ^{8}\left (f x +e \right )\right )}{8}-\frac {3 B \left (\tan ^{8}\left (f x +e \right )\right )}{8}-\frac {i A \left (\tan ^{6}\left (f x +e \right )\right )}{6}-\frac {3 A \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {i B \left (\tan ^{9}\left (f x +e \right )\right )}{9}-\frac {5 B \left (\tan ^{6}\left (f x +e \right )\right )}{6}-i B \left (\tan ^{5}\left (f x +e \right )\right )-A \left (\tan ^{5}\left (f x +e \right )\right )-\frac {3 i A \left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {B \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {5 i A \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {A \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^6,x)

[Out]

1/f*c^6*a^3*(-I*B*tan(f*x+e)^3-1/7*I*B*tan(f*x+e)^7+1/8*I*A*tan(f*x+e)^8-3/8*B*tan(f*x+e)^8-1/6*I*A*tan(f*x+e)
^6-3/7*A*tan(f*x+e)^7+1/9*I*B*tan(f*x+e)^9-5/6*B*tan(f*x+e)^6-I*B*tan(f*x+e)^5-A*tan(f*x+e)^5-3/2*I*A*tan(f*x+
e)^2-1/4*B*tan(f*x+e)^4-5/4*I*A*tan(f*x+e)^4-1/3*A*tan(f*x+e)^3+1/2*B*tan(f*x+e)^2+A*tan(f*x+e))

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maxima [A]  time = 0.78, size = 191, normalized size = 1.41 \[ \frac {280 i \, B a^{3} c^{6} \tan \left (f x + e\right )^{9} + {\left (315 i \, A - 945 \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{8} - 360 \, {\left (3 \, A + i \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{7} + {\left (-420 i \, A - 2100 \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{6} - 2520 \, {\left (A + i \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{5} + {\left (-3150 i \, A - 630 \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{4} - 840 \, {\left (A + 3 i \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{3} + {\left (-3780 i \, A + 1260 \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{2} + 2520 \, A a^{3} c^{6} \tan \left (f x + e\right )}{2520 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^6,x, algorithm="maxima")

[Out]

1/2520*(280*I*B*a^3*c^6*tan(f*x + e)^9 + (315*I*A - 945*B)*a^3*c^6*tan(f*x + e)^8 - 360*(3*A + I*B)*a^3*c^6*ta
n(f*x + e)^7 + (-420*I*A - 2100*B)*a^3*c^6*tan(f*x + e)^6 - 2520*(A + I*B)*a^3*c^6*tan(f*x + e)^5 + (-3150*I*A
- 630*B)*a^3*c^6*tan(f*x + e)^4 - 840*(A + 3*I*B)*a^3*c^6*tan(f*x + e)^3 + (-3780*I*A + 1260*B)*a^3*c^6*tan(f
*x + e)^2 + 2520*A*a^3*c^6*tan(f*x + e))/f

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mupad [B]  time = 8.70, size = 208, normalized size = 1.54 \[ \frac {A\,a^3\,c^6\,\mathrm {tan}\left (e+f\,x\right )+\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (-3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (-B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (5\,A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (-B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{7}-\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (3\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (A-B\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{6}+\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^8\,\left (A+B\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{8}+\frac {B\,a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^9\,1{}\mathrm {i}}{9}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i)^6,x)

[Out]

((a^3*c^6*tan(e + f*x)^3*(A*1i - 3*B)*1i)/3 - (a^3*c^6*tan(e + f*x)^2*(3*A + B*1i)*1i)/2 + a^3*c^6*tan(e + f*x
)^5*(A*1i - B)*1i - (a^3*c^6*tan(e + f*x)^4*(5*A - B*1i)*1i)/4 + (a^3*c^6*tan(e + f*x)^7*(A*3i - B)*1i)/7 + A*
a^3*c^6*tan(e + f*x) - (a^3*c^6*tan(e + f*x)^6*(A - B*5i)*1i)/6 + (a^3*c^6*tan(e + f*x)^8*(A + B*3i)*1i)/8 + (
B*a^3*c^6*tan(e + f*x)^9*1i)/9)/f

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sympy [B]  time = 2.14, size = 345, normalized size = 2.56 \[ \frac {- 96 A a^{3} c^{6} - 32 i B a^{3} c^{6} + \left (- 864 A a^{3} c^{6} e^{2 i e} - 288 i B a^{3} c^{6} e^{2 i e}\right ) e^{2 i f x} + \left (- 3456 A a^{3} c^{6} e^{4 i e} - 1152 i B a^{3} c^{6} e^{4 i e}\right ) e^{4 i f x} + \left (- 2688 A a^{3} c^{6} e^{6 i e} + 2688 i B a^{3} c^{6} e^{6 i e}\right ) e^{6 i f x}}{63 i f e^{18 i e} e^{18 i f x} + 567 i f e^{16 i e} e^{16 i f x} + 2268 i f e^{14 i e} e^{14 i f x} + 5292 i f e^{12 i e} e^{12 i f x} + 7938 i f e^{10 i e} e^{10 i f x} + 7938 i f e^{8 i e} e^{8 i f x} + 5292 i f e^{6 i e} e^{6 i f x} + 2268 i f e^{4 i e} e^{4 i f x} + 567 i f e^{2 i e} e^{2 i f x} + 63 i f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**6,x)

[Out]

(-96*A*a**3*c**6 - 32*I*B*a**3*c**6 + (-864*A*a**3*c**6*exp(2*I*e) - 288*I*B*a**3*c**6*exp(2*I*e))*exp(2*I*f*x
) + (-3456*A*a**3*c**6*exp(4*I*e) - 1152*I*B*a**3*c**6*exp(4*I*e))*exp(4*I*f*x) + (-2688*A*a**3*c**6*exp(6*I*e
) + 2688*I*B*a**3*c**6*exp(6*I*e))*exp(6*I*f*x))/(63*I*f*exp(18*I*e)*exp(18*I*f*x) + 567*I*f*exp(16*I*e)*exp(1
6*I*f*x) + 2268*I*f*exp(14*I*e)*exp(14*I*f*x) + 5292*I*f*exp(12*I*e)*exp(12*I*f*x) + 7938*I*f*exp(10*I*e)*exp(
10*I*f*x) + 7938*I*f*exp(8*I*e)*exp(8*I*f*x) + 5292*I*f*exp(6*I*e)*exp(6*I*f*x) + 2268*I*f*exp(4*I*e)*exp(4*I*
f*x) + 567*I*f*exp(2*I*e)*exp(2*I*f*x) + 63*I*f)

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