3.7.92 \(\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx\) [692]

Optimal. Leaf size=132 \[ \frac {a^3 (i A+B) c^4 (1-i \tan (e+f x))^4}{f}-\frac {4 a^3 (i A+2 B) c^4 (1-i \tan (e+f x))^5}{5 f}+\frac {a^3 (i A+5 B) c^4 (1-i \tan (e+f x))^6}{6 f}-\frac {a^3 B c^4 (1-i \tan (e+f x))^7}{7 f} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 132, 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 = {3669, 78} \begin {gather*} \frac {a^3 c^4 (5 B+i A) (1-i \tan (e+f x))^6}{6 f}-\frac {4 a^3 c^4 (2 B+i A) (1-i \tan (e+f x))^5}{5 f}+\frac {a^3 c^4 (B+i A) (1-i \tan (e+f x))^4}{f}-\frac {a^3 B c^4 (1-i \tan (e+f x))^7}{7 f} \end {gather*}

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])^4,x]

[Out]

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

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Mathematica [A]
time = 1.53, size = 172, normalized size = 1.30 \begin {gather*} \frac {a^3 c^4 \sec (e) \sec ^7(e+f x) (70 (-i A+B) \cos (f x)+70 (-i A+B) \cos (2 e+f x)+175 A \sin (f x)-35 i B \sin (f x)-70 A \sin (2 e+f x)-70 i B \sin (2 e+f x)+147 A \sin (2 e+3 f x)+21 i B \sin (2 e+3 f x)+49 A \sin (4 e+5 f x)+7 i B \sin (4 e+5 f x)+7 A \sin (6 e+7 f x)+i B \sin (6 e+7 f x))}{840 f} \end {gather*}

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])^4,x]

[Out]

(a^3*c^4*Sec[e]*Sec[e + f*x]^7*(70*((-I)*A + B)*Cos[f*x] + 70*((-I)*A + B)*Cos[2*e + f*x] + 175*A*Sin[f*x] - (
35*I)*B*Sin[f*x] - 70*A*Sin[2*e + f*x] - (70*I)*B*Sin[2*e + f*x] + 147*A*Sin[2*e + 3*f*x] + (21*I)*B*Sin[2*e +
3*f*x] + 49*A*Sin[4*e + 5*f*x] + (7*I)*B*Sin[4*e + 5*f*x] + 7*A*Sin[6*e + 7*f*x] + I*B*Sin[6*e + 7*f*x]))/(84
0*f)

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Maple [A]
time = 0.13, size = 159, normalized size = 1.20

method	result	size
risch	\(\frac {16 c^{4} a^{3} \left (105 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+105 B \,{\mathrm e}^{6 i \left (f x +e \right )}+147 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-21 B \,{\mathrm e}^{4 i \left (f x +e \right )}+49 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-7 B \,{\mathrm e}^{2 i \left (f x +e \right )}+7 i A -B \right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}\)	\(108\)
derivativedivides	\(-\frac {i c^{4} a^{3} \left (\frac {B \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {\left (i B +A \right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (-4 B -2 i A +3 i \left (-2 i B +A \right )\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (-4 A +3 i \left (-2 i A -B \right )+5 i B \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (3 i A +3 B -i \left (-2 i B +A \right )\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (3 A -i \left (-2 i A -B \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}+i A \tan \left (f x +e \right )\right )}{f}\)	\(159\)
default	\(-\frac {i c^{4} a^{3} \left (\frac {B \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {\left (i B +A \right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (-4 B -2 i A +3 i \left (-2 i B +A \right )\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (-4 A +3 i \left (-2 i A -B \right )+5 i B \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (3 i A +3 B -i \left (-2 i B +A \right )\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (3 A -i \left (-2 i A -B \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}+i A \tan \left (f x +e \right )\right )}{f}\)	\(159\)
norman	\(\frac {A \,a^{3} c^{4} \tan \left (f x +e \right )}{f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{2 f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6 f}+\frac {\left (-2 i B \,a^{3} c^{4}+A \,a^{3} c^{4}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}+\frac {\left (-i B \,a^{3} c^{4}+2 A \,a^{3} c^{4}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}-\frac {i B \,a^{3} c^{4} \left (\tan ^{7}\left (f x +e \right )\right )}{7 f}\)	\(201\)

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))^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.60, size = 158, normalized size = 1.20 \begin {gather*} -\frac {30 i \, B a^{3} c^{4} \tan \left (f x + e\right )^{7} + 35 \, {\left (i \, A - B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{6} - 42 \, {\left (A - 2 i \, B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{5} + 105 \, {\left (i \, A - B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{4} - 70 \, {\left (2 \, A - i \, B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{3} + 105 \, {\left (i \, A - B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{2} - 210 \, A a^{3} c^{4} \tan \left (f x + e\right )}{210 \, f} \end {gather*}

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))^4,x, algorithm="maxima")

[Out]

-1/210*(30*I*B*a^3*c^4*tan(f*x + e)^7 + 35*(I*A - B)*a^3*c^4*tan(f*x + e)^6 - 42*(A - 2*I*B)*a^3*c^4*tan(f*x +
e)^5 + 105*(I*A - B)*a^3*c^4*tan(f*x + e)^4 - 70*(2*A - I*B)*a^3*c^4*tan(f*x + e)^3 + 105*(I*A - B)*a^3*c^4*t
an(f*x + e)^2 - 210*A*a^3*c^4*tan(f*x + e))/f

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Fricas [A]
time = 4.03, size = 180, normalized size = 1.36 \begin {gather*} -\frac {16 \, {\left (105 \, {\left (-i \, A - B\right )} a^{3} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, {\left (-7 i \, A + B\right )} a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, {\left (-7 i \, A + B\right )} a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-7 i \, A + B\right )} a^{3} c^{4}\right )}}{105 \, {\left (f e^{\left (14 i \, f x + 14 i \, e\right )} + 7 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 21 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 35 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 35 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}

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))^4,x, algorithm="fricas")

[Out]

-16/105*(105*(-I*A - B)*a^3*c^4*e^(6*I*f*x + 6*I*e) + 21*(-7*I*A + B)*a^3*c^4*e^(4*I*f*x + 4*I*e) + 7*(-7*I*A
+ B)*a^3*c^4*e^(2*I*f*x + 2*I*e) + (-7*I*A + B)*a^3*c^4)/(f*e^(14*I*f*x + 14*I*e) + 7*f*e^(12*I*f*x + 12*I*e)
+ 21*f*e^(10*I*f*x + 10*I*e) + 35*f*e^(8*I*f*x + 8*I*e) + 35*f*e^(6*I*f*x + 6*I*e) + 21*f*e^(4*I*f*x + 4*I*e)
+ 7*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (107) = 214\).
time = 0.82, size = 287, normalized size = 2.17 \begin {gather*} \frac {112 i A a^{3} c^{4} - 16 B a^{3} c^{4} + \left (784 i A a^{3} c^{4} e^{2 i e} - 112 B a^{3} c^{4} e^{2 i e}\right ) e^{2 i f x} + \left (2352 i A a^{3} c^{4} e^{4 i e} - 336 B a^{3} c^{4} e^{4 i e}\right ) e^{4 i f x} + \left (1680 i A a^{3} c^{4} e^{6 i e} + 1680 B a^{3} c^{4} e^{6 i e}\right ) e^{6 i f x}}{105 f e^{14 i e} e^{14 i f x} + 735 f e^{12 i e} e^{12 i f x} + 2205 f e^{10 i e} e^{10 i f x} + 3675 f e^{8 i e} e^{8 i f x} + 3675 f e^{6 i e} e^{6 i f x} + 2205 f e^{4 i e} e^{4 i f x} + 735 f e^{2 i e} e^{2 i f x} + 105 f} \end {gather*}

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))**4,x)

[Out]

(112*I*A*a**3*c**4 - 16*B*a**3*c**4 + (784*I*A*a**3*c**4*exp(2*I*e) - 112*B*a**3*c**4*exp(2*I*e))*exp(2*I*f*x)
+ (2352*I*A*a**3*c**4*exp(4*I*e) - 336*B*a**3*c**4*exp(4*I*e))*exp(4*I*f*x) + (1680*I*A*a**3*c**4*exp(6*I*e)
+ 1680*B*a**3*c**4*exp(6*I*e))*exp(6*I*f*x))/(105*f*exp(14*I*e)*exp(14*I*f*x) + 735*f*exp(12*I*e)*exp(12*I*f*x
) + 2205*f*exp(10*I*e)*exp(10*I*f*x) + 3675*f*exp(8*I*e)*exp(8*I*f*x) + 3675*f*exp(6*I*e)*exp(6*I*f*x) + 2205*
f*exp(4*I*e)*exp(4*I*f*x) + 735*f*exp(2*I*e)*exp(2*I*f*x) + 105*f)

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Giac [A]
time = 1.10, size = 228, normalized size = 1.73 \begin {gather*} -\frac {16 \, {\left (-105 i \, A a^{3} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} - 105 \, B a^{3} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} - 147 i \, A a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 21 \, B a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 49 i \, A a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 7 \, B a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 7 i \, A a^{3} c^{4} + B a^{3} c^{4}\right )}}{105 \, {\left (f e^{\left (14 i \, f x + 14 i \, e\right )} + 7 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 21 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 35 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 35 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}

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))^4,x, algorithm="giac")

[Out]

-16/105*(-105*I*A*a^3*c^4*e^(6*I*f*x + 6*I*e) - 105*B*a^3*c^4*e^(6*I*f*x + 6*I*e) - 147*I*A*a^3*c^4*e^(4*I*f*x
+ 4*I*e) + 21*B*a^3*c^4*e^(4*I*f*x + 4*I*e) - 49*I*A*a^3*c^4*e^(2*I*f*x + 2*I*e) + 7*B*a^3*c^4*e^(2*I*f*x + 2
*I*e) - 7*I*A*a^3*c^4 + B*a^3*c^4)/(f*e^(14*I*f*x + 14*I*e) + 7*f*e^(12*I*f*x + 12*I*e) + 21*f*e^(10*I*f*x + 1
0*I*e) + 35*f*e^(8*I*f*x + 8*I*e) + 35*f*e^(6*I*f*x + 6*I*e) + 21*f*e^(4*I*f*x + 4*I*e) + 7*f*e^(2*I*f*x + 2*I
*e) + f)

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Mupad [B]
time = 8.59, size = 156, normalized size = 1.18 \begin {gather*} -\frac {\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (2\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}-A\,a^3\,c^4\,\mathrm {tan}\left (e+f\,x\right )+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (B+A\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6}+\frac {B\,a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^7\,1{}\mathrm {i}}{7}}{f} \end {gather*}

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)^4,x)

[Out]

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

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