3.7.77 \(\int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx\) [677]

Optimal. Leaf size=99 \[ \frac {2 a^2 (i A+B) c^5 (1-i \tan (e+f x))^5}{5 f}-\frac {a^2 (i A+3 B) c^5 (1-i \tan (e+f x))^6}{6 f}+\frac {a^2 B c^5 (1-i \tan (e+f x))^7}{7 f} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 99, 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^2 c^5 (3 B+i A) (1-i \tan (e+f x))^6}{6 f}+\frac {2 a^2 c^5 (B+i A) (1-i \tan (e+f x))^5}{5 f}+\frac {a^2 B c^5 (1-i \tan (e+f x))^7}{7 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5,x]

[Out]

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(254\) vs. \(2(99)=198\).
time = 1.90, size = 254, normalized size = 2.57 \begin {gather*} \frac {a^2 c^5 \sec (e) \sec ^7(e+f x) (35 (-7 i A+3 B) \cos (f x)+35 (-7 i A+3 B) \cos (2 e+f x)-105 i A \cos (2 e+3 f x)+105 B \cos (2 e+3 f x)-105 i A \cos (4 e+3 f x)+105 B \cos (4 e+3 f x)+245 A \sin (f x)+105 i B \sin (f x)-245 A \sin (2 e+f x)-105 i B \sin (2 e+f x)+189 A \sin (2 e+3 f x)+21 i B \sin (2 e+3 f x)-105 A \sin (4 e+3 f x)-105 i B \sin (4 e+3 f x)+98 A \sin (4 e+5 f x)+42 i B \sin (4 e+5 f x)+14 A \sin (6 e+7 f x)+6 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])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5,x]

[Out]

(a^2*c^5*Sec[e]*Sec[e + f*x]^7*(35*((-7*I)*A + 3*B)*Cos[f*x] + 35*((-7*I)*A + 3*B)*Cos[2*e + f*x] - (105*I)*A*
Cos[2*e + 3*f*x] + 105*B*Cos[2*e + 3*f*x] - (105*I)*A*Cos[4*e + 3*f*x] + 105*B*Cos[4*e + 3*f*x] + 245*A*Sin[f*
x] + (105*I)*B*Sin[f*x] - 245*A*Sin[2*e + f*x] - (105*I)*B*Sin[2*e + f*x] + 189*A*Sin[2*e + 3*f*x] + (21*I)*B*
Sin[2*e + 3*f*x] - 105*A*Sin[4*e + 3*f*x] - (105*I)*B*Sin[4*e + 3*f*x] + 98*A*Sin[4*e + 5*f*x] + (42*I)*B*Sin[
4*e + 5*f*x] + 14*A*Sin[6*e + 7*f*x] + (6*I)*B*Sin[6*e + 7*f*x]))/(840*f)

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Maple [A]
time = 0.14, size = 147, normalized size = 1.48

method	result	size
risch	\(\frac {32 c^{5} a^{2} \left (42 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+42 B \,{\mathrm e}^{4 i \left (f x +e \right )}+49 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-21 B \,{\mathrm e}^{2 i \left (f x +e \right )}+7 i A -3 B \right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}\)	\(83\)
derivativedivides	\(-\frac {i c^{5} a^{2} \left (-\frac {B \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {\left (-3 i B -A \right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (i A +4 i \left (i B -A \right )+6 B \right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (-2 i B +2 A \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (-6 i A -4 i \left (i B -A \right )-B \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (i B +3 A \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}+i A \tan \left (f x +e \right )\right )}{f}\)	\(147\)
default	\(-\frac {i c^{5} a^{2} \left (-\frac {B \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {\left (-3 i B -A \right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (i A +4 i \left (i B -A \right )+6 B \right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (-2 i B +2 A \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (-6 i A -4 i \left (i B -A \right )-B \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (i B +3 A \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}+i A \tan \left (f x +e \right )\right )}{f}\)	\(147\)
norman	\(\frac {A \,a^{2} c^{5} \tan \left (f x +e \right )}{f}-\frac {\left (-i A \,a^{2} c^{5}+3 B \,a^{2} c^{5}\right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6 f}-\frac {\left (2 i B \,a^{2} c^{5}+3 A \,a^{2} c^{5}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}-\frac {\left (3 i B \,a^{2} c^{5}+2 A \,a^{2} c^{5}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {\left (-3 i A \,a^{2} c^{5}+B \,a^{2} c^{5}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}-\frac {\left (i A \,a^{2} c^{5}+B \,a^{2} c^{5}\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{2 f}+\frac {i B \,a^{2} c^{5} \left (\tan ^{7}\left (f x +e \right )\right )}{7 f}\)	\(203\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 158, normalized size = 1.60 \begin {gather*} \frac {30 i \, B a^{2} c^{5} \tan \left (f x + e\right )^{7} + 35 \, {\left (i \, A - 3 \, B\right )} a^{2} c^{5} \tan \left (f x + e\right )^{6} - 42 \, {\left (3 \, A + 2 i \, B\right )} a^{2} c^{5} \tan \left (f x + e\right )^{5} + 105 \, {\left (-i \, A - B\right )} a^{2} c^{5} \tan \left (f x + e\right )^{4} - 70 \, {\left (2 \, A + 3 i \, B\right )} a^{2} c^{5} \tan \left (f x + e\right )^{3} + 105 \, {\left (-3 i \, A + B\right )} a^{2} c^{5} \tan \left (f x + e\right )^{2} + 210 \, A a^{2} c^{5} \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))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.44, size = 161, normalized size = 1.63 \begin {gather*} -\frac {32 \, {\left (42 \, {\left (-i \, A - B\right )} a^{2} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, {\left (-7 i \, A + 3 \, B\right )} a^{2} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-7 i \, A + 3 \, B\right )} a^{2} c^{5}\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))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5,x, algorithm="fricas")

[Out]

-32/105*(42*(-I*A - B)*a^2*c^5*e^(4*I*f*x + 4*I*e) + 7*(-7*I*A + 3*B)*a^2*c^5*e^(2*I*f*x + 2*I*e) + (-7*I*A +
3*B)*a^2*c^5)/(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. 243 vs. \(2 (80) = 160\).
time = 0.77, size = 243, normalized size = 2.45 \begin {gather*} \frac {224 i A a^{2} c^{5} - 96 B a^{2} c^{5} + \left (1568 i A a^{2} c^{5} e^{2 i e} - 672 B a^{2} c^{5} e^{2 i e}\right ) e^{2 i f x} + \left (1344 i A a^{2} c^{5} e^{4 i e} + 1344 B a^{2} c^{5} e^{4 i e}\right ) e^{4 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))**2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**5,x)

[Out]

(224*I*A*a**2*c**5 - 96*B*a**2*c**5 + (1568*I*A*a**2*c**5*exp(2*I*e) - 672*B*a**2*c**5*exp(2*I*e))*exp(2*I*f*x
) + (1344*I*A*a**2*c**5*exp(4*I*e) + 1344*B*a**2*c**5*exp(4*I*e))*exp(4*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) + 367
5*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 [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (86) = 172\).
time = 1.19, size = 191, normalized size = 1.93 \begin {gather*} -\frac {32 \, {\left (-42 i \, A a^{2} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} - 42 \, B a^{2} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} - 49 i \, A a^{2} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} + 21 \, B a^{2} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} - 7 i \, A a^{2} c^{5} + 3 \, B a^{2} c^{5}\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))^2*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5,x, algorithm="giac")

[Out]

-32/105*(-42*I*A*a^2*c^5*e^(4*I*f*x + 4*I*e) - 42*B*a^2*c^5*e^(4*I*f*x + 4*I*e) - 49*I*A*a^2*c^5*e^(2*I*f*x +
2*I*e) + 21*B*a^2*c^5*e^(2*I*f*x + 2*I*e) - 7*I*A*a^2*c^5 + 3*B*a^2*c^5)/(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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Mupad [B]
time = 8.68, size = 158, normalized size = 1.60 \begin {gather*} \frac {A\,a^2\,c^5\,\mathrm {tan}\left (e+f\,x\right )+\frac {a^2\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (-3\,B+A\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {a^2\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (-2\,B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}-\frac {a^2\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (3\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\frac {a^2\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^2\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (A+B\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{6}+\frac {B\,a^2\,c^5\,{\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)^2*(c - c*tan(e + f*x)*1i)^5,x)

[Out]

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

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