3.704 \(\int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx\)

Optimal. Leaf size=127 \[ \frac {a^3 (5 B+i A)}{6 c^8 f (\tan (e+f x)+i)^6}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (\tan (e+f x)+i)^7}-\frac {a^3 (B+i A)}{2 c^8 f (\tan (e+f x)+i)^8}+\frac {i a^3 B}{5 c^8 f (\tan (e+f x)+i)^5} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 127, 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} \[ \frac {a^3 (5 B+i A)}{6 c^8 f (\tan (e+f x)+i)^6}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (\tan (e+f x)+i)^7}-\frac {a^3 (B+i A)}{2 c^8 f (\tan (e+f x)+i)^8}+\frac {i a^3 B}{5 c^8 f (\tan (e+f x)+i)^5} \]

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])^8,x]

[Out]

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

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

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Mathematica [A]  time = 10.68, size = 182, normalized size = 1.43 \[ -\frac {i a^3 (\cos (11 e+14 f x)+i \sin (11 e+14 f x)) (56 (55 A+i B) \cos (e+f x)+30 (55 A+9 i B) \cos (3 (e+f x))-280 i A \sin (e+f x)-450 i A \sin (3 (e+f x))-175 i A \sin (5 (e+f x))+385 A \cos (5 (e+f x))+616 B \sin (e+f x)+990 B \sin (3 (e+f x))+385 B \sin (5 (e+f x))+175 i B \cos (5 (e+f x)))}{53760 c^8 f (\cos (f x)+i \sin (f x))^3} \]

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])^8,x]

[Out]

((-1/53760*I)*a^3*(56*(55*A + I*B)*Cos[e + f*x] + 30*(55*A + (9*I)*B)*Cos[3*(e + f*x)] + 385*A*Cos[5*(e + f*x)
] + (175*I)*B*Cos[5*(e + f*x)] - (280*I)*A*Sin[e + f*x] + 616*B*Sin[e + f*x] - (450*I)*A*Sin[3*(e + f*x)] + 99
0*B*Sin[3*(e + f*x)] - (175*I)*A*Sin[5*(e + f*x)] + 385*B*Sin[5*(e + f*x)])*(Cos[11*e + 14*f*x] + I*Sin[11*e +
14*f*x]))/(c^8*f*(Cos[f*x] + I*Sin[f*x])^3)

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fricas [A]  time = 1.67, size = 129, normalized size = 1.02 \[ \frac {{\left (-105 i \, A - 105 \, B\right )} a^{3} e^{\left (16 i \, f x + 16 i \, e\right )} + {\left (-600 i \, A - 360 \, B\right )} a^{3} e^{\left (14 i \, f x + 14 i \, e\right )} + {\left (-1400 i \, A - 280 \, B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} + {\left (-1680 i \, A + 336 \, B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} + {\left (-1050 i \, A + 630 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-280 i \, A + 280 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{53760 \, c^{8} f} \]

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))^8,x, algorithm="fricas")

[Out]

1/53760*((-105*I*A - 105*B)*a^3*e^(16*I*f*x + 16*I*e) + (-600*I*A - 360*B)*a^3*e^(14*I*f*x + 14*I*e) + (-1400*
I*A - 280*B)*a^3*e^(12*I*f*x + 12*I*e) + (-1680*I*A + 336*B)*a^3*e^(10*I*f*x + 10*I*e) + (-1050*I*A + 630*B)*a
^3*e^(8*I*f*x + 8*I*e) + (-280*I*A + 280*B)*a^3*e^(6*I*f*x + 6*I*e))/(c^8*f)

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giac [B]  time = 9.60, size = 525, normalized size = 4.13 \[ -\frac {2 \, {\left (105 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{15} + 525 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 2975 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 140 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 8750 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} + 1190 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} + 22365 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 1596 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 39235 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 4711 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 58075 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 4600 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 63300 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 7380 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 58075 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 4600 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 39235 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 4711 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 22365 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 1596 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 8750 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1190 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2975 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 140 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 525 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{105 \, c^{8} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{16}} \]

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))^8,x, algorithm="giac")

[Out]

-2/105*(105*A*a^3*tan(1/2*f*x + 1/2*e)^15 + 525*I*A*a^3*tan(1/2*f*x + 1/2*e)^14 - 105*B*a^3*tan(1/2*f*x + 1/2*
e)^14 - 2975*A*a^3*tan(1/2*f*x + 1/2*e)^13 - 140*I*B*a^3*tan(1/2*f*x + 1/2*e)^13 - 8750*I*A*a^3*tan(1/2*f*x +
1/2*e)^12 + 1190*B*a^3*tan(1/2*f*x + 1/2*e)^12 + 22365*A*a^3*tan(1/2*f*x + 1/2*e)^11 + 1596*I*B*a^3*tan(1/2*f*
x + 1/2*e)^11 + 39235*I*A*a^3*tan(1/2*f*x + 1/2*e)^10 - 4711*B*a^3*tan(1/2*f*x + 1/2*e)^10 - 58075*A*a^3*tan(1
/2*f*x + 1/2*e)^9 - 4600*I*B*a^3*tan(1/2*f*x + 1/2*e)^9 - 63300*I*A*a^3*tan(1/2*f*x + 1/2*e)^8 + 7380*B*a^3*ta
n(1/2*f*x + 1/2*e)^8 + 58075*A*a^3*tan(1/2*f*x + 1/2*e)^7 + 4600*I*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 39235*I*A*a^
3*tan(1/2*f*x + 1/2*e)^6 - 4711*B*a^3*tan(1/2*f*x + 1/2*e)^6 - 22365*A*a^3*tan(1/2*f*x + 1/2*e)^5 - 1596*I*B*a
^3*tan(1/2*f*x + 1/2*e)^5 - 8750*I*A*a^3*tan(1/2*f*x + 1/2*e)^4 + 1190*B*a^3*tan(1/2*f*x + 1/2*e)^4 + 2975*A*a
^3*tan(1/2*f*x + 1/2*e)^3 + 140*I*B*a^3*tan(1/2*f*x + 1/2*e)^3 + 525*I*A*a^3*tan(1/2*f*x + 1/2*e)^2 - 105*B*a^
3*tan(1/2*f*x + 1/2*e)^2 - 105*A*a^3*tan(1/2*f*x + 1/2*e))/(c^8*f*(tan(1/2*f*x + 1/2*e) + I)^16)

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maple [A]  time = 0.24, size = 90, normalized size = 0.71 \[ \frac {a^{3} \left (-\frac {4 i A +4 B}{8 \left (\tan \left (f x +e \right )+i\right )^{8}}-\frac {8 i B -4 A}{7 \left (\tan \left (f x +e \right )+i\right )^{7}}-\frac {-i A -5 B}{6 \left (\tan \left (f x +e \right )+i\right )^{6}}+\frac {i B}{5 \left (\tan \left (f x +e \right )+i\right )^{5}}\right )}{f \,c^{8}} \]

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))^8,x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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))^8,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 9.47, size = 160, normalized size = 1.26 \[ \frac {-\frac {a^3\,\left (-2\,B+A\,20{}\mathrm {i}\right )}{210}+\frac {a^3\,\mathrm {tan}\left (e+f\,x\right )\,\left (50\,A-B\,16{}\mathrm {i}\right )}{210}+\frac {B\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{5}+\frac {a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (49\,B+A\,35{}\mathrm {i}\right )}{210}}{c^8\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,8{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^6-{\mathrm {tan}\left (e+f\,x\right )}^5\,56{}\mathrm {i}+70\,{\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,56{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,8{}\mathrm {i}+1\right )} \]

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)^8,x)

[Out]

((a^3*tan(e + f*x)*(50*A - B*16i))/210 - (a^3*(A*20i - 2*B))/210 + (B*a^3*tan(e + f*x)^3*1i)/5 + (a^3*tan(e +
f*x)^2*(A*35i + 49*B))/210)/(c^8*f*(tan(e + f*x)^3*56i - 28*tan(e + f*x)^2 - tan(e + f*x)*8i + 70*tan(e + f*x)
^4 - tan(e + f*x)^5*56i - 28*tan(e + f*x)^6 + tan(e + f*x)^7*8i + tan(e + f*x)^8 + 1))

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sympy [A]  time = 1.72, size = 498, normalized size = 3.92 \[ \begin {cases} \frac {\left (- 1803886264320 i A a^{3} c^{40} f^{5} e^{6 i e} + 1803886264320 B a^{3} c^{40} f^{5} e^{6 i e}\right ) e^{6 i f x} + \left (- 6764573491200 i A a^{3} c^{40} f^{5} e^{8 i e} + 4058744094720 B a^{3} c^{40} f^{5} e^{8 i e}\right ) e^{8 i f x} + \left (- 10823317585920 i A a^{3} c^{40} f^{5} e^{10 i e} + 2164663517184 B a^{3} c^{40} f^{5} e^{10 i e}\right ) e^{10 i f x} + \left (- 9019431321600 i A a^{3} c^{40} f^{5} e^{12 i e} - 1803886264320 B a^{3} c^{40} f^{5} e^{12 i e}\right ) e^{12 i f x} + \left (- 3865470566400 i A a^{3} c^{40} f^{5} e^{14 i e} - 2319282339840 B a^{3} c^{40} f^{5} e^{14 i e}\right ) e^{14 i f x} + \left (- 676457349120 i A a^{3} c^{40} f^{5} e^{16 i e} - 676457349120 B a^{3} c^{40} f^{5} e^{16 i e}\right ) e^{16 i f x}}{346346162749440 c^{48} f^{6}} & \text {for}\: 346346162749440 c^{48} f^{6} \neq 0 \\\frac {x \left (A a^{3} e^{16 i e} + 5 A a^{3} e^{14 i e} + 10 A a^{3} e^{12 i e} + 10 A a^{3} e^{10 i e} + 5 A a^{3} e^{8 i e} + A a^{3} e^{6 i e} - i B a^{3} e^{16 i e} - 3 i B a^{3} e^{14 i e} - 2 i B a^{3} e^{12 i e} + 2 i B a^{3} e^{10 i e} + 3 i B a^{3} e^{8 i e} + i B a^{3} e^{6 i e}\right )}{32 c^{8}} & \text {otherwise} \end {cases} \]

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))**8,x)

[Out]

Piecewise((((-1803886264320*I*A*a**3*c**40*f**5*exp(6*I*e) + 1803886264320*B*a**3*c**40*f**5*exp(6*I*e))*exp(6
*I*f*x) + (-6764573491200*I*A*a**3*c**40*f**5*exp(8*I*e) + 4058744094720*B*a**3*c**40*f**5*exp(8*I*e))*exp(8*I
*f*x) + (-10823317585920*I*A*a**3*c**40*f**5*exp(10*I*e) + 2164663517184*B*a**3*c**40*f**5*exp(10*I*e))*exp(10
*I*f*x) + (-9019431321600*I*A*a**3*c**40*f**5*exp(12*I*e) - 1803886264320*B*a**3*c**40*f**5*exp(12*I*e))*exp(1
2*I*f*x) + (-3865470566400*I*A*a**3*c**40*f**5*exp(14*I*e) - 2319282339840*B*a**3*c**40*f**5*exp(14*I*e))*exp(
14*I*f*x) + (-676457349120*I*A*a**3*c**40*f**5*exp(16*I*e) - 676457349120*B*a**3*c**40*f**5*exp(16*I*e))*exp(1
6*I*f*x))/(346346162749440*c**48*f**6), Ne(346346162749440*c**48*f**6, 0)), (x*(A*a**3*exp(16*I*e) + 5*A*a**3*
exp(14*I*e) + 10*A*a**3*exp(12*I*e) + 10*A*a**3*exp(10*I*e) + 5*A*a**3*exp(8*I*e) + A*a**3*exp(6*I*e) - I*B*a*
*3*exp(16*I*e) - 3*I*B*a**3*exp(14*I*e) - 2*I*B*a**3*exp(12*I*e) + 2*I*B*a**3*exp(10*I*e) + 3*I*B*a**3*exp(8*I
*e) + I*B*a**3*exp(6*I*e))/(32*c**8), True))

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