\(\int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx\) [704]

Optimal result

Integrand size = 41, antiderivative size = 127 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=-\frac {a^3 (i A+B)}{2 c^8 f (i+\tan (e+f x))^8}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (i+\tan (e+f x))^7}+\frac {a^3 (i A+5 B)}{6 c^8 f (i+\tan (e+f x))^6}+\frac {i a^3 B}{5 c^8 f (i+\tan (e+f x))^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 127, 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} \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\frac {a^3 (5 B+i A)}{6 c^8 f (\tan (e+f x)+i)^6}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (\tan (e+f x)+i)^7}-\frac {a^3 (B+i A)}{2 c^8 f (\tan (e+f x)+i)^8}+\frac {i a^3 B}{5 c^8 f (\tan (e+f x)+i)^5} \]

[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]

-1/2*(a^3*(I*A + B))/(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 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 \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) \text {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*}

Mathematica [A] (verified)

Time = 5.55 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.66 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\frac {a^3 \left (2 (-10 i A+B)+2 (25 A-8 i B) \tan (e+f x)+7 (5 i A+7 B) \tan ^2(e+f x)+42 i B \tan ^3(e+f x)\right )}{210 c^8 f (i+\tan (e+f x))^8} \]

[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]

(a^3*(2*((-10*I)*A + B) + 2*(25*A - (8*I)*B)*Tan[e + f*x] + 7*((5*I)*A + 7*B)*Tan[e + f*x]^2 + (42*I)*B*Tan[e
+ f*x]^3))/(210*c^8*f*(I + Tan[e + f*x])^8)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {8 i B -4 A}{7 \left (i+\tan \left (f x +e \right )\right )^{7}}-\frac {-i A -5 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}+\frac {i B}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {4 i A +4 B}{8 \left (i+\tan \left (f x +e \right )\right )^{8}}\right )}{f \,c^{8}}\)	\(90\)
default	\(\frac {a^{3} \left (-\frac {8 i B -4 A}{7 \left (i+\tan \left (f x +e \right )\right )^{7}}-\frac {-i A -5 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}+\frac {i B}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {4 i A +4 B}{8 \left (i+\tan \left (f x +e \right )\right )^{8}}\right )}{f \,c^{8}}\)	\(90\)
risch	\(-\frac {a^{3} {\mathrm e}^{16 i \left (f x +e \right )} B}{512 c^{8} f}-\frac {i a^{3} {\mathrm e}^{16 i \left (f x +e \right )} A}{512 c^{8} f}-\frac {3 \,{\mathrm e}^{14 i \left (f x +e \right )} B \,a^{3}}{448 c^{8} f}-\frac {5 i {\mathrm e}^{14 i \left (f x +e \right )} a^{3} A}{448 c^{8} f}-\frac {{\mathrm e}^{12 i \left (f x +e \right )} B \,a^{3}}{192 c^{8} f}-\frac {5 i {\mathrm e}^{12 i \left (f x +e \right )} a^{3} A}{192 c^{8} f}+\frac {{\mathrm e}^{10 i \left (f x +e \right )} B \,a^{3}}{160 c^{8} f}-\frac {i {\mathrm e}^{10 i \left (f x +e \right )} a^{3} A}{32 c^{8} f}+\frac {3 \,{\mathrm e}^{8 i \left (f x +e \right )} B \,a^{3}}{256 c^{8} f}-\frac {5 i {\mathrm e}^{8 i \left (f x +e \right )} a^{3} A}{256 c^{8} f}+\frac {a^{3} {\mathrm e}^{6 i \left (f x +e \right )} B}{192 c^{8} f}-\frac {i a^{3} {\mathrm e}^{6 i \left (f x +e \right )} A}{192 c^{8} f}\)	\(260\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=-\frac {105 \, {\left (i \, A + B\right )} a^{3} e^{\left (16 i \, f x + 16 i \, e\right )} + 120 \, {\left (5 i \, A + 3 \, B\right )} a^{3} e^{\left (14 i \, f x + 14 i \, e\right )} + 280 \, {\left (5 i \, A + B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} + 336 \, {\left (5 i \, A - B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} + 210 \, {\left (5 i \, A - 3 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 280 \, {\left (i \, A - B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{53760 \, c^{8} f} \]

[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 + B)*a^3*e^(16*I*f*x + 16*I*e) + 120*(5*I*A + 3*B)*a^3*e^(14*I*f*x + 14*I*e) + 280*(5*I*A +
B)*a^3*e^(12*I*f*x + 12*I*e) + 336*(5*I*A - B)*a^3*e^(10*I*f*x + 10*I*e) + 210*(5*I*A - 3*B)*a^3*e^(8*I*f*x +
8*I*e) + 280*(I*A - B)*a^3*e^(6*I*f*x + 6*I*e))/(c^8*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. 496 vs. \(2 (104) = 208\).

Time = 0.72 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.91 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\begin {cases} \frac {\left (- 1803886264320 i A a^{3} c^{40} f^{5} e^{6 i e} + 1803886264320 B a^{3} c^{40} f^{5} e^{6 i e}\right ) e^{6 i f x} + \left (- 6764573491200 i A a^{3} c^{40} f^{5} e^{8 i e} + 4058744094720 B a^{3} c^{40} f^{5} e^{8 i e}\right ) e^{8 i f x} + \left (- 10823317585920 i A a^{3} c^{40} f^{5} e^{10 i e} + 2164663517184 B a^{3} c^{40} f^{5} e^{10 i e}\right ) e^{10 i f x} + \left (- 9019431321600 i A a^{3} c^{40} f^{5} e^{12 i e} - 1803886264320 B a^{3} c^{40} f^{5} e^{12 i e}\right ) e^{12 i f x} + \left (- 3865470566400 i A a^{3} c^{40} f^{5} e^{14 i e} - 2319282339840 B a^{3} c^{40} f^{5} e^{14 i e}\right ) e^{14 i f x} + \left (- 676457349120 i A a^{3} c^{40} f^{5} e^{16 i e} - 676457349120 B a^{3} c^{40} f^{5} e^{16 i e}\right ) e^{16 i f x}}{346346162749440 c^{48} f^{6}} & \text {for}\: c^{48} f^{6} \neq 0 \\\frac {x \left (A a^{3} e^{16 i e} + 5 A a^{3} e^{14 i e} + 10 A a^{3} e^{12 i e} + 10 A a^{3} e^{10 i e} + 5 A a^{3} e^{8 i e} + A a^{3} e^{6 i e} - i B a^{3} e^{16 i e} - 3 i B a^{3} e^{14 i e} - 2 i B a^{3} e^{12 i e} + 2 i B a^{3} e^{10 i e} + 3 i B a^{3} e^{8 i e} + i B a^{3} e^{6 i e}\right )}{32 c^{8}} & \text {otherwise} \end {cases} \]

[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(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*e
xp(6*I*e))/(32*c**8), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\text {Exception raised: RuntimeError} \]

[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: expt: undefined: 0 to a negative exponent.

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. 496 vs. \(2 (103) = 206\).

Time = 1.06 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.91 \[ \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 {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}} \]

[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)

Mupad [B] (verification not implemented)

Time = 9.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.26 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\frac {-\frac {a^3\,\left (-2\,B+A\,20{}\mathrm {i}\right )}{210}+\frac {a^3\,\mathrm {tan}\left (e+f\,x\right )\,\left (50\,A-B\,16{}\mathrm {i}\right )}{210}+\frac {B\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{5}+\frac {a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (49\,B+A\,35{}\mathrm {i}\right )}{210}}{c^8\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,8{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^6-{\mathrm {tan}\left (e+f\,x\right )}^5\,56{}\mathrm {i}+70\,{\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,56{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,8{}\mathrm {i}+1\right )} \]

[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))