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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.176473, antiderivative size = 127, normalized size of antiderivative = 1., 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} \[ -\frac{a^3 (5 B+i A)}{4 c^6 f (\tan (e+f x)+i)^4}-\frac{4 a^3 (A-2 i B)}{5 c^6 f (\tan (e+f x)+i)^5}+\frac{2 a^3 (B+i A)}{3 c^6 f (\tan (e+f x)+i)^6}-\frac{i a^3 B}{3 c^6 f (\tan (e+f x)+i)^3} \]

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

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]

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]))))

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

Mathematica [A]  time = 6.05873, size = 112, normalized size = 0.88 \[ \frac{a^3 (\cos (9 e+12 f x)+i \sin (9 e+12 f x)) (-(A+3 i B) (9 \sin (e+f x)+10 \sin (3 (e+f x)))+3 (B-27 i A) \cos (e+f x)+10 (B-3 i A) \cos (3 (e+f x)))}{960 c^6 f (\cos (f x)+i \sin (f x))^3} \]

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])^6,x]

[Out]

(a^3*(3*((-27*I)*A + B)*Cos[e + f*x] + 10*((-3*I)*A + B)*Cos[3*(e + f*x)] - (A + (3*I)*B)*(9*Sin[e + f*x] + 10
*Sin[3*(e + f*x)]))*(Cos[9*e + 12*f*x] + I*Sin[9*e + 12*f*x]))/(960*c^6*f*(Cos[f*x] + I*Sin[f*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 90, normalized size = 0.7 \begin{align*}{\frac{{a}^{3}}{f{c}^{6}} \left ({\frac{-{\frac{i}{3}}B}{ \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}-{\frac{-4\,iA-4\,B}{6\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{6}}}-{\frac{iA+5\,B}{4\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-{\frac{-8\,iB+4\,A}{5\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}} \right ) } \end{align*}

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.38637, size = 258, normalized size = 2.03 \begin{align*} \frac{{\left (-10 i \, A - 10 \, B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-36 i \, A - 12 \, B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-45 i \, A + 15 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-20 i \, A + 20 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{960 \, c^{6} f} \end{align*}

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]

1/960*((-10*I*A - 10*B)*a^3*e^(12*I*f*x + 12*I*e) + (-36*I*A - 12*B)*a^3*e^(10*I*f*x + 10*I*e) + (-45*I*A + 15
*B)*a^3*e^(8*I*f*x + 8*I*e) + (-20*I*A + 20*B)*a^3*e^(6*I*f*x + 6*I*e))/(c^6*f)

________________________________________________________________________________________

Sympy [A]  time = 3.60087, size = 333, normalized size = 2.62 \begin{align*} \begin{cases} \frac{\left (- 491520 i A a^{3} c^{18} f^{3} e^{6 i e} + 491520 B a^{3} c^{18} f^{3} e^{6 i e}\right ) e^{6 i f x} + \left (- 1105920 i A a^{3} c^{18} f^{3} e^{8 i e} + 368640 B a^{3} c^{18} f^{3} e^{8 i e}\right ) e^{8 i f x} + \left (- 884736 i A a^{3} c^{18} f^{3} e^{10 i e} - 294912 B a^{3} c^{18} f^{3} e^{10 i e}\right ) e^{10 i f x} + \left (- 245760 i A a^{3} c^{18} f^{3} e^{12 i e} - 245760 B a^{3} c^{18} f^{3} e^{12 i e}\right ) e^{12 i f x}}{23592960 c^{24} f^{4}} & \text{for}\: 23592960 c^{24} f^{4} \neq 0 \\\frac{x \left (A a^{3} e^{12 i e} + 3 A a^{3} e^{10 i e} + 3 A a^{3} e^{8 i e} + A a^{3} e^{6 i e} - i B a^{3} e^{12 i e} - i B a^{3} e^{10 i e} + i B a^{3} e^{8 i e} + i B a^{3} e^{6 i e}\right )}{8 c^{6}} & \text{otherwise} \end{cases} \end{align*}

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]

Piecewise((((-491520*I*A*a**3*c**18*f**3*exp(6*I*e) + 491520*B*a**3*c**18*f**3*exp(6*I*e))*exp(6*I*f*x) + (-11
05920*I*A*a**3*c**18*f**3*exp(8*I*e) + 368640*B*a**3*c**18*f**3*exp(8*I*e))*exp(8*I*f*x) + (-884736*I*A*a**3*c
**18*f**3*exp(10*I*e) - 294912*B*a**3*c**18*f**3*exp(10*I*e))*exp(10*I*f*x) + (-245760*I*A*a**3*c**18*f**3*exp
(12*I*e) - 245760*B*a**3*c**18*f**3*exp(12*I*e))*exp(12*I*f*x))/(23592960*c**24*f**4), Ne(23592960*c**24*f**4,
0)), (x*(A*a**3*exp(12*I*e) + 3*A*a**3*exp(10*I*e) + 3*A*a**3*exp(8*I*e) + A*a**3*exp(6*I*e) - I*B*a**3*exp(1
2*I*e) - I*B*a**3*exp(10*I*e) + I*B*a**3*exp(8*I*e) + I*B*a**3*exp(6*I*e))/(8*c**6), True))

________________________________________________________________________________________

Giac [B]  time = 1.64067, size = 466, normalized size = 3.67 \begin{align*} -\frac{2 \,{\left (15 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} + 45 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 15 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 215 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} - 390 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 90 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 738 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 24 i \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 746 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 158 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 738 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 24 i \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 390 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 90 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 215 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 45 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 15 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 15 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{15 \, c^{6} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{12}} \end{align*}

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]

-2/15*(15*A*a^3*tan(1/2*f*x + 1/2*e)^11 + 45*I*A*a^3*tan(1/2*f*x + 1/2*e)^10 - 15*B*a^3*tan(1/2*f*x + 1/2*e)^1
0 - 215*A*a^3*tan(1/2*f*x + 1/2*e)^9 - 390*I*A*a^3*tan(1/2*f*x + 1/2*e)^8 + 90*B*a^3*tan(1/2*f*x + 1/2*e)^8 +
738*A*a^3*tan(1/2*f*x + 1/2*e)^7 + 24*I*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 746*I*A*a^3*tan(1/2*f*x + 1/2*e)^6 - 15
8*B*a^3*tan(1/2*f*x + 1/2*e)^6 - 738*A*a^3*tan(1/2*f*x + 1/2*e)^5 - 24*I*B*a^3*tan(1/2*f*x + 1/2*e)^5 - 390*I*
A*a^3*tan(1/2*f*x + 1/2*e)^4 + 90*B*a^3*tan(1/2*f*x + 1/2*e)^4 + 215*A*a^3*tan(1/2*f*x + 1/2*e)^3 + 45*I*A*a^3
*tan(1/2*f*x + 1/2*e)^2 - 15*B*a^3*tan(1/2*f*x + 1/2*e)^2 - 15*A*a^3*tan(1/2*f*x + 1/2*e))/(c^6*f*(tan(1/2*f*x
+ 1/2*e) + I)^12)