3.703 \(\int \frac{(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^7} \, dx\)

Optimal. Leaf size=125 \[ \frac{a^3 (A-5 i B)}{5 c^7 f (\tan (e+f x)+i)^5}-\frac{2 a^3 (2 B+i A)}{3 c^7 f (\tan (e+f x)+i)^6}-\frac{4 a^3 (A-i B)}{7 c^7 f (\tan (e+f x)+i)^7}+\frac{a^3 B}{4 c^7 f (\tan (e+f x)+i)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.174468, antiderivative size = 125, 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} \[ \frac{a^3 (A-5 i B)}{5 c^7 f (\tan (e+f x)+i)^5}-\frac{2 a^3 (2 B+i A)}{3 c^7 f (\tan (e+f x)+i)^6}-\frac{4 a^3 (A-i B)}{7 c^7 f (\tan (e+f x)+i)^7}+\frac{a^3 B}{4 c^7 f (\tan (e+f x)+i)^4} \]

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])^7,x]

[Out]

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

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

Mathematica [A]  time = 7.54606, size = 143, normalized size = 1.14 \[ -\frac{i a^3 (\cos (10 e+13 f x)+i \sin (10 e+13 f x)) (35 (10 A+i B) \cos (2 (e+f x))+20 (5 A+2 i B) \cos (4 (e+f x))-70 i A \sin (2 (e+f x))-40 i A \sin (4 (e+f x))+252 A+175 B \sin (2 (e+f x))+100 B \sin (4 (e+f x)))}{6720 c^7 f (\cos (f x)+i \sin (f x))^3} \]

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])^7,x]

[Out]

((-I/6720)*a^3*(252*A + 35*(10*A + I*B)*Cos[2*(e + f*x)] + 20*(5*A + (2*I)*B)*Cos[4*(e + f*x)] - (70*I)*A*Sin[
2*(e + f*x)] + 175*B*Sin[2*(e + f*x)] - (40*I)*A*Sin[4*(e + f*x)] + 100*B*Sin[4*(e + f*x)])*(Cos[10*e + 13*f*x
] + I*Sin[10*e + 13*f*x]))/(c^7*f*(Cos[f*x] + I*Sin[f*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 89, normalized size = 0.7 \begin{align*}{\frac{{a}^{3}}{f{c}^{7}} \left ( -{\frac{-A+5\,iB}{5\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}-{\frac{-4\,iB+4\,A}{7\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{7}}}-{\frac{4\,iA+8\,B}{6\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{6}}}+{\frac{B}{4\, \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}} \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))^7,x)

[Out]

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

________________________________________________________________________________________

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))^7,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.38774, size = 312, normalized size = 2.5 \begin{align*} \frac{{\left (-30 i \, A - 30 \, B\right )} a^{3} e^{\left (14 i \, f x + 14 i \, e\right )} +{\left (-140 i \, A - 70 \, B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} - 252 i \, A a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-210 i \, A + 105 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-70 i \, A + 70 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{6720 \, c^{7} 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))^7,x, algorithm="fricas")

[Out]

1/6720*((-30*I*A - 30*B)*a^3*e^(14*I*f*x + 14*I*e) + (-140*I*A - 70*B)*a^3*e^(12*I*f*x + 12*I*e) - 252*I*A*a^3
*e^(10*I*f*x + 10*I*e) + (-210*I*A + 105*B)*a^3*e^(8*I*f*x + 8*I*e) + (-70*I*A + 70*B)*a^3*e^(6*I*f*x + 6*I*e)
)/(c^7*f)

________________________________________________________________________________________

Sympy [A]  time = 4.88956, size = 381, normalized size = 3.05 \begin{align*} \begin{cases} \frac{- 396361728 i A a^{3} c^{28} f^{4} e^{10 i e} e^{10 i f x} + \left (- 110100480 i A a^{3} c^{28} f^{4} e^{6 i e} + 110100480 B a^{3} c^{28} f^{4} e^{6 i e}\right ) e^{6 i f x} + \left (- 330301440 i A a^{3} c^{28} f^{4} e^{8 i e} + 165150720 B a^{3} c^{28} f^{4} e^{8 i e}\right ) e^{8 i f x} + \left (- 220200960 i A a^{3} c^{28} f^{4} e^{12 i e} - 110100480 B a^{3} c^{28} f^{4} e^{12 i e}\right ) e^{12 i f x} + \left (- 47185920 i A a^{3} c^{28} f^{4} e^{14 i e} - 47185920 B a^{3} c^{28} f^{4} e^{14 i e}\right ) e^{14 i f x}}{10569646080 c^{35} f^{5}} & \text{for}\: 10569646080 c^{35} f^{5} \neq 0 \\\frac{x \left (A a^{3} e^{14 i e} + 4 A a^{3} e^{12 i e} + 6 A a^{3} e^{10 i e} + 4 A a^{3} e^{8 i e} + A a^{3} e^{6 i e} - i B a^{3} e^{14 i e} - 2 i B a^{3} e^{12 i e} + 2 i B a^{3} e^{8 i e} + i B a^{3} e^{6 i e}\right )}{16 c^{7}} & \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))**7,x)

[Out]

Piecewise(((-396361728*I*A*a**3*c**28*f**4*exp(10*I*e)*exp(10*I*f*x) + (-110100480*I*A*a**3*c**28*f**4*exp(6*I
*e) + 110100480*B*a**3*c**28*f**4*exp(6*I*e))*exp(6*I*f*x) + (-330301440*I*A*a**3*c**28*f**4*exp(8*I*e) + 1651
50720*B*a**3*c**28*f**4*exp(8*I*e))*exp(8*I*f*x) + (-220200960*I*A*a**3*c**28*f**4*exp(12*I*e) - 110100480*B*a
**3*c**28*f**4*exp(12*I*e))*exp(12*I*f*x) + (-47185920*I*A*a**3*c**28*f**4*exp(14*I*e) - 47185920*B*a**3*c**28
*f**4*exp(14*I*e))*exp(14*I*f*x))/(10569646080*c**35*f**5), Ne(10569646080*c**35*f**5, 0)), (x*(A*a**3*exp(14*
I*e) + 4*A*a**3*exp(12*I*e) + 6*A*a**3*exp(10*I*e) + 4*A*a**3*exp(8*I*e) + A*a**3*exp(6*I*e) - I*B*a**3*exp(14
*I*e) - 2*I*B*a**3*exp(12*I*e) + 2*I*B*a**3*exp(8*I*e) + I*B*a**3*exp(6*I*e))/(16*c**7), True))

________________________________________________________________________________________

Giac [B]  time = 1.65522, size = 612, normalized size = 4.9 \begin{align*} -\frac{2 \,{\left (105 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{13} + 420 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{12} - 105 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{12} - 2170 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} - 70 i \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} - 5180 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} + 875 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} + 11431 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 700 i \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 15904 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 2380 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 19436 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 1340 i \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 15904 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 2380 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 11431 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 700 i \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 5180 i \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 875 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 2170 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 70 i \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 420 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^{7} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{14}} \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))^7,x, algorithm="giac")

[Out]

-2/105*(105*A*a^3*tan(1/2*f*x + 1/2*e)^13 + 420*I*A*a^3*tan(1/2*f*x + 1/2*e)^12 - 105*B*a^3*tan(1/2*f*x + 1/2*
e)^12 - 2170*A*a^3*tan(1/2*f*x + 1/2*e)^11 - 70*I*B*a^3*tan(1/2*f*x + 1/2*e)^11 - 5180*I*A*a^3*tan(1/2*f*x + 1
/2*e)^10 + 875*B*a^3*tan(1/2*f*x + 1/2*e)^10 + 11431*A*a^3*tan(1/2*f*x + 1/2*e)^9 + 700*I*B*a^3*tan(1/2*f*x +
1/2*e)^9 + 15904*I*A*a^3*tan(1/2*f*x + 1/2*e)^8 - 2380*B*a^3*tan(1/2*f*x + 1/2*e)^8 - 19436*A*a^3*tan(1/2*f*x
+ 1/2*e)^7 - 1340*I*B*a^3*tan(1/2*f*x + 1/2*e)^7 - 15904*I*A*a^3*tan(1/2*f*x + 1/2*e)^6 + 2380*B*a^3*tan(1/2*f
*x + 1/2*e)^6 + 11431*A*a^3*tan(1/2*f*x + 1/2*e)^5 + 700*I*B*a^3*tan(1/2*f*x + 1/2*e)^5 + 5180*I*A*a^3*tan(1/2
*f*x + 1/2*e)^4 - 875*B*a^3*tan(1/2*f*x + 1/2*e)^4 - 2170*A*a^3*tan(1/2*f*x + 1/2*e)^3 - 70*I*B*a^3*tan(1/2*f*
x + 1/2*e)^3 - 420*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^7*f*(tan(1/2*f*x + 1/2*e) + I)^14)