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

Optimal. Leaf size=154 \[ \frac{i a^6 \tan ^2(e+f x)}{2 c^3 f}+\frac{9 a^6 \tan (e+f x)}{c^3 f}-\frac{80 i a^6}{f \left (c^3-i c^3 \tan (e+f x)\right )}+\frac{40 i a^6 \log (\cos (e+f x))}{c^3 f}-\frac{40 a^6 x}{c^3}+\frac{40 i a^6}{c f (c-i c \tan (e+f x))^2}-\frac{32 i a^6}{3 f (c-i c \tan (e+f x))^3} \]

[Out]

(-40*a^6*x)/c^3 + ((40*I)*a^6*Log[Cos[e + f*x]])/(c^3*f) + (9*a^6*Tan[e + f*x])/(c^3*f) + ((I/2)*a^6*Tan[e + f
*x]^2)/(c^3*f) - (((32*I)/3)*a^6)/(f*(c - I*c*Tan[e + f*x])^3) + ((40*I)*a^6)/(c*f*(c - I*c*Tan[e + f*x])^2) -
((80*I)*a^6)/(f*(c^3 - I*c^3*Tan[e + f*x]))

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Rubi [A]  time = 0.161913, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 43} \[ \frac{i a^6 \tan ^2(e+f x)}{2 c^3 f}+\frac{9 a^6 \tan (e+f x)}{c^3 f}-\frac{80 i a^6}{f \left (c^3-i c^3 \tan (e+f x)\right )}+\frac{40 i a^6 \log (\cos (e+f x))}{c^3 f}-\frac{40 a^6 x}{c^3}+\frac{40 i a^6}{c f (c-i c \tan (e+f x))^2}-\frac{32 i a^6}{3 f (c-i c \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^3,x]

[Out]

(-40*a^6*x)/c^3 + ((40*I)*a^6*Log[Cos[e + f*x]])/(c^3*f) + (9*a^6*Tan[e + f*x])/(c^3*f) + ((I/2)*a^6*Tan[e + f
*x]^2)/(c^3*f) - (((32*I)/3)*a^6)/(f*(c - I*c*Tan[e + f*x])^3) + ((40*I)*a^6)/(c*f*(c - I*c*Tan[e + f*x])^2) -
((80*I)*a^6)/(f*(c^3 - I*c^3*Tan[e + f*x]))

Rule 3522

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] &&  !(IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n]))

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx &=\left (a^6 c^6\right ) \int \frac{\sec ^{12}(e+f x)}{(c-i c \tan (e+f x))^9} \, dx\\ &=\frac{\left (i a^6\right ) \operatorname{Subst}\left (\int \frac{(c-x)^5}{(c+x)^4} \, dx,x,-i c \tan (e+f x)\right )}{c^5 f}\\ &=\frac{\left (i a^6\right ) \operatorname{Subst}\left (\int \left (9 c-x+\frac{32 c^5}{(c+x)^4}-\frac{80 c^4}{(c+x)^3}+\frac{80 c^3}{(c+x)^2}-\frac{40 c^2}{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^5 f}\\ &=-\frac{40 a^6 x}{c^3}+\frac{40 i a^6 \log (\cos (e+f x))}{c^3 f}+\frac{9 a^6 \tan (e+f x)}{c^3 f}+\frac{i a^6 \tan ^2(e+f x)}{2 c^3 f}-\frac{32 i a^6}{3 f (c-i c \tan (e+f x))^3}+\frac{40 i a^6}{c f (c-i c \tan (e+f x))^2}-\frac{80 i a^6}{f \left (c^3-i c^3 \tan (e+f x)\right )}\\ \end{align*}

Mathematica [B]  time = 6.15107, size = 569, normalized size = 3.69 \[ -\frac{a^6 \sec (e) \sec ^2(e+f x) (\cos (3 (e+3 f x))+i \sin (3 (e+3 f x))) \left (-60 i f x \sin (2 e+f x)+70 \sin (2 e+f x)-120 i f x \sin (2 e+3 f x)+11 \sin (2 e+3 f x)-120 i f x \sin (4 e+3 f x)+65 \sin (4 e+3 f x)-60 i f x \sin (4 e+5 f x)-29 \sin (4 e+5 f x)-60 i f x \sin (6 e+5 f x)-2 \sin (6 e+5 f x)+120 f x \cos (2 e+3 f x)+i \cos (2 e+3 f x)+120 f x \cos (4 e+3 f x)+55 i \cos (4 e+3 f x)+60 f x \cos (4 e+5 f x)-25 i \cos (4 e+5 f x)+60 f x \cos (6 e+5 f x)+2 i \cos (6 e+5 f x)-60 i \cos (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )+10 \cos (2 e+f x) \left (-3 i \log \left (\cos ^2(e+f x)\right )+6 f x+11 i\right )+\cos (f x) \left (-30 i \log \left (\cos ^2(e+f x)\right )+60 f x+83 i\right )-60 i \cos (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-30 i \cos (4 e+5 f x) \log \left (\cos ^2(e+f x)\right )-30 i \cos (6 e+5 f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (2 e+f x) \log \left (\cos ^2(e+f x)\right )-60 \sin (2 e+3 f x) \log \left (\cos ^2(e+f x)\right )-60 \sin (4 e+3 f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (4 e+5 f x) \log \left (\cos ^2(e+f x)\right )-30 \sin (6 e+5 f x) \log \left (\cos ^2(e+f x)\right )-60 i f x \sin (f x)+43 \sin (f x)\right )}{12 c^3 f (\cos (f x)+i \sin (f x))^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^3,x]

[Out]

-(a^6*Sec[e]*Sec[e + f*x]^2*(Cos[3*(e + 3*f*x)] + I*Sin[3*(e + 3*f*x)])*(I*Cos[2*e + 3*f*x] + 120*f*x*Cos[2*e
+ 3*f*x] + (55*I)*Cos[4*e + 3*f*x] + 120*f*x*Cos[4*e + 3*f*x] - (25*I)*Cos[4*e + 5*f*x] + 60*f*x*Cos[4*e + 5*f
*x] + (2*I)*Cos[6*e + 5*f*x] + 60*f*x*Cos[6*e + 5*f*x] + 10*Cos[2*e + f*x]*(11*I + 6*f*x - (3*I)*Log[Cos[e + f
*x]^2]) + Cos[f*x]*(83*I + 60*f*x - (30*I)*Log[Cos[e + f*x]^2]) - (60*I)*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2]
- (60*I)*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] - (30*I)*Cos[4*e + 5*f*x]*Log[Cos[e + f*x]^2] - (30*I)*Cos[6*e +
5*f*x]*Log[Cos[e + f*x]^2] + 43*Sin[f*x] - (60*I)*f*x*Sin[f*x] - 30*Log[Cos[e + f*x]^2]*Sin[f*x] + 70*Sin[2*e
+ f*x] - (60*I)*f*x*Sin[2*e + f*x] - 30*Log[Cos[e + f*x]^2]*Sin[2*e + f*x] + 11*Sin[2*e + 3*f*x] - (120*I)*f*
x*Sin[2*e + 3*f*x] - 60*Log[Cos[e + f*x]^2]*Sin[2*e + 3*f*x] + 65*Sin[4*e + 3*f*x] - (120*I)*f*x*Sin[4*e + 3*f
*x] - 60*Log[Cos[e + f*x]^2]*Sin[4*e + 3*f*x] - 29*Sin[4*e + 5*f*x] - (60*I)*f*x*Sin[4*e + 5*f*x] - 30*Log[Cos
[e + f*x]^2]*Sin[4*e + 5*f*x] - 2*Sin[6*e + 5*f*x] - (60*I)*f*x*Sin[6*e + 5*f*x] - 30*Log[Cos[e + f*x]^2]*Sin[
6*e + 5*f*x]))/(12*c^3*f*(Cos[f*x] + I*Sin[f*x])^6)

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Maple [A]  time = 0.039, size = 128, normalized size = 0.8 \begin{align*} 9\,{\frac{{a}^{6}\tan \left ( fx+e \right ) }{{c}^{3}f}}+{\frac{{\frac{i}{2}}{a}^{6} \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{{c}^{3}f}}+80\,{\frac{{a}^{6}}{{c}^{3}f \left ( \tan \left ( fx+e \right ) +i \right ) }}-{\frac{40\,i{a}^{6}}{{c}^{3}f \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}-{\frac{40\,i{a}^{6}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{{c}^{3}f}}-{\frac{32\,{a}^{6}}{3\,{c}^{3}f \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^6/(c-I*c*tan(f*x+e))^3,x)

[Out]

9*a^6*tan(f*x+e)/c^3/f+1/2*I*a^6*tan(f*x+e)^2/c^3/f+80/f*a^6/c^3/(tan(f*x+e)+I)-40*I/f*a^6/c^3/(tan(f*x+e)+I)^
2-40*I/f*a^6/c^3*ln(tan(f*x+e)+I)-32/3/f*a^6/c^3/(tan(f*x+e)+I)^3

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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))^6/(c-I*c*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.35204, size = 468, normalized size = 3.04 \begin{align*} \frac{-4 i \, a^{6} e^{\left (10 i \, f x + 10 i \, e\right )} + 10 i \, a^{6} e^{\left (8 i \, f x + 8 i \, e\right )} - 40 i \, a^{6} e^{\left (6 i \, f x + 6 i \, e\right )} - 126 i \, a^{6} e^{\left (4 i \, f x + 4 i \, e\right )} - 12 i \, a^{6} e^{\left (2 i \, f x + 2 i \, e\right )} + 54 i \, a^{6} +{\left (120 i \, a^{6} e^{\left (4 i \, f x + 4 i \, e\right )} + 240 i \, a^{6} e^{\left (2 i \, f x + 2 i \, e\right )} + 120 i \, a^{6}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{3 \,{\left (c^{3} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, c^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{3} f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^6/(c-I*c*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/3*(-4*I*a^6*e^(10*I*f*x + 10*I*e) + 10*I*a^6*e^(8*I*f*x + 8*I*e) - 40*I*a^6*e^(6*I*f*x + 6*I*e) - 126*I*a^6*
e^(4*I*f*x + 4*I*e) - 12*I*a^6*e^(2*I*f*x + 2*I*e) + 54*I*a^6 + (120*I*a^6*e^(4*I*f*x + 4*I*e) + 240*I*a^6*e^(
2*I*f*x + 2*I*e) + 120*I*a^6)*log(e^(2*I*f*x + 2*I*e) + 1))/(c^3*f*e^(4*I*f*x + 4*I*e) + 2*c^3*f*e^(2*I*f*x +
2*I*e) + c^3*f)

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Sympy [A]  time = 3.27815, size = 223, normalized size = 1.45 \begin{align*} \frac{40 i a^{6} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{3} f} + \frac{\frac{20 i a^{6} e^{- 2 i e} e^{2 i f x}}{c^{3} f} + \frac{18 i a^{6} e^{- 4 i e}}{c^{3} f}}{e^{4 i f x} + 2 e^{- 2 i e} e^{2 i f x} + e^{- 4 i e}} + \frac{\begin{cases} - \frac{4 i a^{6} e^{6 i e} e^{6 i f x}}{3 f} + \frac{6 i a^{6} e^{4 i e} e^{4 i f x}}{f} - \frac{24 i a^{6} e^{2 i e} e^{2 i f x}}{f} & \text{for}\: f \neq 0 \\x \left (8 a^{6} e^{6 i e} - 24 a^{6} e^{4 i e} + 48 a^{6} e^{2 i e}\right ) & \text{otherwise} \end{cases}}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**6/(c-I*c*tan(f*x+e))**3,x)

[Out]

40*I*a**6*log(exp(2*I*f*x) + exp(-2*I*e))/(c**3*f) + (20*I*a**6*exp(-2*I*e)*exp(2*I*f*x)/(c**3*f) + 18*I*a**6*
exp(-4*I*e)/(c**3*f))/(exp(4*I*f*x) + 2*exp(-2*I*e)*exp(2*I*f*x) + exp(-4*I*e)) + Piecewise((-4*I*a**6*exp(6*I
*e)*exp(6*I*f*x)/(3*f) + 6*I*a**6*exp(4*I*e)*exp(4*I*f*x)/f - 24*I*a**6*exp(2*I*e)*exp(2*I*f*x)/f, Ne(f, 0)),
(x*(8*a**6*exp(6*I*e) - 24*a**6*exp(4*I*e) + 48*a**6*exp(2*I*e)), True))/c**3

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Giac [B]  time = 1.83162, size = 389, normalized size = 2.53 \begin{align*} -\frac{2 \,{\left (\frac{120 i \, a^{6} \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}{c^{3}} - \frac{60 i \, a^{6} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{c^{3}} - \frac{60 i \, a^{6} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{c^{3}} - \frac{3 \,{\left (-30 i \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 9 \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 61 i \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 9 \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 30 i \, a^{6}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}^{2} c^{3}} + \frac{-294 i \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 1860 \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 4842 i \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 6680 \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 4842 i \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1860 \, a^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 294 i \, a^{6}}{c^{3}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{6}}\right )}}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^6/(c-I*c*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-2/3*(120*I*a^6*log(tan(1/2*f*x + 1/2*e) + I)/c^3 - 60*I*a^6*log(abs(tan(1/2*f*x + 1/2*e) + 1))/c^3 - 60*I*a^6
*log(abs(tan(1/2*f*x + 1/2*e) - 1))/c^3 - 3*(-30*I*a^6*tan(1/2*f*x + 1/2*e)^4 - 9*a^6*tan(1/2*f*x + 1/2*e)^3 +
61*I*a^6*tan(1/2*f*x + 1/2*e)^2 + 9*a^6*tan(1/2*f*x + 1/2*e) - 30*I*a^6)/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*c^3)
+ (-294*I*a^6*tan(1/2*f*x + 1/2*e)^6 + 1860*a^6*tan(1/2*f*x + 1/2*e)^5 + 4842*I*a^6*tan(1/2*f*x + 1/2*e)^4 -
6680*a^6*tan(1/2*f*x + 1/2*e)^3 - 4842*I*a^6*tan(1/2*f*x + 1/2*e)^2 + 1860*a^6*tan(1/2*f*x + 1/2*e) + 294*I*a^
6)/(c^3*(tan(1/2*f*x + 1/2*e) + I)^6))/f