3.728 \(\int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^5}{(a+i a \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=191 \[ \frac{c^5 (A+8 i B) \tan (e+f x)}{a^3 f}-\frac{8 c^5 (3 A+7 i B)}{a^3 f (-\tan (e+f x)+i)}+\frac{8 c^5 (-3 B+2 i A)}{a^3 f (-\tan (e+f x)+i)^2}+\frac{16 c^5 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{8 c^5 (-4 B+i A) \log (\cos (e+f x))}{a^3 f}-\frac{8 c^5 x (A+4 i B)}{a^3}+\frac{B c^5 \tan ^2(e+f x)}{2 a^3 f} \]

[Out]

(-8*(A + (4*I)*B)*c^5*x)/a^3 - (8*(I*A - 4*B)*c^5*Log[Cos[e + f*x]])/(a^3*f) + (16*(A + I*B)*c^5)/(3*a^3*f*(I
- Tan[e + f*x])^3) + (8*((2*I)*A - 3*B)*c^5)/(a^3*f*(I - Tan[e + f*x])^2) - (8*(3*A + (7*I)*B)*c^5)/(a^3*f*(I
- Tan[e + f*x])) + ((A + (8*I)*B)*c^5*Tan[e + f*x])/(a^3*f) + (B*c^5*Tan[e + f*x]^2)/(2*a^3*f)

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Rubi [A]  time = 0.244538, antiderivative size = 191, 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} \[ \frac{c^5 (A+8 i B) \tan (e+f x)}{a^3 f}-\frac{8 c^5 (3 A+7 i B)}{a^3 f (-\tan (e+f x)+i)}+\frac{8 c^5 (-3 B+2 i A)}{a^3 f (-\tan (e+f x)+i)^2}+\frac{16 c^5 (A+i B)}{3 a^3 f (-\tan (e+f x)+i)^3}-\frac{8 c^5 (-4 B+i A) \log (\cos (e+f x))}{a^3 f}-\frac{8 c^5 x (A+4 i B)}{a^3}+\frac{B c^5 \tan ^2(e+f x)}{2 a^3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(A + (4*I)*B)*c^5*x)/a^3 - (8*(I*A - 4*B)*c^5*Log[Cos[e + f*x]])/(a^3*f) + (16*(A + I*B)*c^5)/(3*a^3*f*(I
- Tan[e + f*x])^3) + (8*((2*I)*A - 3*B)*c^5)/(a^3*f*(I - Tan[e + f*x])^2) - (8*(3*A + (7*I)*B)*c^5)/(a^3*f*(I
- Tan[e + f*x])) + ((A + (8*I)*B)*c^5*Tan[e + f*x])/(a^3*f) + (B*c^5*Tan[e + f*x]^2)/(2*a^3*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]

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+B \tan (e+f x)) (c-i c \tan (e+f x))^5}{(a+i a \tan (e+f x))^3} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(A+B x) (c-i c x)^4}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{(A+8 i B) c^4}{a^4}+\frac{B c^4 x}{a^4}+\frac{16 (A+i B) c^4}{a^4 (-i+x)^4}+\frac{16 (-2 i A+3 B) c^4}{a^4 (-i+x)^3}-\frac{8 (3 A+7 i B) c^4}{a^4 (-i+x)^2}+\frac{8 i (A+4 i B) c^4}{a^4 (-i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{8 (A+4 i B) c^5 x}{a^3}-\frac{8 (i A-4 B) c^5 \log (\cos (e+f x))}{a^3 f}+\frac{16 (A+i B) c^5}{3 a^3 f (i-\tan (e+f x))^3}+\frac{8 (2 i A-3 B) c^5}{a^3 f (i-\tan (e+f x))^2}-\frac{8 (3 A+7 i B) c^5}{a^3 f (i-\tan (e+f x))}+\frac{(A+8 i B) c^5 \tan (e+f x)}{a^3 f}+\frac{B c^5 \tan ^2(e+f x)}{2 a^3 f}\\ \end{align*}

Mathematica [B]  time = 11.3296, size = 1496, normalized size = 7.83 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A + (3*I)*B)*Cos[2*f*x]*Sec[e + f*x]^2*((6*I)*c^5*Cos[e] - 6*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Ta
n[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (((-I)*A + 2*B)*Cos[4*f*x]*Sec[e
+ f*x]^2*(2*c^5*Cos[e] - (2*I)*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x]
+ B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^2*((-I)*A*c^5*Cos[(3*e)/2] + 4*B*c^5*Cos[(3*e)/2]
+ A*c^5*Sin[(3*e)/2] + (4*I)*B*c^5*Sin[(3*e)/2])*(8*Cos[(3*e)/2]*Log[Cos[e + f*x]] + (8*I)*Log[Cos[e + f*x]]*
Sin[(3*e)/2])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Ta
n[e + f*x])^3) + ((A + I*B)*Cos[6*f*x]*Sec[e + f*x]^2*(((2*I)/3)*c^5*Cos[3*e] + (2*c^5*Sin[3*e])/3)*(Cos[f*x]
+ I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e
+ f*x]^4*((B*c^5*Cos[3*e])/2 + (I/2)*B*c^5*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos
[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + (4*I)*B)*Sec[e + f*x]^2*(-8*c^5*f*x*Cos[3*e] - (
8*I)*c^5*f*x*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a
+ I*a*Tan[e + f*x])^3) + ((A + (3*I)*B)*Sec[e + f*x]^2*(6*c^5*Cos[e] + (6*I)*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*
x])^3*Sin[2*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A +
(2*I)*B)*Sec[e + f*x]^2*(-2*c^5*Cos[e] + (2*I)*c^5*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[4*f*x]*(A + B*Tan[e +
f*x]))/(f*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + ((A + I*B)*Sec[e + f*x]^2*((2*c^5*Cos
[3*e])/3 - ((2*I)/3)*c^5*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*Sin[6*f*x]*(A + B*Tan[e + f*x]))/(f*(A*Cos[e + f*
x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((I/2)*A*c^5*Cos[3*
e - f*x] - 4*B*c^5*Cos[3*e - f*x] - (I/2)*A*c^5*Cos[3*e + f*x] + 4*B*c^5*Cos[3*e + f*x] - (A*c^5*Sin[3*e - f*x
])/2 - (4*I)*B*c^5*Sin[3*e - f*x] + (A*c^5*Sin[3*e + f*x])/2 + (4*I)*B*c^5*Sin[3*e + f*x])*(A + B*Tan[e + f*x]
))/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(A*Cos[e + f*x] + B*Sin[e + f*x])*(a + I*a*Tan[e + f*x])^3)
+ (x*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^3*(4*A*c^5*Cos[e] + (16*I)*B*c^5*Cos[e] - 4*A*c^5*Cos[e]^3 - (16*I
)*B*c^5*Cos[e]^3 + (8*I)*A*c^5*Sin[e] - 32*B*c^5*Sin[e] - (16*I)*A*c^5*Cos[e]^2*Sin[e] + 64*B*c^5*Cos[e]^2*Sin
[e] + 24*A*c^5*Cos[e]*Sin[e]^2 + (96*I)*B*c^5*Cos[e]*Sin[e]^2 + (16*I)*A*c^5*Sin[e]^3 - 64*B*c^5*Sin[e]^3 - 4*
A*c^5*Sin[e]*Tan[e] - (16*I)*B*c^5*Sin[e]*Tan[e] - 4*A*c^5*Sin[e]^3*Tan[e] - (16*I)*B*c^5*Sin[e]^3*Tan[e] + I*
(A + (4*I)*B)*(8*c^5*Cos[3*e] + (8*I)*c^5*Sin[3*e])*Tan[e])*(A + B*Tan[e + f*x]))/((A*Cos[e + f*x] + B*Sin[e +
f*x])*(a + I*a*Tan[e + f*x])^3)

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Maple [A]  time = 0.066, size = 244, normalized size = 1.3 \begin{align*}{\frac{A{c}^{5}\tan \left ( fx+e \right ) }{f{a}^{3}}}+{\frac{8\,i{c}^{5}B\tan \left ( fx+e \right ) }{f{a}^{3}}}+{\frac{B{c}^{5} \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{2\,f{a}^{3}}}+{\frac{56\,i{c}^{5}B}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) }}+24\,{\frac{A{c}^{5}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{16\,i{c}^{5}A}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-24\,{\frac{B{c}^{5}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{8\,i{c}^{5}A\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{f{a}^{3}}}-32\,{\frac{B{c}^{5}\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{f{a}^{3}}}-{\frac{16\,A{c}^{5}}{3\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}-{\frac{{\frac{16\,i}{3}}{c}^{5}B}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^3,x)

[Out]

1/f*c^5/a^3*A*tan(f*x+e)+8*I/f*c^5/a^3*B*tan(f*x+e)+1/2*B*c^5*tan(f*x+e)^2/a^3/f+56*I/f*c^5/a^3/(tan(f*x+e)-I)
*B+24/f*c^5/a^3/(tan(f*x+e)-I)*A+16*I/f*c^5/a^3/(tan(f*x+e)-I)^2*A-24/f*c^5/a^3/(tan(f*x+e)-I)^2*B+8*I/f*c^5/a
^3*A*ln(tan(f*x+e)-I)-32/f*c^5/a^3*B*ln(tan(f*x+e)-I)-16/3/f*c^5/a^3/(tan(f*x+e)-I)^3*A-16/3*I/f*c^5/a^3/(tan(
f*x+e)-I)^3*B

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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+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.11626, size = 736, normalized size = 3.85 \begin{align*} -\frac{48 \,{\left (A + 4 i \, B\right )} c^{5} f x e^{\left (10 i \, f x + 10 i \, e\right )} -{\left (8 i \, A - 32 \, B\right )} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} -{\left (-2 i \, A + 8 \, B\right )} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} -{\left (2 i \, A - 2 \, B\right )} c^{5} +{\left (96 \,{\left (A + 4 i \, B\right )} c^{5} f x -{\left (24 i \, A - 96 \, B\right )} c^{5}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (48 \,{\left (A + 4 i \, B\right )} c^{5} f x -{\left (36 i \, A - 144 \, B\right )} c^{5}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} -{\left ({\left (-24 i \, A + 96 \, B\right )} c^{5} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-48 i \, A + 192 \, B\right )} c^{5} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-24 i \, A + 96 \, B\right )} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{3 \,{\left (a^{3} f e^{\left (10 i \, f x + 10 i \, e\right )} + 2 \, a^{3} f e^{\left (8 i \, f x + 8 i \, e\right )} + a^{3} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/3*(48*(A + 4*I*B)*c^5*f*x*e^(10*I*f*x + 10*I*e) - (8*I*A - 32*B)*c^5*e^(4*I*f*x + 4*I*e) - (-2*I*A + 8*B)*c
^5*e^(2*I*f*x + 2*I*e) - (2*I*A - 2*B)*c^5 + (96*(A + 4*I*B)*c^5*f*x - (24*I*A - 96*B)*c^5)*e^(8*I*f*x + 8*I*e
) + (48*(A + 4*I*B)*c^5*f*x - (36*I*A - 144*B)*c^5)*e^(6*I*f*x + 6*I*e) - ((-24*I*A + 96*B)*c^5*e^(10*I*f*x +
10*I*e) + (-48*I*A + 192*B)*c^5*e^(8*I*f*x + 8*I*e) + (-24*I*A + 96*B)*c^5*e^(6*I*f*x + 6*I*e))*log(e^(2*I*f*x
+ 2*I*e) + 1))/(a^3*f*e^(10*I*f*x + 10*I*e) + 2*a^3*f*e^(8*I*f*x + 8*I*e) + a^3*f*e^(6*I*f*x + 6*I*e))

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Sympy [A]  time = 12.5139, size = 415, normalized size = 2.17 \begin{align*} \frac{\frac{\left (2 i A c^{5} - 16 B c^{5}\right ) e^{- 4 i e}}{a^{3} f} + \frac{\left (2 i A c^{5} - 14 B c^{5}\right ) e^{- 2 i e} e^{2 i f x}}{a^{3} f}}{e^{4 i f x} + 2 e^{- 2 i e} e^{2 i f x} + e^{- 4 i e}} + \frac{8 c^{5} \left (- i A + 4 B\right ) \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{3} f} - \frac{\left (\begin{cases} 16 A c^{5} x e^{6 i e} - \frac{6 i A c^{5} e^{4 i e} e^{- 2 i f x}}{f} + \frac{2 i A c^{5} e^{2 i e} e^{- 4 i f x}}{f} - \frac{2 i A c^{5} e^{- 6 i f x}}{3 f} + 64 i B c^{5} x e^{6 i e} + \frac{18 B c^{5} e^{4 i e} e^{- 2 i f x}}{f} - \frac{4 B c^{5} e^{2 i e} e^{- 4 i f x}}{f} + \frac{2 B c^{5} e^{- 6 i f x}}{3 f} & \text{for}\: f \neq 0 \\x \left (16 A c^{5} e^{6 i e} - 12 A c^{5} e^{4 i e} + 8 A c^{5} e^{2 i e} - 4 A c^{5} + 64 i B c^{5} e^{6 i e} - 36 i B c^{5} e^{4 i e} + 16 i B c^{5} e^{2 i e} - 4 i B c^{5}\right ) & \text{otherwise} \end{cases}\right ) e^{- 6 i e}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**5/(a+I*a*tan(f*x+e))**3,x)

[Out]

((2*I*A*c**5 - 16*B*c**5)*exp(-4*I*e)/(a**3*f) + (2*I*A*c**5 - 14*B*c**5)*exp(-2*I*e)*exp(2*I*f*x)/(a**3*f))/(
exp(4*I*f*x) + 2*exp(-2*I*e)*exp(2*I*f*x) + exp(-4*I*e)) + 8*c**5*(-I*A + 4*B)*log(exp(2*I*f*x) + exp(-2*I*e))
/(a**3*f) - Piecewise((16*A*c**5*x*exp(6*I*e) - 6*I*A*c**5*exp(4*I*e)*exp(-2*I*f*x)/f + 2*I*A*c**5*exp(2*I*e)*
exp(-4*I*f*x)/f - 2*I*A*c**5*exp(-6*I*f*x)/(3*f) + 64*I*B*c**5*x*exp(6*I*e) + 18*B*c**5*exp(4*I*e)*exp(-2*I*f*
x)/f - 4*B*c**5*exp(2*I*e)*exp(-4*I*f*x)/f + 2*B*c**5*exp(-6*I*f*x)/(3*f), Ne(f, 0)), (x*(16*A*c**5*exp(6*I*e)
- 12*A*c**5*exp(4*I*e) + 8*A*c**5*exp(2*I*e) - 4*A*c**5 + 64*I*B*c**5*exp(6*I*e) - 36*I*B*c**5*exp(4*I*e) + 1
6*I*B*c**5*exp(2*I*e) - 4*I*B*c**5), True))*exp(-6*I*e)/a**3

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Giac [B]  time = 1.70212, size = 698, normalized size = 3.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^5/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-2/15*(120*(-I*A*c^5 + 4*B*c^5)*log(tan(1/2*f*x + 1/2*e) - I)/a^3 - 60*(-I*A*c^5 + 4*B*c^5)*log(abs(tan(1/2*f*
x + 1/2*e) + 1))/a^3 + 60*(I*A*c^5 - 4*B*c^5)*log(abs(tan(1/2*f*x + 1/2*e) - 1))/a^3 - 15*(6*I*A*c^5*tan(1/2*f
*x + 1/2*e)^4 - 24*B*c^5*tan(1/2*f*x + 1/2*e)^4 - A*c^5*tan(1/2*f*x + 1/2*e)^3 - 8*I*B*c^5*tan(1/2*f*x + 1/2*e
)^3 - 12*I*A*c^5*tan(1/2*f*x + 1/2*e)^2 + 49*B*c^5*tan(1/2*f*x + 1/2*e)^2 + A*c^5*tan(1/2*f*x + 1/2*e) + 8*I*B
*c^5*tan(1/2*f*x + 1/2*e) + 6*I*A*c^5 - 24*B*c^5)/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*a^3) + (294*I*A*c^5*tan(1/2*
f*x + 1/2*e)^6 - 1176*B*c^5*tan(1/2*f*x + 1/2*e)^6 + 1884*A*c^5*tan(1/2*f*x + 1/2*e)^5 + 7416*I*B*c^5*tan(1/2*
f*x + 1/2*e)^5 - 4890*I*A*c^5*tan(1/2*f*x + 1/2*e)^4 + 19320*B*c^5*tan(1/2*f*x + 1/2*e)^4 - 6920*A*c^5*tan(1/2
*f*x + 1/2*e)^3 - 26480*I*B*c^5*tan(1/2*f*x + 1/2*e)^3 + 4890*I*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 19320*B*c^5*tan
(1/2*f*x + 1/2*e)^2 + 1884*A*c^5*tan(1/2*f*x + 1/2*e) + 7416*I*B*c^5*tan(1/2*f*x + 1/2*e) - 294*I*A*c^5 + 1176
*B*c^5)/(a^3*(tan(1/2*f*x + 1/2*e) - I)^6))/f