3.267 \(\int \frac{(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=98 \[ -\frac{f \cosh (c+d x)}{a d^2}+\frac{i f \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac{i (e+f x) \sinh ^2(c+d x)}{2 a d}+\frac{(e+f x) \sinh (c+d x)}{a d}-\frac{i f x}{4 a d} \]

[Out]

((-I/4)*f*x)/(a*d) - (f*Cosh[c + d*x])/(a*d^2) + ((e + f*x)*Sinh[c + d*x])/(a*d) + ((I/4)*f*Cosh[c + d*x]*Sinh
[c + d*x])/(a*d^2) - ((I/2)*(e + f*x)*Sinh[c + d*x]^2)/(a*d)

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Rubi [A]  time = 0.0995435, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {5563, 3296, 2638, 5446, 2635, 8} \[ -\frac{f \cosh (c+d x)}{a d^2}+\frac{i f \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac{i (e+f x) \sinh ^2(c+d x)}{2 a d}+\frac{(e+f x) \sinh (c+d x)}{a d}-\frac{i f x}{4 a d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

((-I/4)*f*x)/(a*d) - (f*Cosh[c + d*x])/(a*d^2) + ((e + f*x)*Sinh[c + d*x])/(a*d) + ((I/4)*f*Cosh[c + d*x]*Sinh
[c + d*x])/(a*d^2) - ((I/2)*(e + f*x)*Sinh[c + d*x]^2)/(a*d)

Rule 5563

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(n -
2)*Sinh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 + b^2, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(e+f x) \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{a}+\frac{\int (e+f x) \cosh (c+d x) \, dx}{a}\\ &=\frac{(e+f x) \sinh (c+d x)}{a d}-\frac{i (e+f x) \sinh ^2(c+d x)}{2 a d}+\frac{(i f) \int \sinh ^2(c+d x) \, dx}{2 a d}-\frac{f \int \sinh (c+d x) \, dx}{a d}\\ &=-\frac{f \cosh (c+d x)}{a d^2}+\frac{(e+f x) \sinh (c+d x)}{a d}+\frac{i f \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac{i (e+f x) \sinh ^2(c+d x)}{2 a d}-\frac{(i f) \int 1 \, dx}{4 a d}\\ &=-\frac{i f x}{4 a d}-\frac{f \cosh (c+d x)}{a d^2}+\frac{(e+f x) \sinh (c+d x)}{a d}+\frac{i f \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac{i (e+f x) \sinh ^2(c+d x)}{2 a d}\\ \end{align*}

Mathematica [A]  time = 1.13317, size = 60, normalized size = 0.61 \[ \frac{d (e+f x) (4 \sinh (c+d x)-i \cosh (2 (c+d x)))+i f (\sinh (c+d x)+4 i) \cosh (c+d x)}{4 a d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(I*f*Cosh[c + d*x]*(4*I + Sinh[c + d*x]) + d*(e + f*x)*((-I)*Cosh[2*(c + d*x)] + 4*Sinh[c + d*x]))/(4*a*d^2)

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Maple [A]  time = 0.121, size = 113, normalized size = 1.2 \begin{align*}{\frac{-{\frac{i}{16}} \left ( 2\,dfx+2\,de-f \right ){{\rm e}^{2\,dx+2\,c}}}{a{d}^{2}}}+{\frac{ \left ( dfx+de-f \right ){{\rm e}^{dx+c}}}{2\,a{d}^{2}}}-{\frac{ \left ( dfx+de+f \right ){{\rm e}^{-dx-c}}}{2\,a{d}^{2}}}-{\frac{{\frac{i}{16}} \left ( 2\,dfx+2\,de+f \right ){{\rm e}^{-2\,dx-2\,c}}}{a{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)

[Out]

-1/16*I*(2*d*f*x+2*d*e-f)/a/d^2*exp(2*d*x+2*c)+1/2*(d*f*x+d*e-f)/a/d^2*exp(d*x+c)-1/2*(d*f*x+d*e+f)/a/d^2*exp(
-d*x-c)-1/16*I*(2*d*f*x+2*d*e+f)/a/d^2*exp(-2*d*x-2*c)

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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((f*x+e)*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.14459, size = 242, normalized size = 2.47 \begin{align*} \frac{{\left (-2 i \, d f x - 2 i \, d e +{\left (-2 i \, d f x - 2 i \, d e + i \, f\right )} e^{\left (4 \, d x + 4 \, c\right )} + 8 \,{\left (d f x + d e - f\right )} e^{\left (3 \, d x + 3 \, c\right )} - 8 \,{\left (d f x + d e + f\right )} e^{\left (d x + c\right )} - i \, f\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, a d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/16*(-2*I*d*f*x - 2*I*d*e + (-2*I*d*f*x - 2*I*d*e + I*f)*e^(4*d*x + 4*c) + 8*(d*f*x + d*e - f)*e^(3*d*x + 3*c
) - 8*(d*f*x + d*e + f)*e^(d*x + c) - I*f)*e^(-2*d*x - 2*c)/(a*d^2)

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Sympy [A]  time = 2.07587, size = 330, normalized size = 3.37 \begin{align*} \begin{cases} \frac{\left (\left (- 512 a^{5} d^{9} e e^{4 c} - 512 a^{5} d^{9} f x e^{4 c} - 512 a^{5} d^{8} f e^{4 c}\right ) e^{- d x} + \left (512 a^{5} d^{9} e e^{6 c} + 512 a^{5} d^{9} f x e^{6 c} - 512 a^{5} d^{8} f e^{6 c}\right ) e^{d x} + \left (- 128 i a^{5} d^{9} e e^{3 c} - 128 i a^{5} d^{9} f x e^{3 c} - 64 i a^{5} d^{8} f e^{3 c}\right ) e^{- 2 d x} + \left (- 128 i a^{5} d^{9} e e^{7 c} - 128 i a^{5} d^{9} f x e^{7 c} + 64 i a^{5} d^{8} f e^{7 c}\right ) e^{2 d x}\right ) e^{- 5 c}}{1024 a^{6} d^{10}} & \text{for}\: 1024 a^{6} d^{10} e^{5 c} \neq 0 \\- \frac{x^{2} \left (i f e^{4 c} - 2 f e^{3 c} - 2 f e^{c} - i f\right ) e^{- 2 c}}{8 a} - \frac{x \left (i e e^{4 c} - 2 e e^{3 c} - 2 e e^{c} - i e\right ) e^{- 2 c}}{4 a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

Piecewise((((-512*a**5*d**9*e*exp(4*c) - 512*a**5*d**9*f*x*exp(4*c) - 512*a**5*d**8*f*exp(4*c))*exp(-d*x) + (5
12*a**5*d**9*e*exp(6*c) + 512*a**5*d**9*f*x*exp(6*c) - 512*a**5*d**8*f*exp(6*c))*exp(d*x) + (-128*I*a**5*d**9*
e*exp(3*c) - 128*I*a**5*d**9*f*x*exp(3*c) - 64*I*a**5*d**8*f*exp(3*c))*exp(-2*d*x) + (-128*I*a**5*d**9*e*exp(7
*c) - 128*I*a**5*d**9*f*x*exp(7*c) + 64*I*a**5*d**8*f*exp(7*c))*exp(2*d*x))*exp(-5*c)/(1024*a**6*d**10), Ne(10
24*a**6*d**10*exp(5*c), 0)), (-x**2*(I*f*exp(4*c) - 2*f*exp(3*c) - 2*f*exp(c) - I*f)*exp(-2*c)/(8*a) - x*(I*e*
exp(4*c) - 2*e*exp(3*c) - 2*e*exp(c) - I*e)*exp(-2*c)/(4*a), True))

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Giac [B]  time = 1.19529, size = 332, normalized size = 3.39 \begin{align*} \frac{-2 i \, d f x e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d f x e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d f x e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d f x e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d f x e^{\left (d x + 2 \, c\right )} - 2 \, d f x e^{c} - 2 i \, d e^{\left (5 \, d x + 6 \, c + 1\right )} + i \, f e^{\left (5 \, d x + 6 \, c\right )} + 6 \, d e^{\left (4 \, d x + 5 \, c + 1\right )} - 7 \, f e^{\left (4 \, d x + 5 \, c\right )} - 8 i \, d e^{\left (3 \, d x + 4 \, c + 1\right )} + 8 i \, f e^{\left (3 \, d x + 4 \, c\right )} - 8 \, d e^{\left (2 \, d x + 3 \, c + 1\right )} - 8 \, f e^{\left (2 \, d x + 3 \, c\right )} + 6 i \, d e^{\left (d x + 2 \, c + 1\right )} + 7 i \, f e^{\left (d x + 2 \, c\right )} - 2 \, d e^{\left (c + 1\right )} - f e^{c}}{16 \, a d^{2} e^{\left (3 \, d x + 4 \, c\right )} - 16 i \, a d^{2} e^{\left (2 \, d x + 3 \, c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

(-2*I*d*f*x*e^(5*d*x + 6*c) + 6*d*f*x*e^(4*d*x + 5*c) - 8*I*d*f*x*e^(3*d*x + 4*c) - 8*d*f*x*e^(2*d*x + 3*c) +
6*I*d*f*x*e^(d*x + 2*c) - 2*d*f*x*e^c - 2*I*d*e^(5*d*x + 6*c + 1) + I*f*e^(5*d*x + 6*c) + 6*d*e^(4*d*x + 5*c +
1) - 7*f*e^(4*d*x + 5*c) - 8*I*d*e^(3*d*x + 4*c + 1) + 8*I*f*e^(3*d*x + 4*c) - 8*d*e^(2*d*x + 3*c + 1) - 8*f*
e^(2*d*x + 3*c) + 6*I*d*e^(d*x + 2*c + 1) + 7*I*f*e^(d*x + 2*c) - 2*d*e^(c + 1) - f*e^c)/(16*a*d^2*e^(3*d*x +
4*c) - 16*I*a*d^2*e^(2*d*x + 3*c))