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

Optimal. Leaf size=131 \[ -\frac{i \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f}+\frac{\cosh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\sinh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f} \]

[Out]

(Cosh[c - (d*e)/f]*CoshIntegral[(d*e)/f + d*x])/(a*f) - ((I/2)*CoshIntegral[(2*d*e)/f + 2*d*x]*Sinh[2*c - (2*d
*e)/f])/(a*f) + (Sinh[c - (d*e)/f]*SinhIntegral[(d*e)/f + d*x])/(a*f) - ((I/2)*Cosh[2*c - (2*d*e)/f]*SinhInteg
ral[(2*d*e)/f + 2*d*x])/(a*f)

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Rubi [A]  time = 0.324126, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {5563, 3303, 3298, 3301, 5448, 12} \[ -\frac{i \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f}+\frac{\cosh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\sinh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[c - (d*e)/f]*CoshIntegral[(d*e)/f + d*x])/(a*f) - ((I/2)*CoshIntegral[(2*d*e)/f + 2*d*x]*Sinh[2*c - (2*d
*e)/f])/(a*f) + (Sinh[c - (d*e)/f]*SinhIntegral[(d*e)/f + d*x])/(a*f) - ((I/2)*Cosh[2*c - (2*d*e)/f]*SinhInteg
ral[(2*d*e)/f + 2*d*x])/(a*f)

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 3303

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

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5448

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

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

Mathematica [A]  time = 0.418634, size = 112, normalized size = 0.85 \[ \frac{2 \cosh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (d \left (\frac{e}{f}+x\right )\right )-i \left (\sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d (e+f x)}{f}\right )+2 i \sinh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+\cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d (e+f x)}{f}\right )\right )}{2 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.114, size = 180, normalized size = 1.4 \begin{align*} -{\frac{1}{2\,af}{{\rm e}^{-{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,dx+c-{\frac{cf-de}{f}} \right ) }-{\frac{1}{2\,af}{{\rm e}^{{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-dx-c-{\frac{-cf+de}{f}} \right ) }+{\frac{{\frac{i}{4}}}{af}{{\rm e}^{2\,{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-2\,dx-2\,c-2\,{\frac{-cf+de}{f}} \right ) }-{\frac{{\frac{i}{4}}}{af}{{\rm e}^{-2\,{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,2\,dx+2\,c-2\,{\frac{cf-de}{f}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a/f*exp(-(c*f-d*e)/f)*Ei(1,d*x+c-(c*f-d*e)/f)-1/2/a/f*exp((c*f-d*e)/f)*Ei(1,-d*x-c-(-c*f+d*e)/f)+1/4*I/a/
f*exp(2*(c*f-d*e)/f)*Ei(1,-2*d*x-2*c-2*(-c*f+d*e)/f)-1/4*I/a/f*exp(-2*(c*f-d*e)/f)*Ei(1,2*d*x+2*c-2*(c*f-d*e)/
f)

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.12151, size = 258, normalized size = 1.97 \begin{align*} \frac{i \,{\rm Ei}\left (-\frac{2 \,{\left (d f x + d e\right )}}{f}\right ) e^{\left (\frac{2 \,{\left (d e - c f\right )}}{f}\right )} + 2 \,{\rm Ei}\left (-\frac{d f x + d e}{f}\right ) e^{\left (\frac{d e - c f}{f}\right )} + 2 \,{\rm Ei}\left (\frac{d f x + d e}{f}\right ) e^{\left (-\frac{d e - c f}{f}\right )} - i \,{\rm Ei}\left (\frac{2 \,{\left (d f x + d e\right )}}{f}\right ) e^{\left (-\frac{2 \,{\left (d e - c f\right )}}{f}\right )}}{4 \, a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18696, size = 208, normalized size = 1.59 \begin{align*} -\frac{{\left (i \,{\rm Ei}\left (\frac{2 \,{\left (d f x + d e\right )}}{f}\right ) e^{\left (4 \, c - \frac{2 \, d e}{f}\right )} - 2 \,{\rm Ei}\left (\frac{d f x + d e}{f}\right ) e^{\left (3 \, c - \frac{d e}{f}\right )} - 2 \,{\rm Ei}\left (-\frac{d f x + d e}{f}\right ) e^{\left (c + \frac{d e}{f}\right )} - i \,{\rm Ei}\left (-\frac{2 \,{\left (d f x + d e\right )}}{f}\right ) e^{\left (\frac{2 \, d e}{f}\right )} + 3 i \, e^{\left (2 \, c\right )} \log \left (f x + e\right ) - 3 i \, e^{\left (2 \, c\right )} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-2 \, c\right )}}{4 \, a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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