3.270 \(\int \frac{\cosh ^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\)

Optimal. Leaf size=180 \[ \frac{d \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f^2}-\frac{i d \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}-\frac{i d \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}+\frac{d \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f^2}+\frac{i \sinh (2 c+2 d x)}{2 a f (e+f x)}-\frac{\cosh (c+d x)}{a f (e+f x)} \]

[Out]

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

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Rubi [A]  time = 0.391807, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {5563, 3297, 3303, 3298, 3301, 5448, 12} \[ \frac{d \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f^2}-\frac{i d \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}-\frac{i d \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}+\frac{d \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f^2}+\frac{i \sinh (2 c+2 d x)}{2 a f (e+f x)}-\frac{\cosh (c+d x)}{a f (e+f x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]

[Out]

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

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 3297

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

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

Mathematica [A]  time = 0.712969, size = 212, normalized size = 1.18 \[ \frac{2 d (e+f x) \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (d \left (\frac{e}{f}+x\right )\right )-2 i d (e+f x) \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d (e+f x)}{f}\right )-2 i d e \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d (e+f x)}{f}\right )-2 i d f x \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d (e+f x)}{f}\right )+2 d e \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+2 d f x \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+i f \sinh (2 (c+d x))-2 f \cosh (c+d x)}{2 a f^2 (e+f x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]

[Out]

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

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Maple [A]  time = 0.125, size = 299, normalized size = 1.7 \begin{align*} -{\frac{d{{\rm e}^{-dx-c}}}{2\,af \left ( dfx+de \right ) }}+{\frac{d}{2\,a{f}^{2}}{{\rm e}^{-{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,dx+c-{\frac{cf-de}{f}} \right ) }-{\frac{d{{\rm e}^{dx+c}}}{2\,a{f}^{2}} \left ({\frac{de}{f}}+dx \right ) ^{-1}}-{\frac{d}{2\,a{f}^{2}}{{\rm e}^{{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-dx-c-{\frac{-cf+de}{f}} \right ) }+{\frac{{\frac{i}{4}}d{{\rm e}^{2\,dx+2\,c}}}{a{f}^{2}} \left ({\frac{de}{f}}+dx \right ) ^{-1}}+{\frac{{\frac{i}{2}}d}{a{f}^{2}}{{\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}}d{{\rm e}^{-2\,dx-2\,c}}}{af \left ( dfx+de \right ) }}+{\frac{{\frac{i}{2}}d}{a{f}^{2}}{{\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)^2/(a+I*a*sinh(d*x+c)),x)

[Out]

-1/2/a*d*exp(-d*x-c)/f/(d*f*x+d*e)+1/2/a*d/f^2*exp(-(c*f-d*e)/f)*Ei(1,d*x+c-(c*f-d*e)/f)-1/2*d/a/f^2*exp(d*x+c
)/(d*e/f+d*x)-1/2*d/a/f^2*exp((c*f-d*e)/f)*Ei(1,-d*x-c-(-c*f+d*e)/f)+1/4*I*d/a/f^2*exp(2*d*x+2*c)/(d*e/f+d*x)+
1/2*I*d/a/f^2*exp(2*(c*f-d*e)/f)*Ei(1,-2*d*x-2*c-2*(-c*f+d*e)/f)-1/4*I/a*d*exp(-2*d*x-2*c)/f/(d*f*x+d*e)+1/2*I
/a*d/f^2*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)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.21271, size = 510, normalized size = 2.83 \begin{align*} \frac{{\left (i \, f e^{\left (4 \, d x + 4 \, c\right )} - 2 \, f e^{\left (3 \, d x + 3 \, c\right )} +{\left ({\left (-2 i \, d f x - 2 i \, d e\right )}{\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 \,{\left (d f x + d e\right )}{\rm Ei}\left (-\frac{d f x + d e}{f}\right ) e^{\left (\frac{d e - c f}{f}\right )} + 2 \,{\left (d f x + d e\right )}{\rm Ei}\left (\frac{d f x + d e}{f}\right ) e^{\left (-\frac{d e - c f}{f}\right )} +{\left (-2 i \, d f x - 2 i \, d e\right )}{\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 )}\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, f e^{\left (d x + c\right )} - i \, f\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{4 \,{\left (a f^{3} x + a e f^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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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)**2/(a+I*a*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError