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

Optimal. Leaf size=76 \[ -\frac{i \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\log (e+f x)}{a f} \]

[Out]

Log[e + f*x]/(a*f) - (I*CoshIntegral[(d*e)/f + d*x]*Sinh[c - (d*e)/f])/(a*f) - (I*Cosh[c - (d*e)/f]*SinhIntegr
al[(d*e)/f + d*x])/(a*f)

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Rubi [A]  time = 0.204152, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {5563, 31, 3303, 3298, 3301} \[ -\frac{i \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\log (e+f x)}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[e + f*x]/(a*f) - (I*CoshIntegral[(d*e)/f + d*x]*Sinh[c - (d*e)/f])/(a*f) - (I*Cosh[c - (d*e)/f]*SinhIntegr
al[(d*e)/f + 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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]

Rubi steps

\begin{align*} \int \frac{\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=-\frac{i \int \frac{\sinh (c+d x)}{e+f x} \, dx}{a}+\frac{\int \frac{1}{e+f x} \, dx}{a}\\ &=\frac{\log (e+f x)}{a f}-\frac{\left (i \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}-\frac{\left (i \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}\\ &=\frac{\log (e+f x)}{a f}-\frac{i \text{Chi}\left (\frac{d e}{f}+d x\right ) \sinh \left (c-\frac{d e}{f}\right )}{a f}-\frac{i \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}\\ \end{align*}

Mathematica [A]  time = 0.320749, size = 62, normalized size = 0.82 \[ \frac{-i \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (d \left (\frac{e}{f}+x\right )\right )-i \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+\log (e+f x)}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.09, size = 103, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( fx+e \right ) }{af}}+{\frac{{\frac{i}{2}}}{af}{{\rm e}^{{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-dx-c-{\frac{-cf+de}{f}} \right ) }-{\frac{{\frac{i}{2}}}{af}{{\rm e}^{-{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,dx+c-{\frac{cf-de}{f}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.51494, size = 103, normalized size = 1.36 \begin{align*} -\frac{i \, e^{\left (-c + \frac{d e}{f}\right )} E_{1}\left (\frac{{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac{i \, e^{\left (c - \frac{d e}{f}\right )} E_{1}\left (-\frac{{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac{\log \left (f x + e\right )}{a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*e^(-c + d*e/f)*exp_integral_e(1, (f*x + e)*d/f)/(a*f) + 1/2*I*e^(c - d*e/f)*exp_integral_e(1, -(f*x + e
)*d/f)/(a*f) + log(f*x + e)/(a*f)

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Fricas [A]  time = 2.18993, size = 159, normalized size = 2.09 \begin{align*} \frac{i \,{\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{d f x + d e}{f}\right ) e^{\left (-\frac{d e - c f}{f}\right )} + 2 \, \log \left (\frac{f x + e}{f}\right )}{2 \, a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.18928, size = 109, normalized size = 1.43 \begin{align*} -\frac{{\left (i \,{\rm Ei}\left (\frac{d f x + d e}{f}\right ) e^{\left (2 \, c - \frac{d e}{f}\right )} - i \,{\rm Ei}\left (-\frac{d f x + d e}{f}\right ) e^{\left (\frac{d e}{f}\right )} - 2 \, e^{c} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-c\right )}}{2 \, a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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