3.26 \(\int \frac{\cosh (a+\frac{b}{x})}{x} \, dx\)

Optimal. Leaf size=21 \[ -\cosh (a) \text{Chi}\left (\frac{b}{x}\right )-\sinh (a) \text{Shi}\left (\frac{b}{x}\right ) \]

[Out]

-(Cosh[a]*CoshIntegral[b/x]) - Sinh[a]*SinhIntegral[b/x]

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Rubi [A]  time = 0.0318961, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5319, 5317, 5316} \[ -\cosh (a) \text{Chi}\left (\frac{b}{x}\right )-\sinh (a) \text{Shi}\left (\frac{b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b/x]/x,x]

[Out]

-(Cosh[a]*CoshIntegral[b/x]) - Sinh[a]*SinhIntegral[b/x]

Rule 5319

Int[Cosh[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Cosh[c], Int[Cosh[d*x^n]/x, x], x] + Dist[Sinh[c], In
t[Sinh[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 5317

Int[Cosh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CoshIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 5316

Int[Sinh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinhIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rubi steps

\begin{align*} \int \frac{\cosh \left (a+\frac{b}{x}\right )}{x} \, dx &=\cosh (a) \int \frac{\cosh \left (\frac{b}{x}\right )}{x} \, dx+\sinh (a) \int \frac{\sinh \left (\frac{b}{x}\right )}{x} \, dx\\ &=-\cosh (a) \text{Chi}\left (\frac{b}{x}\right )-\sinh (a) \text{Shi}\left (\frac{b}{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0099736, size = 21, normalized size = 1. \[ -\cosh (a) \text{Chi}\left (\frac{b}{x}\right )-\sinh (a) \text{Shi}\left (\frac{b}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b/x]/x,x]

[Out]

-(Cosh[a]*CoshIntegral[b/x]) - Sinh[a]*SinhIntegral[b/x]

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Maple [A]  time = 0.026, size = 27, normalized size = 1.3 \begin{align*}{\frac{{{\rm e}^{-a}}}{2}{\it Ei} \left ( 1,{\frac{b}{x}} \right ) }+{\frac{{{\rm e}^{a}}}{2}{\it Ei} \left ( 1,-{\frac{b}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a+b/x)/x,x)

[Out]

1/2*exp(-a)*Ei(1,b/x)+1/2*exp(a)*Ei(1,-b/x)

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Maxima [A]  time = 1.32185, size = 32, normalized size = 1.52 \begin{align*} -\frac{1}{2} \,{\rm Ei}\left (-\frac{b}{x}\right ) e^{\left (-a\right )} - \frac{1}{2} \,{\rm Ei}\left (\frac{b}{x}\right ) e^{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x)/x,x, algorithm="maxima")

[Out]

-1/2*Ei(-b/x)*e^(-a) - 1/2*Ei(b/x)*e^a

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Fricas [A]  time = 1.72666, size = 95, normalized size = 4.52 \begin{align*} -\frac{1}{2} \,{\left ({\rm Ei}\left (\frac{b}{x}\right ) +{\rm Ei}\left (-\frac{b}{x}\right )\right )} \cosh \left (a\right ) - \frac{1}{2} \,{\left ({\rm Ei}\left (\frac{b}{x}\right ) -{\rm Ei}\left (-\frac{b}{x}\right )\right )} \sinh \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x)/x,x, algorithm="fricas")

[Out]

-1/2*(Ei(b/x) + Ei(-b/x))*cosh(a) - 1/2*(Ei(b/x) - Ei(-b/x))*sinh(a)

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Sympy [A]  time = 1.39757, size = 17, normalized size = 0.81 \begin{align*} - \sinh{\left (a \right )} \operatorname{Shi}{\left (\frac{b}{x} \right )} - \cosh{\left (a \right )} \operatorname{Chi}\left (\frac{b}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x)/x,x)

[Out]

-sinh(a)*Shi(b/x) - cosh(a)*Chi(b/x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (a + \frac{b}{x}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(a+b/x)/x,x, algorithm="giac")

[Out]

integrate(cosh(a + b/x)/x, x)