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]

-Chi(b/x)*cosh(a)-Shi(b/x)*sinh(a)

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 5316

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

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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ -\cosh (a) \text {Chi}\left (\frac {b}{x}\right )-\sinh (a) \text {Shi}\left (\frac {b}{x}\right ) \]

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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fricas [A]  time = 0.50, size = 39, normalized size = 1.86 \[ -\frac {1}{2} \, {\left ({\rm Ei}\left (\frac {b}{x}\right ) + {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \relax (a) - \frac {1}{2} \, {\left ({\rm Ei}\left (\frac {b}{x}\right ) - {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \relax (a) \]

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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giac [B]  time = 0.14, size = 43, normalized size = 2.05 \[ -\frac {b {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )} + b {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 27, normalized size = 1.29 \[ \frac {{\mathrm e}^{-a} \Ei \left (1, \frac {b}{x}\right )}{2}+\frac {{\mathrm e}^{a} \Ei \left (1, -\frac {b}{x}\right )}{2} \]

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 = 0.35, size = 24, normalized size = 1.14 \[ -\frac {1}{2} \, {\rm Ei}\left (-\frac {b}{x}\right ) e^{\left (-a\right )} - \frac {1}{2} \, {\rm Ei}\left (\frac {b}{x}\right ) e^{a} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ -\mathrm {cosh}\relax (a)\,\mathrm {coshint}\left (\frac {b}{x}\right )-\mathrm {sinh}\relax (a)\,\mathrm {sinhint}\left (\frac {b}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b/x)/x,x)

[Out]

- cosh(a)*coshint(b/x) - sinh(a)*sinhint(b/x)

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sympy [A]  time = 1.08, size = 17, normalized size = 0.81 \[ - \sinh {\relax (a )} \operatorname {Shi}{\left (\frac {b}{x} \right )} - \cosh {\relax (a )} \operatorname {Chi}\left (\frac {b}{x}\right ) \]

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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