3.1.32 \(\int \frac {\sinh (a+\frac {b}{x})}{x} \, dx\) [32]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {5426, 5425, 5424} \begin {gather*} \sinh (a) \left (-\text {Chi}\left (\frac {b}{x}\right )\right )-\cosh (a) \text {Shi}\left (\frac {b}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x]/x,x]

[Out]

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

Rule 5424

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

Rule 5425

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

Rule 5426

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

Rubi steps

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

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b/x]/x,x]

[Out]

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

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Maple [A]
time = 0.78, size = 27, normalized size = 1.29

method	result	size
risch	\(-\frac {{\mathrm e}^{-a} \expIntegral \left (1, \frac {b}{x}\right )}{2}+\frac {{\mathrm e}^{a} \expIntegral \left (1, -\frac {b}{x}\right )}{2}\)	\(27\)
meijerg	\(-\cosh \left (a \right ) \hyperbolicSineIntegral \left (\frac {b}{x}\right )-\frac {\sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {2 \hyperbolicCosineIntegral \left (\frac {b}{x}\right )-2 \ln \left (\frac {b}{x}\right )-2 \gamma }{\sqrt {\pi }}+\frac {2 \gamma -2 \ln \left (x \right )+2 \ln \left (i b \right )}{\sqrt {\pi }}\right )}{2}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b/x)/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 24, normalized size = 1.14 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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 = 0.33, size = 39, normalized size = 1.86 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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 = 0.61, size = 17, normalized size = 0.81 \begin {gather*} - \sinh {\left (a \right )} \operatorname {Chi}\left (\frac {b}{x}\right ) - \cosh {\left (a \right )} \operatorname {Shi}{\left (\frac {b}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sinh(a)*Chi(b/x) - cosh(a)*Shi(b/x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).
time = 0.44, size = 44, normalized size = 2.10 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\mathrm {sinh}\left (a\right )\,\mathrm {coshint}\left (\frac {b}{x}\right )-\mathrm {cosh}\left (a\right )\,\mathrm {sinhint}\left (\frac {b}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b/x)/x,x)

[Out]

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

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