Optimal. Leaf size=33 \[ -b \cosh (a) \text{Chi}\left (\frac{b}{x}\right )-b \sinh (a) \text{Shi}\left (\frac{b}{x}\right )+x \sinh \left (a+\frac{b}{x}\right ) \]
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Rubi [A] time = 0.0757435, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5302, 3297, 3303, 3298, 3301} \[ -b \cosh (a) \text{Chi}\left (\frac{b}{x}\right )-b \sinh (a) \text{Shi}\left (\frac{b}{x}\right )+x \sinh \left (a+\frac{b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5302
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \sinh \left (a+\frac{b}{x}\right ) \, dx &=-\operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=x \sinh \left (a+\frac{b}{x}\right )-b \operatorname{Subst}\left (\int \frac{\cosh (a+b x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=x \sinh \left (a+\frac{b}{x}\right )-(b \cosh (a)) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,\frac{1}{x}\right )-(b \sinh (a)) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=-b \cosh (a) \text{Chi}\left (\frac{b}{x}\right )+x \sinh \left (a+\frac{b}{x}\right )-b \sinh (a) \text{Shi}\left (\frac{b}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0196536, size = 33, normalized size = 1. \[ -b \cosh (a) \text{Chi}\left (\frac{b}{x}\right )-b \sinh (a) \text{Shi}\left (\frac{b}{x}\right )+x \sinh \left (a+\frac{b}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 56, normalized size = 1.7 \begin{align*}{\frac{b{{\rm e}^{-a}}}{2}{\it Ei} \left ( 1,{\frac{b}{x}} \right ) }-{\frac{x}{2}{{\rm e}^{-{\frac{ax+b}{x}}}}}+{\frac{{{\rm e}^{a}}b}{2}{\it Ei} \left ( 1,-{\frac{b}{x}} \right ) }+{\frac{x}{2}{{\rm e}^{{\frac{ax+b}{x}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15812, size = 49, normalized size = 1.48 \begin{align*} -\frac{1}{2} \,{\left ({\rm Ei}\left (-\frac{b}{x}\right ) e^{\left (-a\right )} +{\rm Ei}\left (\frac{b}{x}\right ) e^{a}\right )} b + x \sinh \left (a + \frac{b}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6999, size = 135, normalized size = 4.09 \begin{align*} -\frac{1}{2} \,{\left (b{\rm Ei}\left (\frac{b}{x}\right ) + b{\rm Ei}\left (-\frac{b}{x}\right )\right )} \cosh \left (a\right ) - \frac{1}{2} \,{\left (b{\rm Ei}\left (\frac{b}{x}\right ) - b{\rm Ei}\left (-\frac{b}{x}\right )\right )} \sinh \left (a\right ) + x \sinh \left (\frac{a x + b}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (a + \frac{b}{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (a + \frac{b}{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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