Optimal. Leaf size=13 \[ -\frac{\cosh \left (a+\frac{b}{x}\right )}{b} \]
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Rubi [A] time = 0.0169405, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5320, 2638} \[ -\frac{\cosh \left (a+\frac{b}{x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 5320
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+\frac{b}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0045134, size = 13, normalized size = 1. \[ -\frac{\cosh \left (a+\frac{b}{x}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{b}\cosh \left ( a+{\frac{b}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05249, size = 18, normalized size = 1.38 \begin{align*} -\frac{\cosh \left (a + \frac{b}{x}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62389, size = 30, normalized size = 2.31 \begin{align*} -\frac{\cosh \left (\frac{a x + b}{x}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.56414, size = 15, normalized size = 1.15 \begin{align*} \begin{cases} - \frac{\cosh{\left (a + \frac{b}{x} \right )}}{b} & \text{for}\: b \neq 0 \\- \frac{\sinh{\left (a \right )}}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25665, size = 34, normalized size = 2.62 \begin{align*} -\frac{e^{\left (a + \frac{b}{x}\right )} + e^{\left (-a - \frac{b}{x}\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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