Optimal. Leaf size=29 \[ \frac{\sinh \left (a+\frac{b}{x}\right )}{b^2}-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x} \]
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Rubi [A] time = 0.0302479, antiderivative size = 29, 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 = {5320, 3296, 2637} \[ \frac{\sinh \left (a+\frac{b}{x}\right )}{b^2}-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 5320
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+\frac{b}{x}\right )}{x^3} \, dx &=-\operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x}+\frac{\operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\frac{1}{x}\right )}{b}\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x}+\frac{\sinh \left (a+\frac{b}{x}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0255047, size = 29, normalized size = 1. \[ \frac{x \sinh \left (a+\frac{b}{x}\right )-b \cosh \left (a+\frac{b}{x}\right )}{b^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 44, normalized size = 1.5 \begin{align*} -{\frac{1}{{b}^{2}} \left ( \left ( a+{\frac{b}{x}} \right ) \cosh \left ( a+{\frac{b}{x}} \right ) -\sinh \left ( a+{\frac{b}{x}} \right ) -a\cosh \left ( a+{\frac{b}{x}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.18438, size = 65, normalized size = 2.24 \begin{align*} -\frac{1}{4} \, b{\left (\frac{e^{\left (-a\right )} \Gamma \left (3, \frac{b}{x}\right )}{b^{3}} - \frac{e^{a} \Gamma \left (3, -\frac{b}{x}\right )}{b^{3}}\right )} - \frac{\sinh \left (a + \frac{b}{x}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63522, size = 73, normalized size = 2.52 \begin{align*} -\frac{b \cosh \left (\frac{a x + b}{x}\right ) - x \sinh \left (\frac{a x + b}{x}\right )}{b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.08758, size = 29, normalized size = 1. \begin{align*} \begin{cases} - \frac{\cosh{\left (a + \frac{b}{x} \right )}}{b x} + \frac{\sinh{\left (a + \frac{b}{x} \right )}}{b^{2}} & \text{for}\: b \neq 0 \\- \frac{\sinh{\left (a \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \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 \frac{\sinh \left (a + \frac{b}{x}\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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