Optimal. Leaf size=15 \[ -\frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b} \]
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Rubi [A] time = 0.0171619, antiderivative size = 15, 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 = {5321, 2637} \[ -\frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5321
Rule 2637
Rubi steps
\begin{align*} \int \frac{\cosh \left (a+\frac{b}{x^2}\right )}{x^3} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0045576, size = 15, normalized size = 1. \[ -\frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b}\sinh \left ( a+{\frac{b}{{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00659, size = 18, normalized size = 1.2 \begin{align*} -\frac{\sinh \left (a + \frac{b}{x^{2}}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66986, size = 41, normalized size = 2.73 \begin{align*} -\frac{\sinh \left (\frac{a x^{2} + b}{x^{2}}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.36162, size = 22, normalized size = 1.47 \begin{align*} \begin{cases} - \frac{\sinh{\left (a + \frac{b}{x^{2}} \right )}}{2 b} & \text{for}\: b \neq 0 \\- \frac{\cosh{\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 [A] time = 1.32703, size = 36, normalized size = 2.4 \begin{align*} -\frac{e^{\left (a + \frac{b}{x^{2}}\right )} - e^{\left (-a - \frac{b}{x^{2}}\right )}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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