Optimal. Leaf size=34 \[ \frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b^2}-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^2} \]
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Rubi [A] time = 0.0335038, antiderivative size = 34, 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} \[ \frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b^2}-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^2} \]
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^2}\right )}{x^5} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^2}+\frac{\operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\frac{1}{x^2}\right )}{2 b}\\ &=-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^2}+\frac{\sinh \left (a+\frac{b}{x^2}\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0273929, size = 34, normalized size = 1. \[ \frac{x^2 \sinh \left (a+\frac{b}{x^2}\right )-b \cosh \left (a+\frac{b}{x^2}\right )}{2 b^2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 55, normalized size = 1.6 \begin{align*} -{\frac{-{x}^{2}+b}{4\,{x}^{2}{b}^{2}}{{\rm e}^{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}-{\frac{{x}^{2}+b}{4\,{x}^{2}{b}^{2}}{{\rm e}^{-{\frac{a{x}^{2}+b}{{x}^{2}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.31073, size = 65, normalized size = 1.91 \begin{align*} -\frac{1}{8} \, b{\left (\frac{e^{\left (-a\right )} \Gamma \left (3, \frac{b}{x^{2}}\right )}{b^{3}} - \frac{e^{a} \Gamma \left (3, -\frac{b}{x^{2}}\right )}{b^{3}}\right )} - \frac{\sinh \left (a + \frac{b}{x^{2}}\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71071, size = 93, normalized size = 2.74 \begin{align*} \frac{x^{2} \sinh \left (\frac{a x^{2} + b}{x^{2}}\right ) - b \cosh \left (\frac{a x^{2} + b}{x^{2}}\right )}{2 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.0673, size = 37, normalized size = 1.09 \begin{align*} \begin{cases} - \frac{\cosh{\left (a + \frac{b}{x^{2}} \right )}}{2 b x^{2}} + \frac{\sinh{\left (a + \frac{b}{x^{2}} \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\- \frac{\sinh{\left (a \right )}}{4 x^{4}} & \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^{2}}\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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