Optimal. Leaf size=47 \[ \frac{\sinh \left (a+\frac{b}{x^2}\right )}{b^2 x^2}-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{b^3}-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^4} \]
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Rubi [A] time = 0.0586295, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5320, 3296, 2638} \[ \frac{\sinh \left (a+\frac{b}{x^2}\right )}{b^2 x^2}-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{b^3}-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^4} \]
Antiderivative was successfully verified.
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Rule 5320
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+\frac{b}{x^2}\right )}{x^7} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^4}+\frac{\operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,\frac{1}{x^2}\right )}{b}\\ &=-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^4}+\frac{\sinh \left (a+\frac{b}{x^2}\right )}{b^2 x^2}-\frac{\operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\frac{1}{x^2}\right )}{b^2}\\ &=-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{b^3}-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b x^4}+\frac{\sinh \left (a+\frac{b}{x^2}\right )}{b^2 x^2}\\ \end{align*}
Mathematica [A] time = 0.043287, size = 44, normalized size = 0.94 \[ \frac{2 b x^2 \sinh \left (a+\frac{b}{x^2}\right )-\left (b^2+2 x^4\right ) \cosh \left (a+\frac{b}{x^2}\right )}{2 b^3 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 73, normalized size = 1.6 \begin{align*} -{\frac{2\,{x}^{4}-2\,b{x}^{2}+{b}^{2}}{4\,{b}^{3}{x}^{4}}{{\rm e}^{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}-{\frac{2\,{x}^{4}+2\,b{x}^{2}+{b}^{2}}{4\,{b}^{3}{x}^{4}}{{\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 = 1.1839, size = 63, normalized size = 1.34 \begin{align*} -\frac{1}{12} \, b{\left (\frac{e^{\left (-a\right )} \Gamma \left (4, \frac{b}{x^{2}}\right )}{b^{4}} + \frac{e^{a} \Gamma \left (4, -\frac{b}{x^{2}}\right )}{b^{4}}\right )} - \frac{\sinh \left (a + \frac{b}{x^{2}}\right )}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65089, size = 115, normalized size = 2.45 \begin{align*} \frac{2 \, b x^{2} \sinh \left (\frac{a x^{2} + b}{x^{2}}\right ) -{\left (2 \, x^{4} + b^{2}\right )} \cosh \left (\frac{a x^{2} + b}{x^{2}}\right )}{2 \, b^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 48.2079, size = 51, normalized size = 1.09 \begin{align*} \begin{cases} - \frac{\cosh{\left (a + \frac{b}{x^{2}} \right )}}{2 b x^{4}} + \frac{\sinh{\left (a + \frac{b}{x^{2}} \right )}}{b^{2} x^{2}} - \frac{\cosh{\left (a + \frac{b}{x^{2}} \right )}}{b^{3}} & \text{for}\: b \neq 0 \\- \frac{\sinh{\left (a \right )}}{6 x^{6}} & \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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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