Optimal. Leaf size=75 \[ \frac{e^{-a} \left (b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},b x^n\right )}{2 n x^2}-\frac{e^a \left (-b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-b x^n\right )}{2 n x^2} \]
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Rubi [A] time = 0.062377, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5360, 2218} \[ \frac{e^{-a} \left (b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},b x^n\right )}{2 n x^2}-\frac{e^a \left (-b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-b x^n\right )}{2 n x^2} \]
Antiderivative was successfully verified.
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Rule 5360
Rule 2218
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+b x^n\right )}{x^3} \, dx &=-\left (\frac{1}{2} \int \frac{e^{-a-b x^n}}{x^3} \, dx\right )+\frac{1}{2} \int \frac{e^{a+b x^n}}{x^3} \, dx\\ &=-\frac{e^a \left (-b x^n\right )^{2/n} \Gamma \left (-\frac{2}{n},-b x^n\right )}{2 n x^2}+\frac{e^{-a} \left (b x^n\right )^{2/n} \Gamma \left (-\frac{2}{n},b x^n\right )}{2 n x^2}\\ \end{align*}
Mathematica [A] time = 0.0717494, size = 72, normalized size = 0.96 \[ \frac{(\cosh (a)-\sinh (a)) \left (b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},b x^n\right )-(\sinh (a)+\cosh (a)) \left (-b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-b x^n\right )}{2 n x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.039, size = 77, normalized size = 1. \begin{align*} -{\frac{\sinh \left ( a \right ) }{2\,{x}^{2}}{\mbox{$_1$F$_2$}(-{n}^{-1};\,{\frac{1}{2}},1-{n}^{-1};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}}+{\frac{{x}^{-2+n}b\cosh \left ( a \right ) }{-2+n}{\mbox{$_1$F$_2$}({\frac{1}{2}}-{n}^{-1};\,{\frac{3}{2}},{\frac{3}{2}}-{n}^{-1};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1991, size = 93, normalized size = 1.24 \begin{align*} \frac{\left (b x^{n}\right )^{\frac{2}{n}} e^{\left (-a\right )} \Gamma \left (-\frac{2}{n}, b x^{n}\right )}{2 \, n x^{2}} - \frac{\left (-b x^{n}\right )^{\frac{2}{n}} e^{a} \Gamma \left (-\frac{2}{n}, -b x^{n}\right )}{2 \, n x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sinh \left (b x^{n} + a\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b x^{n} \right )}}{x^{3}}\, dx \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 (b x^{n} + a\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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