Optimal. Leaf size=75 \[ \frac{e^{-a} x^3 \left (b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},b x^n\right )}{2 n}-\frac{e^a x^3 \left (-b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b x^n\right )}{2 n} \]
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Rubi [A] time = 0.0738682, 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} \[ \frac{e^{-a} x^3 \left (b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},b x^n\right )}{2 n}-\frac{e^a x^3 \left (-b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 5360
Rule 2218
Rubi steps
\begin{align*} \int x^2 \sinh \left (a+b x^n\right ) \, dx &=-\left (\frac{1}{2} \int e^{-a-b x^n} x^2 \, dx\right )+\frac{1}{2} \int e^{a+b x^n} x^2 \, dx\\ &=-\frac{e^a x^3 \left (-b x^n\right )^{-3/n} \Gamma \left (\frac{3}{n},-b x^n\right )}{2 n}+\frac{e^{-a} x^3 \left (b x^n\right )^{-3/n} \Gamma \left (\frac{3}{n},b x^n\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0947994, size = 88, normalized size = 1.17 \[ -\frac{x^3 \left (-b^2 x^{2 n}\right )^{-3/n} \left ((\sinh (a)+\cosh (a)) \left (b x^n\right )^{3/n} \text{Gamma}\left (\frac{3}{n},-b x^n\right )-(\cosh (a)-\sinh (a)) \left (-b x^n\right )^{3/n} \text{Gamma}\left (\frac{3}{n},b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.09, size = 77, normalized size = 1. \begin{align*}{\frac{{x}^{3}\sinh \left ( a \right ) }{3}{\mbox{$_1$F$_2$}({\frac{3}{2\,n}};\,{\frac{1}{2}},1+{\frac{3}{2\,n}};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}}+{\frac{{x}^{n+3}b\cosh \left ( a \right ) }{n+3}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{3}{2\,n}};\,{\frac{3}{2}},{\frac{3}{2}}+{\frac{3}{2\,n}};\,{\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 = 1.2102, size = 99, normalized size = 1.32 \begin{align*} \frac{x^{3} e^{\left (-a\right )} \Gamma \left (\frac{3}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\frac{3}{n}} n} - \frac{x^{3} e^{a} \Gamma \left (\frac{3}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\frac{3}{n}} n} \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 (x^{2} \sinh \left (b x^{n} + a\right ), 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 x^{2} \sinh{\left (a + b x^{n} \right )}\, 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 x^{2} \sinh \left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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