Optimal. Leaf size=99 \[ \frac{e^{-a} (e x)^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )}{2 e n}-\frac{e^a (e x)^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )}{2 e n} \]
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Rubi [A] time = 0.0705051, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5360, 2218} \[ \frac{e^{-a} (e x)^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )}{2 e n}-\frac{e^a (e x)^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )}{2 e n} \]
Antiderivative was successfully verified.
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Rule 5360
Rule 2218
Rubi steps
\begin{align*} \int (e x)^m \sinh \left (a+b x^n\right ) \, dx &=-\left (\frac{1}{2} \int e^{-a-b x^n} (e x)^m \, dx\right )+\frac{1}{2} \int e^{a+b x^n} (e x)^m \, dx\\ &=-\frac{e^a (e x)^{1+m} \left (-b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-b x^n\right )}{2 e n}+\frac{e^{-a} (e x)^{1+m} \left (b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},b x^n\right )}{2 e n}\\ \end{align*}
Mathematica [A] time = 0.170415, size = 102, normalized size = 1.03 \[ -\frac{x (e x)^m \left (-b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \left ((\sinh (a)+\cosh (a)) \left (b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )-(\cosh (a)-\sinh (a)) \left (-b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.22, size = 115, normalized size = 1.2 \begin{align*}{\frac{ \left ( ex \right ) ^{m}x\sinh \left ( a \right ) }{1+m}{\mbox{$_1$F$_2$}({\frac{m}{2\,n}}+{\frac{1}{2\,n}};\,{\frac{1}{2}},1+{\frac{m}{2\,n}}+{\frac{1}{2\,n}};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}}+{\frac{ \left ( ex \right ) ^{m}{x}^{n+1}b\cosh \left ( a \right ) }{m+n+1}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{m}{2\,n}}+{\frac{1}{2\,n}};\,{\frac{3}{2}},{\frac{3}{2}}+{\frac{m}{2\,n}}+{\frac{1}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \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 \left (e x\right )^{m} \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 \left (e x\right )^{m} \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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