Optimal. Leaf size=67 \[ -\frac{1}{2} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )-\frac{1}{2} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right ) \]
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Rubi [A] time = 0.0875178, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5350, 3308, 2181} \[ -\frac{1}{2} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )-\frac{1}{2} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5350
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int (e x)^m \sinh \left (a+\frac{b}{x}\right ) \, dx &=-\left (\left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sinh (a+b x) \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\frac{1}{2} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{-i (i a+i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{i (i a+i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} b e^a \left (-\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac{b}{x}\right )-\frac{1}{2} b e^{-a} \left (\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac{b}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0752309, size = 63, normalized size = 0.94 \[ -\frac{1}{2} b (e x)^m \left ((\sinh (a)+\cosh (a)) \left (-\frac{b}{x}\right )^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )+(\cosh (a)-\sinh (a)) \left (\frac{b}{x}\right )^m \text{Gamma}\left (-m-1,\frac{b}{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.034, size = 70, normalized size = 1. \begin{align*}{\frac{ \left ( ex \right ) ^{m}b\cosh \left ( a \right ) }{m}{\mbox{$_1$F$_2$}(-{\frac{m}{2}};\,{\frac{3}{2}},1-{\frac{m}{2}};\,{\frac{{b}^{2}}{4\,{x}^{2}}})}}+{\frac{ \left ( ex \right ) ^{m}x\sinh \left ( a \right ) }{1+m}{\mbox{$_1$F$_2$}(-{\frac{1}{2}}-{\frac{m}{2}};\,{\frac{1}{2}},{\frac{1}{2}}-{\frac{m}{2}};\,{\frac{{b}^{2}}{4\,{x}^{2}}})}} \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 (a + \frac{b}{x}\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 (\frac{a x + b}{x}\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 + \frac{b}{x} \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 (a + \frac{b}{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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