Optimal. Leaf size=146 \[ -\frac{1}{8} e^{3 a} b 3^{m+1} \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{3 b}{x}\right )+\frac{3}{8} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )+\frac{3}{8} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right )-\frac{1}{8} e^{-3 a} b 3^{m+1} \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{3 b}{x}\right ) \]
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Rubi [A] time = 0.250817, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5350, 3312, 3308, 2181} \[ -\frac{1}{8} e^{3 a} b 3^{m+1} \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{3 b}{x}\right )+\frac{3}{8} e^a b \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{b}{x}\right )+\frac{3}{8} e^{-a} b \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{b}{x}\right )-\frac{1}{8} e^{-3 a} b 3^{m+1} \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{3 b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5350
Rule 3312
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int (e x)^m \sinh ^3\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 ^3(a+b x) \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (i \left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int \left (\frac{3}{4} i x^{-2-m} \sinh (a+b x)-\frac{1}{4} i x^{-2-m} \sinh (3 a+3 b x)\right ) \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\frac{1}{4} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \sinh (3 a+3 b x) \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{4} \left (3 \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 )\\ &=-\left (\frac{1}{8} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{-i (3 i a+3 i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{8} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{i (3 i a+3 i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 \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}{8} \left (3 \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}{8} 3^{1+m} b e^{3 a} \left (-\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac{3 b}{x}\right )+\frac{3}{8} b e^a \left (-\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac{b}{x}\right )+\frac{3}{8} b e^{-a} \left (\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac{b}{x}\right )-\frac{1}{8} 3^{1+m} b e^{-3 a} \left (\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac{3 b}{x}\right )\\ \end{align*}
Mathematica [F] time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \sinh \left ( a+{\frac{b}{x}} \right ) \right ) ^{3}\, dx \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 )^{3}\,{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 )^{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 \left (e x\right )^{m} \sinh ^{3}{\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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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