Optimal. Leaf size=90 \[ -e^{2 a} b 2^{m-1} \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{2 b}{x}\right )+e^{-2 a} b 2^{m-1} \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{2 b}{x}\right )-\frac{x (e x)^m}{2 (m+1)} \]
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Rubi [A] time = 0.157582, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5350, 3312, 3307, 2181} \[ -e^{2 a} b 2^{m-1} \left (-\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,-\frac{2 b}{x}\right )+e^{-2 a} b 2^{m-1} \left (\frac{b}{x}\right )^m (e x)^m \text{Gamma}\left (-m-1,\frac{2 b}{x}\right )-\frac{x (e x)^m}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 5350
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int (e x)^m \sinh ^2\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 ^2(a+b x) \, dx,x,\frac{1}{x}\right )\right )\\ &=\left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int \left (\frac{x^{-2-m}}{2}-\frac{1}{2} x^{-2-m} \cosh (2 a+2 b x)\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x (e x)^m}{2 (1+m)}-\frac{1}{2} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int x^{-2-m} \cosh (2 a+2 b x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x (e x)^m}{2 (1+m)}-\frac{1}{4} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{-i (2 i a+2 i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (\left (\frac{1}{x}\right )^m (e x)^m\right ) \operatorname{Subst}\left (\int e^{i (2 i a+2 i b x)} x^{-2-m} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{x (e x)^m}{2 (1+m)}-2^{-1+m} b e^{2 a} \left (-\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,-\frac{2 b}{x}\right )+2^{-1+m} b e^{-2 a} \left (\frac{b}{x}\right )^m (e x)^m \Gamma \left (-1-m,\frac{2 b}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.25195, size = 88, normalized size = 0.98 \[ -\frac{(e x)^m \left (b 2^m (m+1) (\sinh (a)+\cosh (a))^2 \left (-\frac{b}{x}\right )^m \text{Gamma}\left (-m-1,-\frac{2 b}{x}\right )-b 2^m (m+1) (\cosh (a)-\sinh (a))^2 \left (\frac{b}{x}\right )^m \text{Gamma}\left (-m-1,\frac{2 b}{x}\right )+x\right )}{2 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \sinh \left ( a+{\frac{b}{x}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 )^{2}, 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 ^{2}{\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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