Optimal. Leaf size=59 \[ \frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{2 b}+\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{2 b} \]
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Rubi [A] time = 0.0701193, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3308, 2181} \[ \frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{2 b}+\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^m \sinh (a+b x) \, dx &=\frac{1}{2} \int e^{-i (i a+i b x)} x^m \, dx-\frac{1}{2} \int e^{i (i a+i b x)} x^m \, dx\\ &=\frac{e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b}+\frac{e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.016536, size = 53, normalized size = 0.9 \[ \frac{e^{-a} x^m \left (e^{2 a} (-b x)^{-m} \text{Gamma}(m+1,-b x)+(b x)^{-m} \text{Gamma}(m+1,b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.034, size = 73, normalized size = 1.2 \begin{align*}{\frac{{x}^{1+m}\sinh \left ( a \right ) }{1+m}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{m}{2}};\,{\frac{1}{2}},{\frac{3}{2}}+{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}}+{\frac{b{x}^{2+m}\cosh \left ( a \right ) }{2+m}{\mbox{$_1$F$_2$}(1+{\frac{m}{2}};\,{\frac{3}{2}},2+{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31893, size = 74, normalized size = 1.25 \begin{align*} \frac{1}{2} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) - \frac{1}{2} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.70492, size = 225, normalized size = 3.81 \begin{align*} \frac{\cosh \left (m \log \left (b\right ) + a\right ) \Gamma \left (m + 1, b x\right ) + \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) - \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) - \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \left (b\right ) + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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^{m} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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