Optimal. Leaf size=59 \[ \frac{1}{2} e^{-a} b^2 x^m (b x)^{-m} \text{Gamma}(m-2,b x)-\frac{1}{2} e^a b^2 x^m (-b x)^{-m} \text{Gamma}(m-2,-b x) \]
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Rubi [A] time = 0.0708014, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3308, 2181} \[ \frac{1}{2} e^{-a} b^2 x^m (b x)^{-m} \text{Gamma}(m-2,b x)-\frac{1}{2} e^a b^2 x^m (-b x)^{-m} \text{Gamma}(m-2,-b x) \]
Antiderivative was successfully verified.
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Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^{-3+m} \sinh (a+b x) \, dx &=\frac{1}{2} \int e^{-i (i a+i b x)} x^{-3+m} \, dx-\frac{1}{2} \int e^{i (i a+i b x)} x^{-3+m} \, dx\\ &=-\frac{1}{2} b^2 e^a x^m (-b x)^{-m} \Gamma (-2+m,-b x)+\frac{1}{2} b^2 e^{-a} x^m (b x)^{-m} \Gamma (-2+m,b x)\\ \end{align*}
Mathematica [A] time = 0.0217781, size = 54, normalized size = 0.92 \[ \frac{1}{2} e^{-a} b^2 x^m \left ((b x)^{-m} \text{Gamma}(m-2,b x)-e^{2 a} (-b x)^{-m} \text{Gamma}(m-2,-b x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.03, size = 71, normalized size = 1.2 \begin{align*}{\frac{{x}^{-2+m}\sinh \left ( a \right ) }{-2+m}{\mbox{$_1$F$_2$}(-1+{\frac{m}{2}};\,{\frac{1}{2}},{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}}+{\frac{b{x}^{-1+m}\cosh \left ( a \right ) }{-1+m}{\mbox{$_1$F$_2$}(-{\frac{1}{2}}+{\frac{m}{2}};\,{\frac{3}{2}},{\frac{1}{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 = 1.32526, size = 74, normalized size = 1.25 \begin{align*} \frac{1}{2} \, \left (b x\right )^{-m + 2} x^{m - 2} e^{\left (-a\right )} \Gamma \left (m - 2, b x\right ) - \frac{1}{2} \, \left (-b x\right )^{-m + 2} x^{m - 2} e^{a} \Gamma \left (m - 2, -b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69793, size = 258, normalized size = 4.37 \begin{align*} \frac{\cosh \left ({\left (m - 3\right )} \log \left (b\right ) + a\right ) \Gamma \left (m - 2, b x\right ) + \cosh \left ({\left (m - 3\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m - 2, -b x\right ) - \Gamma \left (m - 2, -b x\right ) \sinh \left ({\left (m - 3\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m - 2, b x\right ) \sinh \left ({\left (m - 3\right )} \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 - 3} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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