Optimal. Leaf size=49 \[ \frac{1}{2} e^{-a} x^m (b x)^{-m} \text{Gamma}(m,b x)-\frac{1}{2} e^a x^m (-b x)^{-m} \text{Gamma}(m,-b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0694422, antiderivative size = 49, 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} \[ \frac{1}{2} e^{-a} x^m (b x)^{-m} \text{Gamma}(m,b x)-\frac{1}{2} e^a x^m (-b x)^{-m} \text{Gamma}(m,-b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^{-1+m} \sinh (a+b x) \, dx &=\frac{1}{2} \int e^{-i (i a+i b x)} x^{-1+m} \, dx-\frac{1}{2} \int e^{i (i a+i b x)} x^{-1+m} \, dx\\ &=-\frac{1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)+\frac{1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x)\\ \end{align*}
Mathematica [A] time = 0.022222, size = 49, normalized size = 1. \[ \frac{1}{2} e^{-a} x^m (b x)^{-m} \text{Gamma}(m,b x)-\frac{1}{2} e^a x^m (-b x)^{-m} \text{Gamma}(m,-b x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.041, size = 67, normalized size = 1.4 \begin{align*}{\frac{{x}^{m}\sinh \left ( a \right ) }{m}{\mbox{$_1$F$_2$}({\frac{m}{2}};\,{\frac{1}{2}},1+{\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{3}{2}}+{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.21694, size = 58, normalized size = 1.18 \begin{align*} \frac{x^{m} e^{\left (-a\right )} \Gamma \left (m, b x\right )}{2 \, \left (b x\right )^{m}} - \frac{x^{m} e^{a} \Gamma \left (m, -b x\right )}{2 \, \left (-b x\right )^{m}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.69251, size = 236, normalized size = 4.82 \begin{align*} \frac{\cosh \left ({\left (m - 1\right )} \log \left (b\right ) + a\right ) \Gamma \left (m, b x\right ) + \cosh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m, -b x\right ) - \Gamma \left (m, -b x\right ) \sinh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m, b x\right ) \sinh \left ({\left (m - 1\right )} \log \left (b\right ) + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 1} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]