Optimal. Leaf size=70 \[ \frac{e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}+\frac{e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.117687, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5448, 12, 3308, 2181} \[ \frac{e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}+\frac{e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5448
Rule 12
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^m \cosh (a+b x) \sinh (a+b x) \, dx &=\int \frac{1}{2} x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{2} \int x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx-\frac{1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx\\ &=\frac{2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac{2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0459349, size = 66, normalized size = 0.94 \[ \frac{e^{-2 a} 2^{-m-3} x^m \left (-b^2 x^2\right )^{-m} \left (e^{4 a} (b x)^m \text{Gamma}(m+1,-2 b x)+(-b x)^m \text{Gamma}(m+1,2 b x)\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.26501, size = 80, normalized size = 1.14 \begin{align*} \frac{1}{4} \, \left (2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (m + 1, 2 \, b x\right ) - \frac{1}{4} \, \left (-2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (2 \, a\right )} \Gamma \left (m + 1, -2 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.20564, size = 258, normalized size = 3.69 \begin{align*} \frac{\cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) + \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) - \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) - \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right )}{8 \, 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} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]