Optimal. Leaf size=134 \[ \frac{e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-3 b x)}{8 b}+\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{8 b}+\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{8 b}+\frac{e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,3 b x)}{8 b} \]
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Rubi [A] time = 0.196015, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5448, 3308, 2181} \[ \frac{e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-3 b x)}{8 b}+\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{8 b}+\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{8 b}+\frac{e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,3 b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^m \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\int \left (\frac{1}{4} x^m \sinh (a+b x)+\frac{1}{4} x^m \sinh (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int x^m \sinh (a+b x) \, dx+\frac{1}{4} \int x^m \sinh (3 a+3 b x) \, dx\\ &=\frac{1}{8} \int e^{-i (i a+i b x)} x^m \, dx-\frac{1}{8} \int e^{i (i a+i b x)} x^m \, dx+\frac{1}{8} \int e^{-i (3 i a+3 i b x)} x^m \, dx-\frac{1}{8} \int e^{i (3 i a+3 i b x)} x^m \, dx\\ &=\frac{3^{-1-m} e^{3 a} x^m (-b x)^{-m} \Gamma (1+m,-3 b x)}{8 b}+\frac{e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{8 b}+\frac{e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{8 b}+\frac{3^{-1-m} e^{-3 a} x^m (b x)^{-m} \Gamma (1+m,3 b x)}{8 b}\\ \end{align*}
Mathematica [A] time = 0.173345, size = 114, normalized size = 0.85 \[ \frac{e^{-3 a} x^m \left (3^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{6 a} (b x)^m \text{Gamma}(m+1,-3 b x)+(-b x)^m \text{Gamma}(m+1,3 b x)\right )+3 e^{2 a} \left (e^{2 a} (-b x)^{-m} \text{Gamma}(m+1,-b x)+(b x)^{-m} \text{Gamma}(m+1,b x)\right )\right )}{24 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}\sinh \left ( bx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3533, size = 153, normalized size = 1.14 \begin{align*} \frac{1}{8} \, \left (3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (m + 1, 3 \, b x\right ) + \frac{1}{8} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) - \frac{1}{8} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) - \frac{1}{8} \, \left (-3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (m + 1, -3 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2133, size = 486, normalized size = 3.63 \begin{align*} \frac{\cosh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) \Gamma \left (m + 1, 3 \, b x\right ) + 3 \, \cosh \left (m \log \left (b\right ) + a\right ) \Gamma \left (m + 1, b x\right ) + 3 \, \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) + \cosh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) \Gamma \left (m + 1, -3 \, b x\right ) - \Gamma \left (m + 1, 3 \, b x\right ) \sinh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) - 3 \, \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) - \Gamma \left (m + 1, -3 \, b x\right ) \sinh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) - 3 \, \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \left (b\right ) + a\right )}{24 \, b} \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 x^{m} \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + 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 x^{m} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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