Optimal. Leaf size=155 \[ \frac{e^{6 a} 2^{-m-7} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-6 b x)}{b}-\frac{3 e^{2 a} 2^{-m-7} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}-\frac{3 e^{-2 a} 2^{-m-7} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}+\frac{e^{-6 a} 2^{-m-7} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,6 b x)}{b} \]
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Rubi [A] time = 0.243486, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3308, 2181} \[ \frac{e^{6 a} 2^{-m-7} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-6 b x)}{b}-\frac{3 e^{2 a} 2^{-m-7} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}-\frac{3 e^{-2 a} 2^{-m-7} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}+\frac{e^{-6 a} 2^{-m-7} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,6 b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^m \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} x^m \sinh (2 a+2 b x)+\frac{1}{32} x^m \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int x^m \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{64} \int e^{-i (6 i a+6 i b x)} x^m \, dx-\frac{1}{64} \int e^{i (6 i a+6 i b x)} x^m \, dx-\frac{3}{64} \int e^{-i (2 i a+2 i b x)} x^m \, dx+\frac{3}{64} \int e^{i (2 i a+2 i b x)} x^m \, dx\\ &=\frac{2^{-7-m} 3^{-1-m} e^{6 a} x^m (-b x)^{-m} \Gamma (1+m,-6 b x)}{b}-\frac{3\ 2^{-7-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}-\frac{3\ 2^{-7-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\frac{2^{-7-m} 3^{-1-m} e^{-6 a} x^m (b x)^{-m} \Gamma (1+m,6 b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.152997, size = 119, normalized size = 0.77 \[ \frac{e^{-6 a} 2^{-m-7} 3^{-m-1} x^m \left (-b^2 x^2\right )^{-m} \left ((-b x)^m \left (\text{Gamma}(m+1,6 b x)-e^{4 a} 3^{m+2} \text{Gamma}(m+1,2 b x)\right )+e^{12 a} (b x)^m \text{Gamma}(m+1,-6 b x)-e^{8 a} 3^{m+2} (b x)^m \text{Gamma}(m+1,-2 b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19729, size = 158, normalized size = 1.02 \begin{align*} \frac{1}{64} \, \left (6 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-6 \, a\right )} \Gamma \left (m + 1, 6 \, b x\right ) - \frac{3}{64} \, \left (2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (m + 1, 2 \, b x\right ) + \frac{3}{64} \, \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}{64} \, \left (-6 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (6 \, a\right )} \Gamma \left (m + 1, -6 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8499, size = 520, normalized size = 3.35 \begin{align*} \frac{\cosh \left (m \log \left (6 \, b\right ) + 6 \, a\right ) \Gamma \left (m + 1, 6 \, b x\right ) - 9 \, \cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) - 9 \, \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) + \cosh \left (m \log \left (-6 \, b\right ) - 6 \, a\right ) \Gamma \left (m + 1, -6 \, b x\right ) - \Gamma \left (m + 1, 6 \, b x\right ) \sinh \left (m \log \left (6 \, b\right ) + 6 \, a\right ) + 9 \, \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) + 9 \, \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) - \Gamma \left (m + 1, -6 \, b x\right ) \sinh \left (m \log \left (-6 \, b\right ) - 6 \, a\right )}{384 \, 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} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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