Optimal. Leaf size=141 \[ \frac{e^{4 a} 2^{-2 (m+3)} x^m (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{2 a} 2^{-m-4} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}-\frac{e^{-2 a} 2^{-m-4} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}+\frac{e^{-4 a} 2^{-2 (m+3)} x^m (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b} \]
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Rubi [A] time = 0.210249, antiderivative size = 141, 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} \[ \frac{e^{4 a} 2^{-2 (m+3)} x^m (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{2 a} 2^{-m-4} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}-\frac{e^{-2 a} 2^{-m-4} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}+\frac{e^{-4 a} 2^{-2 (m+3)} x^m (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^m \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{1}{4} x^m \sinh (2 a+2 b x)+\frac{1}{8} x^m \sinh (4 a+4 b x)\right ) \, dx\\ &=\frac{1}{8} \int x^m \sinh (4 a+4 b x) \, dx-\frac{1}{4} \int x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{16} \int e^{-i (4 i a+4 i b x)} x^m \, dx-\frac{1}{16} \int e^{i (4 i a+4 i b x)} x^m \, dx-\frac{1}{8} \int e^{-i (2 i a+2 i b x)} x^m \, dx+\frac{1}{8} \int e^{i (2 i a+2 i b x)} x^m \, dx\\ &=\frac{4^{-3-m} e^{4 a} x^m (-b x)^{-m} \Gamma (1+m,-4 b x)}{b}-\frac{2^{-4-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}-\frac{2^{-4-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\frac{4^{-3-m} e^{-4 a} x^m (b x)^{-m} \Gamma (1+m,4 b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.136832, size = 112, normalized size = 0.79 \[ \frac{e^{-4 a} 4^{-m-3} x^m \left (-b^2 x^2\right )^{-m} \left ((-b x)^m \left (\text{Gamma}(m+1,4 b x)-e^{2 a} 2^{m+2} \text{Gamma}(m+1,2 b x)\right )+e^{8 a} (b x)^m \text{Gamma}(m+1,-4 b x)-e^{6 a} 2^{m+2} (b x)^m \text{Gamma}(m+1,-2 b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\cosh \left ( bx+a \right ) \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 = 1.22765, size = 158, normalized size = 1.12 \begin{align*} \frac{1}{16} \, \left (4 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-4 \, a\right )} \Gamma \left (m + 1, 4 \, b x\right ) - \frac{1}{8} \, \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}{8} \, \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}{16} \, \left (-4 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (4 \, a\right )} \Gamma \left (m + 1, -4 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86706, size = 518, normalized size = 3.67 \begin{align*} \frac{\cosh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) \Gamma \left (m + 1, 4 \, b x\right ) - 4 \, \cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) - 4 \, \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) + \cosh \left (m \log \left (-4 \, b\right ) - 4 \, a\right ) \Gamma \left (m + 1, -4 \, b x\right ) - \Gamma \left (m + 1, 4 \, b x\right ) \sinh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) + 4 \, \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) + 4 \, \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) - \Gamma \left (m + 1, -4 \, b x\right ) \sinh \left (m \log \left (-4 \, b\right ) - 4 \, a\right )}{64 \, 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 ^{3}{\left (a + b x \right )} \cosh{\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 ) \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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