Optimal. Leaf size=209 \[ \frac{e^{5 a} 5^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-5 b x)}{32 b}+\frac{e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-3 b x)}{32 b}-\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{16 b}+\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{16 b}-\frac{e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,3 b x)}{32 b}-\frac{e^{-5 a} 5^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,5 b x)}{32 b} \]
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Rubi [A] time = 0.28686, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3307, 2181} \[ \frac{e^{5 a} 5^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-5 b x)}{32 b}+\frac{e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-3 b x)}{32 b}-\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{16 b}+\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{16 b}-\frac{e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,3 b x)}{32 b}-\frac{e^{-5 a} 5^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,5 b x)}{32 b} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{1}{8} x^m \cosh (a+b x)+\frac{1}{16} x^m \cosh (3 a+3 b x)+\frac{1}{16} x^m \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac{1}{16} \int x^m \cosh (3 a+3 b x) \, dx+\frac{1}{16} \int x^m \cosh (5 a+5 b x) \, dx-\frac{1}{8} \int x^m \cosh (a+b x) \, dx\\ &=\frac{1}{32} \int e^{-i (3 i a+3 i b x)} x^m \, dx+\frac{1}{32} \int e^{i (3 i a+3 i b x)} x^m \, dx+\frac{1}{32} \int e^{-i (5 i a+5 i b x)} x^m \, dx+\frac{1}{32} \int e^{i (5 i a+5 i b x)} x^m \, dx-\frac{1}{16} \int e^{-i (i a+i b x)} x^m \, dx-\frac{1}{16} \int e^{i (i a+i b x)} x^m \, dx\\ &=\frac{5^{-1-m} e^{5 a} x^m (-b x)^{-m} \Gamma (1+m,-5 b x)}{32 b}+\frac{3^{-1-m} e^{3 a} x^m (-b x)^{-m} \Gamma (1+m,-3 b x)}{32 b}-\frac{e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{16 b}+\frac{e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{16 b}-\frac{3^{-1-m} e^{-3 a} x^m (b x)^{-m} \Gamma (1+m,3 b x)}{32 b}-\frac{5^{-1-m} e^{-5 a} x^m (b x)^{-m} \Gamma (1+m,5 b x)}{32 b}\\ \end{align*}
Mathematica [A] time = 0.294794, size = 175, normalized size = 0.84 \[ \frac{e^{-5 a} x^m \left (5 e^{2 a} 3^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{6 a} (b x)^m \text{Gamma}(m+1,-3 b x)-(-b x)^m \text{Gamma}(m+1,3 b x)\right )+3\ 5^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{10 a} (b x)^m \text{Gamma}(m+1,-5 b x)-(-b x)^m \text{Gamma}(m+1,5 b x)\right )-30 e^{6 a} (-b x)^{-m} \text{Gamma}(m+1,-b x)+30 e^{4 a} (b x)^{-m} \text{Gamma}(m+1,b x)\right )}{480 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, 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 ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36135, size = 231, normalized size = 1.11 \begin{align*} -\frac{1}{32} \, \left (5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-5 \, a\right )} \Gamma \left (m + 1, 5 \, b x\right ) - \frac{1}{32} \, \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}{16} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) + \frac{1}{16} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) - \frac{1}{32} \, \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}{32} \, \left (-5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (5 \, a\right )} \Gamma \left (m + 1, -5 \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00391, size = 764, normalized size = 3.66 \begin{align*} -\frac{3 \, \cosh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) \Gamma \left (m + 1, 5 \, b x\right ) + 5 \, \cosh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) \Gamma \left (m + 1, 3 \, b x\right ) - 30 \, \cosh \left (m \log \left (b\right ) + a\right ) \Gamma \left (m + 1, b x\right ) + 30 \, \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) - 5 \, \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 (-5 \, b\right ) - 5 \, a\right ) \Gamma \left (m + 1, -5 \, b x\right ) - 3 \, \Gamma \left (m + 1, 5 \, b x\right ) \sinh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) - 5 \, \Gamma \left (m + 1, 3 \, b x\right ) \sinh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) - 30 \, \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) + 5 \, \Gamma \left (m + 1, -3 \, b x\right ) \sinh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) + 3 \, \Gamma \left (m + 1, -5 \, b x\right ) \sinh \left (m \log \left (-5 \, b\right ) - 5 \, a\right ) + 30 \, \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \left (b\right ) + a\right )}{480 \, 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 ^{2}{\left (a + b x \right )} \cosh ^{3}{\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 )^{3} \sinh \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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