Optimal. Leaf size=85 \[ \frac{e^{4 a} 2^{-2 (m+3)} x^m (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{-4 a} 2^{-2 (m+3)} x^m (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b}-\frac{x^{m+1}}{8 (m+1)} \]
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Rubi [A] time = 0.12789, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5448, 3307, 2181} \[ \frac{e^{4 a} 2^{-2 (m+3)} x^m (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{-4 a} 2^{-2 (m+3)} x^m (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b}-\frac{x^{m+1}}{8 (m+1)} \]
Antiderivative was successfully verified.
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Rule 5448
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^m \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{x^m}{8}+\frac{1}{8} x^m \cosh (4 a+4 b x)\right ) \, dx\\ &=-\frac{x^{1+m}}{8 (1+m)}+\frac{1}{8} \int x^m \cosh (4 a+4 b x) \, dx\\ &=-\frac{x^{1+m}}{8 (1+m)}+\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{x^{1+m}}{8 (1+m)}+\frac{4^{-3-m} e^{4 a} x^m (-b x)^{-m} \Gamma (1+m,-4 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 = 0.205554, size = 76, normalized size = 0.89 \[ \frac{1}{64} x^m \left (\frac{e^{4 a} 4^{-m} (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{-4 a} 4^{-m} (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b}-\frac{8 x}{m+1}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88802, size = 377, normalized size = 4.44 \begin{align*} -\frac{8 \, b x \cosh \left (m \log \left (x\right )\right ) +{\left (m + 1\right )} \cosh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) \Gamma \left (m + 1, 4 \, b x\right ) -{\left (m + 1\right )} \cosh \left (m \log \left (-4 \, b\right ) - 4 \, a\right ) \Gamma \left (m + 1, -4 \, b x\right ) -{\left (m + 1\right )} \Gamma \left (m + 1, 4 \, b x\right ) \sinh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) +{\left (m + 1\right )} \Gamma \left (m + 1, -4 \, b x\right ) \sinh \left (m \log \left (-4 \, b\right ) - 4 \, a\right ) + 8 \, b x \sinh \left (m \log \left (x\right )\right )}{64 \,{\left (b m + b\right )}} \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 ^{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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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