Optimal. Leaf size=139 \[ \frac {e^{4 a} 2^{-2 (m+3)} x^m (-b x)^{-m} \Gamma (m+1,-4 b x)}{b}+\frac {e^{2 a} 2^{-m-4} x^m (-b x)^{-m} \Gamma (m+1,-2 b x)}{b}+\frac {e^{-2 a} 2^{-m-4} x^m (b x)^{-m} \Gamma (m+1,2 b x)}{b}+\frac {e^{-4 a} 2^{-2 (m+3)} x^m (b x)^{-m} \Gamma (m+1,4 b x)}{b} \]
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Rubi [A] time = 0.24, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 2181
Rule 3308
Rule 5448
Rubi steps
\begin {align*} \int x^m \cosh ^3(a+b x) \sinh (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*}
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Mathematica [A] time = 0.13, size = 110, normalized size = 0.79 \[ \frac {e^{-4 a} 4^{-m-3} x^m \left (-b^2 x^2\right )^{-m} \left ((-b x)^m \left (e^{2 a} 2^{m+2} \Gamma (m+1,2 b x)+\Gamma (m+1,4 b x)\right )+e^{8 a} (b x)^m \Gamma (m+1,-4 b x)+e^{6 a} 2^{m+2} (b x)^m \Gamma (m+1,-2 b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 172, normalized size = 1.24 \[ \frac {\cosh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) \Gamma \left (m + 1, 4 \, b x\right ) + 4 \, \cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) + 4 \, \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) + \cosh \left (m \log \left (-4 \, b\right ) - 4 \, a\right ) \Gamma \left (m + 1, -4 \, b x\right ) - \Gamma \left (m + 1, 4 \, b x\right ) \sinh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) - 4 \, \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) - 4 \, \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) - \Gamma \left (m + 1, -4 \, b x\right ) \sinh \left (m \log \left (-4 \, b\right ) - 4 \, a\right )}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 117, normalized size = 0.84 \[ \frac {1}{16} \, \left (4 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-4 \, a\right )} \Gamma \left (m + 1, 4 \, b x\right ) + \frac {1}{8} \, \left (2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (m + 1, 2 \, b x\right ) - \frac {1}{8} \, \left (-2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (2 \, a\right )} \Gamma \left (m + 1, -2 \, b x\right ) - \frac {1}{16} \, \left (-4 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (4 \, a\right )} \Gamma \left (m + 1, -4 \, b x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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