Optimal. Leaf size=209 \[ \frac {5^{-1-m} e^{5 a} x^m (-b x)^{-m} \Gamma (1+m,-5 b x)}{32 b}+\frac {3^{-1-m} e^{3 a} x^m (-b x)^{-m} \Gamma (1+m,-3 b x)}{32 b}-\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{16 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{16 b}-\frac {3^{-1-m} e^{-3 a} x^m (b x)^{-m} \Gamma (1+m,3 b x)}{32 b}-\frac {5^{-1-m} e^{-5 a} x^m (b x)^{-m} \Gamma (1+m,5 b x)}{32 b} \]
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Rubi [A]
time = 0.21, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5556, 3388,
2212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 5556
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*}
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Mathematica [A]
time = 1.91, size = 185, normalized size = 0.89 \begin {gather*} \frac {x^m \left (-30 e^a (-b x)^{-m} \Gamma (1+m,-b x)+30 e^{-a} (b x)^{-m} \Gamma (1+m,b x)+5\ 3^{-m} \left (-b^2 x^2\right )^{-m} \left (-(-b x)^m \Gamma (1+m,3 b x) (\cosh (a)-\sinh (a))^3+(b x)^m \Gamma (1+m,-3 b x) (\cosh (a)+\sinh (a))^3\right )+3\ 5^{-m} \left (-b^2 x^2\right )^{-m} \left (-(-b x)^m \Gamma (1+m,5 b x) (\cosh (a)-\sinh (a))^5+(b x)^m \Gamma (1+m,-5 b x) (\cosh (a)+\sinh (a))^5\right )\right )}{480 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.39, size = 0, normalized size = 0.00 \[\int x^{m} \left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.12, size = 171, normalized size = 0.82 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.12, size = 248, normalized size = 1.19 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \int x^{m} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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