Optimal. Leaf size=85 \[ \frac {e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \Gamma (m+1,-2 b x)}{b}-\frac {e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \Gamma (m+1,2 b x)}{b}-\frac {x^{m+1}}{2 (m+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3312, 3307, 2181} \[ \frac {e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \text {Gamma}(m+1,-2 b x)}{b}-\frac {e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \text {Gamma}(m+1,2 b x)}{b}-\frac {x^{m+1}}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rubi steps
\begin {align*} \int x^m \sinh ^2(a+b x) \, dx &=-\int \left (\frac {x^m}{2}-\frac {1}{2} x^m \cosh (2 a+2 b x)\right ) \, dx\\ &=-\frac {x^{1+m}}{2 (1+m)}+\frac {1}{2} \int x^m \cosh (2 a+2 b x) \, dx\\ &=-\frac {x^{1+m}}{2 (1+m)}+\frac {1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx+\frac {1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx\\ &=-\frac {x^{1+m}}{2 (1+m)}+\frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}-\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 76, normalized size = 0.89 \[ \frac {1}{8} x^m \left (\frac {e^{2 a} 2^{-m} (-b x)^{-m} \Gamma (m+1,-2 b x)}{b}-\frac {e^{-2 a} 2^{-m} (b x)^{-m} \Gamma (m+1,2 b x)}{b}-\frac {4 x}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 122, normalized size = 1.44 \[ -\frac {4 \, b x \cosh \left (m \log \relax (x)\right ) + {\left (m + 1\right )} \cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) - {\left (m + 1\right )} \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) - {\left (m + 1\right )} \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) + {\left (m + 1\right )} \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) + 4 \, b x \sinh \left (m \log \relax (x)\right )}{8 \, {\left (b m + b\right )}} \]
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 \[ \int x^{m} \sinh \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\sinh ^{2}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 71, normalized size = 0.84 \[ -\frac {1}{4} \, \left (2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (m + 1, 2 \, b x\right ) - \frac {1}{4} \, \left (-2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (2 \, a\right )} \Gamma \left (m + 1, -2 \, b x\right ) - \frac {x^{m + 1}}{2 \, {\left (m + 1\right )}} \]
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 \[ \int x^m\,{\mathrm {sinh}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{m} \sinh ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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