Optimal. Leaf size=83 \[ e^{2 a} b 2^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m-1,-2 b x)-e^{-2 a} b 2^{-m-1} x^m (b x)^{-m} \text{Gamma}(m-1,2 b x)+\frac{x^{m-1}}{2 (1-m)} \]
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Rubi [A] time = 0.136826, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3312, 3307, 2181} \[ e^{2 a} b 2^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m-1,-2 b x)-e^{-2 a} b 2^{-m-1} x^m (b x)^{-m} \text{Gamma}(m-1,2 b x)+\frac{x^{m-1}}{2 (1-m)} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^{-2+m} \sinh ^2(a+b x) \, dx &=-\int \left (\frac{x^{-2+m}}{2}-\frac{1}{2} x^{-2+m} \cosh (2 a+2 b x)\right ) \, dx\\ &=\frac{x^{-1+m}}{2 (1-m)}+\frac{1}{2} \int x^{-2+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^{-2+m} \, dx+\frac{1}{4} \int e^{i (2 i a+2 i b x)} x^{-2+m} \, dx\\ &=\frac{x^{-1+m}}{2 (1-m)}+2^{-1-m} b e^{2 a} x^m (-b x)^{-m} \Gamma (-1+m,-2 b x)-2^{-1-m} b e^{-2 a} x^m (b x)^{-m} \Gamma (-1+m,2 b x)\\ \end{align*}
Mathematica [A] time = 0.100091, size = 72, normalized size = 0.87 \[ \frac{1}{2} x^m \left (e^{2 a} b 2^{-m} (-b x)^{-m} \text{Gamma}(m-1,-2 b x)-e^{-2 a} b 2^{-m} (b x)^{-m} \text{Gamma}(m-1,2 b x)+\frac{1}{x-m x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{x}^{-2+m} \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 = 2.79577, size = 424, normalized size = 5.11 \begin{align*} -\frac{4 \, b x \cosh \left ({\left (m - 2\right )} \log \left (x\right )\right ) +{\left (m - 1\right )} \cosh \left ({\left (m - 2\right )} \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m - 1, 2 \, b x\right ) -{\left (m - 1\right )} \cosh \left ({\left (m - 2\right )} \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 ({\left (m - 2\right )} \log \left (2 \, b\right ) + 2 \, a\right ) +{\left (m - 1\right )} \Gamma \left (m - 1, -2 \, b x\right ) \sinh \left ({\left (m - 2\right )} \log \left (-2 \, b\right ) - 2 \, a\right ) + 4 \, b x \sinh \left ({\left (m - 2\right )} \log \left (x\right )\right )}{8 \,{\left (b m - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 - 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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