Integrand size = 14, antiderivative size = 84 \[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=\frac {x^{-2+m}}{2 (2-m)}-2^{-m} b^2 e^{2 a} x^m (-b x)^{-m} \Gamma (-2+m,-2 b x)-2^{-m} b^2 e^{-2 a} x^m (b x)^{-m} \Gamma (-2+m,2 b x) \]
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Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3393, 3388, 2212} \[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=-e^{2 a} b^2 2^{-m} x^m (-b x)^{-m} \Gamma (m-2,-2 b x)-e^{-2 a} b^2 2^{-m} x^m (b x)^{-m} \Gamma (m-2,2 b x)+\frac {x^{m-2}}{2 (2-m)} \]
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Rule 2212
Rule 3388
Rule 3393
Rubi steps \begin{align*} \text {integral}& = -\int \left (\frac {x^{-3+m}}{2}-\frac {1}{2} x^{-3+m} \cosh (2 a+2 b x)\right ) \, dx \\ & = \frac {x^{-2+m}}{2 (2-m)}+\frac {1}{2} \int x^{-3+m} \cosh (2 a+2 b x) \, dx \\ & = \frac {x^{-2+m}}{2 (2-m)}+\frac {1}{4} \int e^{-i (2 i a+2 i b x)} x^{-3+m} \, dx+\frac {1}{4} \int e^{i (2 i a+2 i b x)} x^{-3+m} \, dx \\ & = \frac {x^{-2+m}}{2 (2-m)}-2^{-m} b^2 e^{2 a} x^m (-b x)^{-m} \Gamma (-2+m,-2 b x)-2^{-m} b^2 e^{-2 a} x^m (b x)^{-m} \Gamma (-2+m,2 b x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=x^m \left (\frac {1}{(4-2 m) x^2}-2^{-m} b^2 e^{2 a} (-b x)^{-m} \Gamma (-2+m,-2 b x)-2^{-m} b^2 e^{-2 a} (b x)^{-m} \Gamma (-2+m,2 b x)\right ) \]
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\[\int x^{m -3} \sinh \left (b x +a \right )^{2}d x\]
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none
Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.62 \[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=-\frac {4 \, b x \cosh \left ({\left (m - 3\right )} \log \left (x\right )\right ) + {\left (m - 2\right )} \cosh \left ({\left (m - 3\right )} \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m - 2, 2 \, b x\right ) - {\left (m - 2\right )} \cosh \left ({\left (m - 3\right )} \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m - 2, -2 \, b x\right ) - {\left (m - 2\right )} \Gamma \left (m - 2, 2 \, b x\right ) \sinh \left ({\left (m - 3\right )} \log \left (2 \, b\right ) + 2 \, a\right ) + {\left (m - 2\right )} \Gamma \left (m - 2, -2 \, b x\right ) \sinh \left ({\left (m - 3\right )} \log \left (-2 \, b\right ) - 2 \, a\right ) + 4 \, b x \sinh \left ({\left (m - 3\right )} \log \left (x\right )\right )}{8 \, {\left (b m - 2 \, b\right )}} \]
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\[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=\int x^{m - 3} \sinh ^{2}{\left (a + b x \right )}\, dx \]
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Exception generated. \[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=\int { x^{m - 3} \sinh \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^{-3+m} \sinh ^2(a+b x) \, dx=\int x^{m-3}\,{\mathrm {sinh}\left (a+b\,x\right )}^2 \,d x \]
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