Integrand size = 12, antiderivative size = 49 \[ \int x^{-1+m} \sinh (a+b x) \, dx=-\frac {1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)+\frac {1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x) \]
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Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3389, 2212} \[ \int x^{-1+m} \sinh (a+b x) \, dx=\frac {1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x)-\frac {1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x) \]
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Rule 2212
Rule 3389
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int e^{-i (i a+i b x)} x^{-1+m} \, dx-\frac {1}{2} \int e^{i (i a+i b x)} x^{-1+m} \, dx \\ & = -\frac {1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)+\frac {1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int x^{-1+m} \sinh (a+b x) \, dx=-\frac {1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)+\frac {1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.37
method | result | size |
meijerg | \(\frac {x^{m} \operatorname {hypergeom}\left (\left [\frac {m}{2}\right ], \left [\frac {1}{2}, 1+\frac {m}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \sinh \left (a \right )}{m}+\frac {b \,x^{1+m} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {m}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \cosh \left (a \right )}{1+m}\) | \(67\) |
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none
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.59 \[ \int x^{-1+m} \sinh (a+b x) \, dx=\frac {\cosh \left ({\left (m - 1\right )} \log \left (b\right ) + a\right ) \Gamma \left (m, b x\right ) + \cosh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m, -b x\right ) - \Gamma \left (m, -b x\right ) \sinh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m, b x\right ) \sinh \left ({\left (m - 1\right )} \log \left (b\right ) + a\right )}{2 \, b} \]
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Exception generated. \[ \int x^{-1+m} \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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none
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int x^{-1+m} \sinh (a+b x) \, dx=\frac {x^{m} e^{\left (-a\right )} \Gamma \left (m, b x\right )}{2 \, \left (b x\right )^{m}} - \frac {x^{m} e^{a} \Gamma \left (m, -b x\right )}{2 \, \left (-b x\right )^{m}} \]
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\[ \int x^{-1+m} \sinh (a+b x) \, dx=\int { x^{m - 1} \sinh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^{-1+m} \sinh (a+b x) \, dx=\int x^{m-1}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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