Integrand size = 12, antiderivative size = 55 \[ \int x^{-2+m} \cosh (a+b x) \, dx=\frac {1}{2} b e^a x^m (-b x)^{-m} \Gamma (-1+m,-b x)-\frac {1}{2} b e^{-a} x^m (b x)^{-m} \Gamma (-1+m,b x) \]
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Time = 0.05 (sec) , antiderivative size = 55, 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 = {3388, 2212} \[ \int x^{-2+m} \cosh (a+b x) \, dx=\frac {1}{2} e^a b x^m (-b x)^{-m} \Gamma (m-1,-b x)-\frac {1}{2} e^{-a} b x^m (b x)^{-m} \Gamma (m-1,b x) \]
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Rule 2212
Rule 3388
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int e^{-i (i a+i b x)} x^{-2+m} \, dx+\frac {1}{2} \int e^{i (i a+i b x)} x^{-2+m} \, dx \\ & = \frac {1}{2} b e^a x^m (-b x)^{-m} \Gamma (-1+m,-b x)-\frac {1}{2} b e^{-a} x^m (b x)^{-m} \Gamma (-1+m,b x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int x^{-2+m} \cosh (a+b x) \, dx=\frac {1}{2} b e^{-a} x^m \left (e^{2 a} (-b x)^{-m} \Gamma (-1+m,-b x)-(b x)^{-m} \Gamma (-1+m,b x)\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.22
method | result | size |
meijerg | \(\frac {x^{-1+m} \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {m}{2}\right ], \left [\frac {1}{2}, \frac {1}{2}+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \cosh \left (a \right )}{-1+m}+\frac {b \,x^{m} \operatorname {hypergeom}\left (\left [\frac {m}{2}\right ], \left [\frac {3}{2}, 1+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \sinh \left (a \right )}{m}\) | \(67\) |
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Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.56 \[ \int x^{-2+m} \cosh (a+b x) \, dx=-\frac {\cosh \left ({\left (m - 2\right )} \log \left (b\right ) + a\right ) \Gamma \left (m - 1, b x\right ) - \cosh \left ({\left (m - 2\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m - 1, -b x\right ) + \Gamma \left (m - 1, -b x\right ) \sinh \left ({\left (m - 2\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m - 1, b x\right ) \sinh \left ({\left (m - 2\right )} \log \left (b\right ) + a\right )}{2 \, b} \]
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Exception generated. \[ \int x^{-2+m} \cosh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int x^{-2+m} \cosh (a+b x) \, dx=-\frac {1}{2} \, \left (b x\right )^{-m + 1} x^{m - 1} e^{\left (-a\right )} \Gamma \left (m - 1, b x\right ) - \frac {1}{2} \, \left (-b x\right )^{-m + 1} x^{m - 1} e^{a} \Gamma \left (m - 1, -b x\right ) \]
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\[ \int x^{-2+m} \cosh (a+b x) \, dx=\int { x^{m - 2} \cosh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^{-2+m} \cosh (a+b x) \, dx=\int x^{m-2}\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]
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