Integrand size = 12, antiderivative size = 59 \[ \int x^{3+m} \cosh (a+b x) \, dx=-\frac {e^a x^m (-b x)^{-m} \Gamma (4+m,-b x)}{2 b^4}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (4+m,b x)}{2 b^4} \]
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Time = 0.05 (sec) , antiderivative size = 59, 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} \[ \int x^{3+m} \cosh (a+b x) \, dx=-\frac {e^a x^m (-b x)^{-m} \Gamma (m+4,-b x)}{2 b^4}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+4,b x)}{2 b^4} \]
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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^{3+m} \, dx+\frac {1}{2} \int e^{i (i a+i b x)} x^{3+m} \, dx \\ & = -\frac {e^a x^m (-b x)^{-m} \Gamma (4+m,-b x)}{2 b^4}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (4+m,b x)}{2 b^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int x^{3+m} \cosh (a+b x) \, dx=-\frac {e^a x^m (-b x)^{-m} \Gamma (4+m,-b x)+e^{-a} x^m (b x)^{-m} \Gamma (4+m,b x)}{2 b^4} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24
method | result | size |
meijerg | \(\frac {x^{4+m} \operatorname {hypergeom}\left (\left [2+\frac {m}{2}\right ], \left [\frac {1}{2}, 3+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \cosh \left (a \right )}{4+m}+\frac {b \,x^{5+m} \operatorname {hypergeom}\left (\left [\frac {5}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {7}{2}+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \sinh \left (a \right )}{5+m}\) | \(73\) |
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Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.46 \[ \int x^{3+m} \cosh (a+b x) \, dx=-\frac {\cosh \left ({\left (m + 3\right )} \log \left (b\right ) + a\right ) \Gamma \left (m + 4, b x\right ) - \cosh \left ({\left (m + 3\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m + 4, -b x\right ) + \Gamma \left (m + 4, -b x\right ) \sinh \left ({\left (m + 3\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m + 4, b x\right ) \sinh \left ({\left (m + 3\right )} \log \left (b\right ) + a\right )}{2 \, b} \]
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Exception generated. \[ \int x^{3+m} \cosh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93 \[ \int x^{3+m} \cosh (a+b x) \, dx=-\frac {1}{2} \, \left (b x\right )^{-m - 4} x^{m + 4} e^{\left (-a\right )} \Gamma \left (m + 4, b x\right ) - \frac {1}{2} \, \left (-b x\right )^{-m - 4} x^{m + 4} e^{a} \Gamma \left (m + 4, -b x\right ) \]
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\[ \int x^{3+m} \cosh (a+b x) \, dx=\int { x^{m + 3} \cosh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^{3+m} \cosh (a+b x) \, dx=\int x^{m+3}\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]
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