Integrand size = 10, antiderivative size = 75 \[ \int x \sinh \left (a+b x^n\right ) \, dx=-\frac {e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )}{2 n}+\frac {e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )}{2 n} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5468, 2250} \[ \int x \sinh \left (a+b x^n\right ) \, dx=\frac {e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )}{2 n}-\frac {e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )}{2 n} \]
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Rule 2250
Rule 5468
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int e^{-a-b x^n} x \, dx\right )+\frac {1}{2} \int e^{a+b x^n} x \, dx \\ & = -\frac {e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )}{2 n}+\frac {e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )}{2 n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int x \sinh \left (a+b x^n\right ) \, dx=\frac {e^{-a} x^2 \left (-e^{2 a} \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )+\left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )\right )}{2 n} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.55 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92
method | result | size |
meijerg | \(\frac {x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{n}\right ], \left [\frac {1}{2}, 1+\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{2}+\frac {x^{2+n} b \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {1}{n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )}{2+n}\) | \(69\) |
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\[ \int x \sinh \left (a+b x^n\right ) \, dx=\int { x \sinh \left (b x^{n} + a\right ) \,d x } \]
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\[ \int x \sinh \left (a+b x^n\right ) \, dx=\int x \sinh {\left (a + b x^{n} \right )}\, dx \]
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none
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int x \sinh \left (a+b x^n\right ) \, dx=\frac {x^{2} e^{\left (-a\right )} \Gamma \left (\frac {2}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\frac {2}{n}} n} - \frac {x^{2} e^{a} \Gamma \left (\frac {2}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\frac {2}{n}} n} \]
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\[ \int x \sinh \left (a+b x^n\right ) \, dx=\int { x \sinh \left (b x^{n} + a\right ) \,d x } \]
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Timed out. \[ \int x \sinh \left (a+b x^n\right ) \, dx=\int x\,\mathrm {sinh}\left (a+b\,x^n\right ) \,d x \]
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