Integrand size = 8, antiderivative size = 59 \[ \int x \text {arcsinh}(a x)^n \, dx=\frac {2^{-3-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))}{a^2}+\frac {2^{-3-n} \Gamma (1+n,2 \text {arcsinh}(a x))}{a^2} \]
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Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5780, 5556, 12, 3389, 2212} \[ \int x \text {arcsinh}(a x)^n \, dx=\frac {2^{-n-3} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-2 \text {arcsinh}(a x))}{a^2}+\frac {2^{-n-3} \Gamma (n+1,2 \text {arcsinh}(a x))}{a^2} \]
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Rule 12
Rule 2212
Rule 3389
Rule 5556
Rule 5780
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \cosh (x) \sinh (x) \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {1}{2} x^n \sinh (2 x) \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\text {arcsinh}(a x)\right )}{2 a^2} \\ & = -\frac {\text {Subst}\left (\int e^{-2 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{4 a^2}+\frac {\text {Subst}\left (\int e^{2 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{4 a^2} \\ & = \frac {2^{-3-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))}{a^2}+\frac {2^{-3-n} \Gamma (1+n,2 \text {arcsinh}(a x))}{a^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int x \text {arcsinh}(a x)^n \, dx=\frac {2^{-3-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))+2^{-3-n} \Gamma (1+n,2 \text {arcsinh}(a x))}{a^2} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\operatorname {arcsinh}\left (a x \right )^{2+n} \operatorname {hypergeom}\left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \operatorname {arcsinh}\left (a x \right )^{2}\right )}{a^{2} \left (2+n \right )}\) | \(38\) |
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\[ \int x \text {arcsinh}(a x)^n \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
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\[ \int x \text {arcsinh}(a x)^n \, dx=\int x \operatorname {asinh}^{n}{\left (a x \right )}\, dx \]
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\[ \int x \text {arcsinh}(a x)^n \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
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\[ \int x \text {arcsinh}(a x)^n \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
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Timed out. \[ \int x \text {arcsinh}(a x)^n \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]
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