Integrand size = 23, antiderivative size = 113 \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {3^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))}{8 a^4}-\frac {3 (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{8 a^4}-\frac {3 \Gamma (1+n,\text {arcsinh}(a x))}{8 a^4}+\frac {3^{-1-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{8 a^4} \]
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Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5819, 3393, 3389, 2212} \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {3^{-n-1} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-3 \text {arcsinh}(a x))}{8 a^4}-\frac {3 \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-\text {arcsinh}(a x))}{8 a^4}-\frac {3 \Gamma (n+1,\text {arcsinh}(a x))}{8 a^4}+\frac {3^{-n-1} \Gamma (n+1,3 \text {arcsinh}(a x))}{8 a^4} \]
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Rule 2212
Rule 3389
Rule 3393
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \sinh ^3(x) \, dx,x,\text {arcsinh}(a x)\right )}{a^4} \\ & = \frac {i \text {Subst}\left (\int \left (\frac {3}{4} i x^n \sinh (x)-\frac {1}{4} i x^n \sinh (3 x)\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int x^n \sinh (3 x) \, dx,x,\text {arcsinh}(a x)\right )}{4 a^4}-\frac {3 \text {Subst}\left (\int x^n \sinh (x) \, dx,x,\text {arcsinh}(a x)\right )}{4 a^4} \\ & = -\frac {\text {Subst}\left (\int e^{-3 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4}+\frac {\text {Subst}\left (\int e^{3 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4}+\frac {3 \text {Subst}\left (\int e^{-x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4}-\frac {3 \text {Subst}\left (\int e^x x^n \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4} \\ & = \frac {3^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))}{8 a^4}-\frac {3 (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{8 a^4}-\frac {3 \Gamma (1+n,\text {arcsinh}(a x))}{8 a^4}+\frac {3^{-1-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{8 a^4} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {3^{-1-n} (-\text {arcsinh}(a x))^{-n} \left (\text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))-3^{2+n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))+(-\text {arcsinh}(a x))^n \left (-3^{2+n} \Gamma (1+n,\text {arcsinh}(a x))+\Gamma (1+n,3 \text {arcsinh}(a x))\right )\right )}{8 a^4} \]
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\[\int \frac {x^{3} \operatorname {arcsinh}\left (a x \right )^{n}}{\sqrt {a^{2} x^{2}+1}}d x\]
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\[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{3} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^{3} \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \]
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\[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{3} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \]
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