Optimal. Leaf size=80 \[ \frac{2^{-n-3} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^3}-\frac{2^{-n-3} \text{Gamma}\left (n+1,2 \sinh ^{-1}(a x)\right )}{a^3}-\frac{\sinh ^{-1}(a x)^{n+1}}{2 a^3 (n+1)} \]
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Rubi [A] time = 0.193296, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5779, 3312, 3307, 2181} \[ \frac{2^{-n-3} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^3}-\frac{2^{-n-3} \text{Gamma}\left (n+1,2 \sinh ^{-1}(a x)\right )}{a^3}-\frac{\sinh ^{-1}(a x)^{n+1}}{2 a^3 (n+1)} \]
Antiderivative was successfully verified.
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Rule 5779
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int \frac{x^2 \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sinh ^2(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{x^n}{2}-\frac{1}{2} x^n \cosh (2 x)\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\sinh ^{-1}(a x)^{1+n}}{2 a^3 (1+n)}+\frac{\operatorname{Subst}\left (\int x^n \cosh (2 x) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac{\sinh ^{-1}(a x)^{1+n}}{2 a^3 (1+n)}+\frac{\operatorname{Subst}\left (\int e^{-2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac{\operatorname{Subst}\left (\int e^{2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac{\sinh ^{-1}(a x)^{1+n}}{2 a^3 (1+n)}+\frac{2^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^3}-\frac{2^{-3-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.206218, size = 86, normalized size = 1.08 \[ \frac{2^{-n-3} \left (-\sinh ^{-1}(a x)\right )^{-n} \left ((n+1) \sinh ^{-1}(a x)^n \text{Gamma}\left (n+1,-2 \sinh ^{-1}(a x)\right )-\left (-\sinh ^{-1}(a x)\right )^n \left ((n+1) \text{Gamma}\left (n+1,2 \sinh ^{-1}(a x)\right )+2^{n+2} \sinh ^{-1}(a x)^{n+1}\right )\right )}{a^3 (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{n}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{asinh}^{n}{\left (a x \right )}}{\sqrt{a^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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