Optimal. Leaf size=113 \[ \frac{3^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac{3 \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{8 a^4}-\frac{3 \text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-n-1} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^4} \]
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Rubi [A] time = 0.252861, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5779, 3312, 3308, 2181} \[ \frac{3^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac{3 \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{8 a^4}-\frac{3 \text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-n-1} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 5779
Rule 3312
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int \frac{x^3 \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sinh ^3(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{i \operatorname{Subst}\left (\int \left (\frac{3}{4} i x^n \sinh (x)-\frac{1}{4} i x^n \sinh (3 x)\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int x^n \sinh (3 x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}-\frac{3 \operatorname{Subst}\left (\int x^n \sinh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac{\operatorname{Subst}\left (\int e^{3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac{3 \operatorname{Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac{3 \operatorname{Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{3^{-1-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-3 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac{3 \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{8 a^4}-\frac{3 \Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-1-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{8 a^4}\\ \end{align*}
Mathematica [A] time = 0.189915, size = 100, normalized size = 0.88 \[ \frac{3^{-n-1} \left (-\sinh ^{-1}(a x)\right )^{-n} \left (\left (-\sinh ^{-1}(a x)\right )^n \left (\text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )-3^{n+2} \text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )\right )+\sinh ^{-1}(a x)^n \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )-3^{n+2} \sinh ^{-1}(a x)^n \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \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^{3} \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^{3} \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^{3} \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^{3} \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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