Optimal. Leaf size=113 \[ \frac {3^{-1-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-3 \sinh ^{-1}(a x)\right )}{8 a^3}-\frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac {3^{-1-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{8 a^3} \]
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Rubi [A]
time = 0.10, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5780, 5556,
3388, 2212} \begin {gather*} \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^3}-\frac {\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {\text {Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac {3^{-n-1} \text {Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 5556
Rule 5780
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}(a x)^n \, dx &=\frac {\text {Subst}\left (\int x^n \cosh (x) \sinh ^2(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{4} x^n \cosh (x)+\frac {1}{4} x^n \cosh (3 x)\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\text {Subst}\left (\int x^n \cosh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {Subst}\left (\int x^n \cosh (3 x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=\frac {\text {Subst}\left (\int e^{-3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac {\text {Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac {\text {Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {\text {Subst}\left (\int e^{3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^3}\\ &=\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^3}-\frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{8 a^3}+\frac {\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac {3^{-1-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 97, normalized size = 0.86 \begin {gather*} \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 )-\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )+\Gamma \left (1+n,\sinh ^{-1}(a x)\right )-3^{-1-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{8 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.33, size = 0, normalized size = 0.00 \[\int x^{2} \arcsinh \left (a x \right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {asinh}^{n}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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