Optimal. Leaf size=80 \[ -\frac {\sinh ^{-1}(a x)^{1+n}}{2 a^3 (1+n)}+\frac {2^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^3}-\frac {2^{-3-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^3} \]
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Rubi [A]
time = 0.13, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {5819, 3393,
3388, 2212} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3393
Rule 5819
Rubi steps
\begin {align*} \int \frac {x^2 \sinh ^{-1}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\text {Subst}\left (\int x^n \sinh ^2(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\text {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 {\text {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 {\text {Subst}\left (\int e^{-2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac {\text {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*}
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Mathematica [A]
time = 0.15, size = 86, normalized size = 1.08 \begin {gather*} \frac {2^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \left ((1+n) \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )-\left (-\sinh ^{-1}(a x)\right )^n \left (2^{2+n} \sinh ^{-1}(a x)^{1+n}+(1+n) \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )\right )\right )}{a^3 (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \arcsinh \left (a x \right )^{n}}{\sqrt {a^{2} x^{2}+1}}\, dx\]
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 \frac {x^{2} \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, 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 \frac {x^2\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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