Optimal. Leaf size=27 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^3}-\frac {\log \left (\sinh ^{-1}(a x)\right )}{2 a^3} \]
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Rubi [A] time = 0.14, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5779, 3312, 3301} \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^3}-\frac {\log \left (\sinh ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5779
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\log \left (\sinh ^{-1}(a x)\right )}{2 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^3}\\ &=\frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )}{2 a^3}-\frac {\log \left (\sinh ^{-1}(a x)\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.81 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a x)\right )-\log \left (\sinh ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.89 \[ \frac {\Chi \left (2 \arcsinh \left (a x \right )\right )}{2 a^{3}}-\frac {\ln \left (\arcsinh \left (a x \right )\right )}{2 a^{3}} \]
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 \[ \int \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^2}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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