Optimal. Leaf size=101 \[ -\frac {2 x \sqrt {a^2 x^4+1}}{9 a}+\frac {\left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{9 a^{3/2} \sqrt {a^2 x^4+1}}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5902, 12, 321, 220} \[ -\frac {2 x \sqrt {a^2 x^4+1}}{9 a}+\frac {\left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{9 a^{3/2} \sqrt {a^2 x^4+1}}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 220
Rule 321
Rule 5902
Rubi steps
\begin {align*} \int x^2 \sinh ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )-\frac {1}{3} \int \frac {2 a x^4}{\sqrt {1+a^2 x^4}} \, dx\\ &=\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )-\frac {1}{3} (2 a) \int \frac {x^4}{\sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )+\frac {2 \int \frac {1}{\sqrt {1+a^2 x^4}} \, dx}{9 a}\\ &=-\frac {2 x \sqrt {1+a^2 x^4}}{9 a}+\frac {1}{3} x^3 \sinh ^{-1}\left (a x^2\right )+\frac {\left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{9 a^{3/2} \sqrt {1+a^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 75, normalized size = 0.74 \[ \frac {1}{9} \left (-\frac {2 \left (a^2 x^5+x\right )}{a \sqrt {a^2 x^4+1}}-\frac {2 \sqrt {i a} F\left (\left .i \sinh ^{-1}\left (\sqrt {i a} x\right )\right |-1\right )}{a^2}+3 x^3 \sinh ^{-1}\left (a x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arsinh}\left (a x^{2}\right ), x\right ) \]
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 \[ \int x^{2} \operatorname {arsinh}\left (a x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 89, normalized size = 0.88 \[ \frac {x^{3} \arcsinh \left (a \,x^{2}\right )}{3}-\frac {2 a \left (\frac {x \sqrt {a^{2} x^{4}+1}}{3 a^{2}}-\frac {\sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \EllipticF \left (x \sqrt {i a}, i\right )}{3 a^{2} \sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\right )}{3} \]
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 \[ \frac {1}{3} \, x^{3} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - \frac {2}{9} \, x^{3} - 2 \, a \int \frac {x^{4}}{3 \, {\left (a^{3} x^{6} + a x^{2} + {\left (a^{2} x^{4} + 1\right )}^{\frac {3}{2}}\right )}}\,{d x} - \frac {i \, \sqrt {2} {\left (\log \left (\frac {i \, \sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right ) - \log \left (-\frac {i \, \sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right )\right )}}{12 \, a^{\frac {3}{2}}} - \frac {i \, \sqrt {2} {\left (\log \left (\frac {i \, \sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right ) - \log \left (-\frac {i \, \sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right )\right )}}{12 \, a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{12 \, a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{12 \, a^{\frac {3}{2}}} \]
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 \[ \int x^2\,\mathrm {asinh}\left (a\,x^2\right ) \,d x \]
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 \[ \int x^{2} \operatorname {asinh}{\left (a x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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