Optimal. Leaf size=18 \[ \frac {1}{3} \log \left (\sqrt {x^6+1}+x^3\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.44, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {275, 215} \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1+x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \sinh ^{-1}\left (x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 0.44 \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (\sqrt {x^6+1}+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 7, normalized size = 0.39 \begin {gather*} \frac {\arcsinh \left (x^{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 33, normalized size = 1.83 \begin {gather*} \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \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 = -0.06 \begin {gather*} \int \frac {x^2}{\sqrt {x^6+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 5, normalized size = 0.28 \begin {gather*} \frac {\operatorname {asinh}{\left (x^{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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