Optimal. Leaf size=14 \[ \frac {1}{6} \tan ^{-1}\left (\frac {1}{2} \left (-3+x^3\right )\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 632, 210}
\begin {gather*} \frac {1}{6} \text {ArcTan}\left (\frac {1}{2} \left (x^3-3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 1366
Rubi steps
\begin {align*} \int \frac {x^2}{13-6 x^3+x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{13-6 x+x^2} \, dx,x,x^3\right )\\ &=-\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,2 \left (-3+x^3\right )\right )\right )\\ &=\frac {1}{6} \tan ^{-1}\left (\frac {1}{2} \left (-3+x^3\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{6} \tan ^{-1}\left (\frac {1}{2} \left (-3+x^3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 11, normalized size = 0.79
method | result | size |
default | \(\frac {\arctan \left (\frac {x^{3}}{2}-\frac {3}{2}\right )}{6}\) | \(11\) |
risch | \(\frac {\arctan \left (\frac {x^{3}}{2}-\frac {3}{2}\right )}{6}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.14, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {1}{2} \, x^{3} - \frac {3}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {1}{2} \, x^{3} - \frac {3}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 10, normalized size = 0.71 \begin {gather*} \frac {\operatorname {atan}{\left (\frac {x^{3}}{2} - \frac {3}{2} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {1}{2} \, x^{3} - \frac {3}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 10, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x^3}{2}-\frac {3}{2}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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