Optimal. Leaf size=25 \[ -x+\frac {x^3}{3}+\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1607, 470, 327,
209} \begin {gather*} \frac {\text {ArcTan}\left (\sqrt {2} x\right )}{\sqrt {2}}+\frac {x^3}{3}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 470
Rule 1607
Rubi steps
\begin {align*} \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx &=\int \frac {x^2 \left (-1+2 x^2\right )}{1+2 x^2} \, dx\\ &=\frac {x^3}{3}-2 \int \frac {x^2}{1+2 x^2} \, dx\\ &=-x+\frac {x^3}{3}+\int \frac {1}{1+2 x^2} \, dx\\ &=-x+\frac {x^3}{3}+\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -x+\frac {x^3}{3}+\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 21, normalized size = 0.84
method | result | size |
default | \(-x +\frac {x^{3}}{3}+\frac {\arctan \left (x \sqrt {2}\right ) \sqrt {2}}{2}\) | \(21\) |
risch | \(-x +\frac {x^{3}}{3}+\frac {\arctan \left (x \sqrt {2}\right ) \sqrt {2}}{2}\) | \(21\) |
meijerg | \(\frac {\sqrt {2}\, \left (-\frac {2 x \sqrt {2}\, \left (-10 x^{2}+15\right )}{15}+2 \arctan \left (x \sqrt {2}\right )\right )}{8}-\frac {\sqrt {2}\, \left (2 x \sqrt {2}-2 \arctan \left (x \sqrt {2}\right )\right )}{8}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.78, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 0.80 \begin {gather*} \frac {x^{3}}{3} - x + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.24, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 20, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\right )}{2}-x+\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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