Optimal. Leaf size=25 \[ \frac {x^3}{3}-x+\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]
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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 = {1593, 459, 321, 203} \begin {gather*} \frac {x^3}{3}-x+\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 459
Rule 1593
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*} \frac {x^3}{3}-x+\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.74, 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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giac [A] time = 0.35, 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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maple [A] time = 0.00, size = 21, normalized size = 0.84 \begin {gather*} \frac {x^{3}}{3}-x +\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x \right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, 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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sympy [A] time = 0.14, 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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