Optimal. Leaf size=15 \[ \frac {1}{2} \log \left (1-x^2+x^4\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1601}
\begin {gather*} \frac {1}{2} \log \left (x^4-x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 1601
Rubi steps
\begin {align*} \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx &=\frac {1}{2} \log \left (1-x^2+x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (1-x^2+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.72, size = 13, normalized size = 0.87 \begin {gather*} \frac {\text {Log}\left [1-x^2+x^4\right ]}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 14, normalized size = 0.93
method | result | size |
default | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{2}\) | \(14\) |
norman | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{2}\) | \(14\) |
risch | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{2}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (x^{4} - x^{2} + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (x^{4}-x^{2}+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (x^4-x^2+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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