Optimal. Leaf size=25 \[ \frac {1}{2} \log \left (x^2+x+1\right )-\frac {1}{2} \log \left (x^2-x+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1164, 628} \begin {gather*} \frac {1}{2} \log \left (x^2+x+1\right )-\frac {1}{2} \log \left (x^2-x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1164
Rubi steps
\begin {align*} \int \frac {1-x^2}{1+x^2+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1+2 x}{-1-x-x^2} \, dx\right )-\frac {1}{2} \int \frac {1-2 x}{-1+x-x^2} \, dx\\ &=-\frac {1}{2} \log \left (1-x+x^2\right )+\frac {1}{2} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (x^2+x+1\right )-\frac {1}{2} \log \left (x^2-x+1\right ) \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 {1-x^2}{1+x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.56, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 1.40 \begin {gather*} \frac {1}{4} \, \log \left ({\left | x + \frac {1}{x + \frac {1}{x}} + \frac {1}{x} + 2 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x + \frac {1}{x + \frac {1}{x}} + \frac {1}{x} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.88 \begin {gather*} -\frac {\ln \left (x^{2}-x +1\right )}{2}+\frac {\ln \left (x^{2}+x +1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (x^{2} - x + 1\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 = 0.40 \begin {gather*} \mathrm {atanh}\left (\frac {x}{x^2+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 0.76 \begin {gather*} - \frac {\log {\left (x^{2} - x + 1 \right )}}{2} + \frac {\log {\left (x^{2} + x + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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