Optimal. Leaf size=51 \[ \frac {\log \left (3 x^2+2 \sqrt {3} x+2\right )}{4 \sqrt {3}}-\frac {\log \left (3 x^2-2 \sqrt {3} x+2\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1165, 628} \begin {gather*} \frac {\log \left (3 x^2+2 \sqrt {3} x+2\right )}{4 \sqrt {3}}-\frac {\log \left (3 x^2-2 \sqrt {3} x+2\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1165
Rubi steps
\begin {align*} \int \frac {2-3 x^2}{4+9 x^4} \, dx &=-\frac {\int \frac {\frac {2}{\sqrt {3}}+2 x}{-\frac {2}{3}-\frac {2 x}{\sqrt {3}}-x^2} \, dx}{4 \sqrt {3}}-\frac {\int \frac {\frac {2}{\sqrt {3}}-2 x}{-\frac {2}{3}+\frac {2 x}{\sqrt {3}}-x^2} \, dx}{4 \sqrt {3}}\\ &=-\frac {\log \left (2-2 \sqrt {3} x+3 x^2\right )}{4 \sqrt {3}}+\frac {\log \left (2+2 \sqrt {3} x+3 x^2\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.86 \begin {gather*} \frac {\log \left (3 x^2+2 \sqrt {3} x+2\right )-\log \left (-3 x^2+2 \sqrt {3} x-2\right )}{4 \sqrt {3}} \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 {2-3 x^2}{4+9 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.55, size = 42, normalized size = 0.82 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (\frac {9 \, x^{4} + 24 \, x^{2} + 4 \, \sqrt {3} {\left (3 \, x^{3} + 2 \, x\right )} + 4}{9 \, x^{4} + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 40, normalized size = 0.78 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}} x + \frac {2}{3}\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}} x + \frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 82, normalized size = 1.61 \begin {gather*} -\frac {\sqrt {6}\, \sqrt {2}\, \ln \left (\frac {x^{2}-\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}{x^{2}+\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}\right )}{48}+\frac {\sqrt {6}\, \sqrt {2}\, \ln \left (\frac {x^{2}+\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}{x^{2}-\frac {\sqrt {6}\, \sqrt {2}\, x}{3}+\frac {2}{3}}\right )}{48} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 39, normalized size = 0.76 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (3 \, x^{2} + 2 \, \sqrt {3} x + 2\right ) - \frac {1}{12} \, \sqrt {3} \log \left (3 \, x^{2} - 2 \, \sqrt {3} x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.43, size = 21, normalized size = 0.41 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x}{3\,x^2+2}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 49, normalized size = 0.96 \begin {gather*} - \frac {\sqrt {3} \log {\left (x^{2} - \frac {2 \sqrt {3} x}{3} + \frac {2}{3} \right )}}{12} + \frac {\sqrt {3} \log {\left (x^{2} + \frac {2 \sqrt {3} x}{3} + \frac {2}{3} \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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