Optimal. Leaf size=21 \[ \frac {5 x}{2 \left (1-x^2\right )}+\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {28, 385, 207} \begin {gather*} \frac {5 x}{2 \left (1-x^2\right )}+\frac {1}{2} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 207
Rule 385
Rubi steps
\begin {align*} \int \frac {3+2 x^2}{1-2 x^2+x^4} \, dx &=\int \frac {3+2 x^2}{\left (-1+x^2\right )^2} \, dx\\ &=\frac {5 x}{2 \left (1-x^2\right )}-\frac {1}{2} \int \frac {1}{-1+x^2} \, dx\\ &=\frac {5 x}{2 \left (1-x^2\right )}+\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.29 \begin {gather*} \frac {1}{4} \left (-\frac {10 x}{x^2-1}-\log (1-x)+\log (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 {3+2 x^2}{1-2 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.03, size = 34, normalized size = 1.62 \begin {gather*} \frac {{\left (x^{2} - 1\right )} \log \left (x + 1\right ) - {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 10 \, x}{4 \, {\left (x^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 25, normalized size = 1.19 \begin {gather*} -\frac {5 \, x}{2 \, {\left (x^{2} - 1\right )}} + \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 1.33 \begin {gather*} \frac {\ln \left (x +1\right )}{4}-\frac {\ln \left (x -1\right )}{4}-\frac {5}{4 \left (x +1\right )}-\frac {5}{4 \left (x -1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 23, normalized size = 1.10 \begin {gather*} -\frac {5 \, x}{2 \, {\left (x^{2} - 1\right )}} + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 17, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atanh}\relax (x)}{2}-\frac {5\,x}{2\,\left (x^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 22, normalized size = 1.05 \begin {gather*} - \frac {5 x}{2 x^{2} - 2} - \frac {\log {\left (x - 1 \right )}}{4} + \frac {\log {\left (x + 1 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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