Optimal. Leaf size=34 \[ \frac {3}{2 (1-x)}-\frac {5}{4} \log (1-x)+2 \log (x)-\frac {3}{4} \log (x+1) \]
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Rubi [A] time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1612} \[ \frac {3}{2 (1-x)}-\frac {5}{4} \log (1-x)+2 \log (x)-\frac {3}{4} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 1612
Rubi steps
\begin {align*} \int \frac {2+x^2}{(-1+x)^2 x (1+x)} \, dx &=\int \left (\frac {3}{2 (-1+x)^2}-\frac {5}{4 (-1+x)}+\frac {2}{x}-\frac {3}{4 (1+x)}\right ) \, dx\\ &=\frac {3}{2 (1-x)}-\frac {5}{4} \log (1-x)+2 \log (x)-\frac {3}{4} \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.94 \[ -\frac {3}{2 (x-1)}-\frac {5}{4} \log (1-x)+2 \log (x)-\frac {3}{4} \log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 34, normalized size = 1.00 \[ -\frac {3 \, {\left (x - 1\right )} \log \left (x + 1\right ) + 5 \, {\left (x - 1\right )} \log \left (x - 1\right ) - 8 \, {\left (x - 1\right )} \log \relax (x) + 6}{4 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 34, normalized size = 1.00 \[ -\frac {3}{2 \, {\left (x - 1\right )}} + 2 \, \log \left ({\left | -\frac {1}{x - 1} - 1 \right |}\right ) - \frac {3}{4} \, \log \left ({\left | -\frac {2}{x - 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.74 \[ 2 \ln \relax (x )-\frac {5 \ln \left (x -1\right )}{4}-\frac {3 \ln \left (x +1\right )}{4}-\frac {3}{2 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 24, normalized size = 0.71 \[ -\frac {3}{2 \, {\left (x - 1\right )}} - \frac {3}{4} \, \log \left (x + 1\right ) - \frac {5}{4} \, \log \left (x - 1\right ) + 2 \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.11, size = 26, normalized size = 0.76 \[ 2\,\ln \relax (x)-\frac {3\,\ln \left (x+1\right )}{4}-\frac {5\,\ln \left (x-1\right )}{4}-\frac {3}{2\,\left (x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 27, normalized size = 0.79 \[ 2 \log {\relax (x )} - \frac {5 \log {\left (x - 1 \right )}}{4} - \frac {3 \log {\left (x + 1 \right )}}{4} - \frac {3}{2 x - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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