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 = 0.0574547, antiderivative size = 34, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0166073, 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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Maple [A] time = 0.007, size = 25, normalized size = 0.7 \begin{align*} -{\frac{3}{2\,x-2}}-{\frac{5\,\ln \left ( x-1 \right ) }{4}}+2\,\ln \left ( x \right ) -{\frac{3\,\ln \left ( 1+x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984737, size = 32, normalized size = 0.94 \begin{align*} -\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 \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53738, size = 112, normalized size = 3.29 \begin{align*} -\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 \left (x\right ) + 6}{4 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.133397, size = 27, normalized size = 0.79 \begin{align*} 2 \log{\left (x \right )} - \frac{5 \log{\left (x - 1 \right )}}{4} - \frac{3 \log{\left (x + 1 \right )}}{4} - \frac{3}{2 x - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15256, size = 46, normalized size = 1.35 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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