Optimal. Leaf size=24 \[ \frac {1}{1+x}+\frac {3}{2} \log (1-x)-\frac {1}{2} \log (1+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, 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 = {907}
\begin {gather*} \frac {1}{x+1}+\frac {3}{2} \log (1-x)-\frac {1}{2} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 907
Rubi steps
\begin {align*} \int \frac {3+2 x+x^2}{(-1+x) (1+x)^2} \, dx &=\int \left (\frac {3}{2 (-1+x)}-\frac {1}{(1+x)^2}-\frac {1}{2 (1+x)}\right ) \, dx\\ &=\frac {1}{1+x}+\frac {3}{2} \log (1-x)-\frac {1}{2} \log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{1+x}+\frac {3}{2} \log (-1+x)-\frac {1}{2} \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 19, normalized size = 0.79
method | result | size |
default | \(\frac {3 \ln \left (-1+x \right )}{2}+\frac {1}{1+x}-\frac {\ln \left (1+x \right )}{2}\) | \(19\) |
norman | \(\frac {3 \ln \left (-1+x \right )}{2}+\frac {1}{1+x}-\frac {\ln \left (1+x \right )}{2}\) | \(19\) |
risch | \(\frac {3 \ln \left (-1+x \right )}{2}+\frac {1}{1+x}-\frac {\ln \left (1+x \right )}{2}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 6.61, size = 18, normalized size = 0.75 \begin {gather*} \frac {1}{x + 1} - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {3}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 26, normalized size = 1.08 \begin {gather*} -\frac {{\left (x + 1\right )} \log \left (x + 1\right ) - 3 \, {\left (x + 1\right )} \log \left (x - 1\right ) - 2}{2 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 0.79 \begin {gather*} \frac {3 \log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} + \frac {1}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{x + 1} + \log \left ({\left | x + 1 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | -\frac {2}{x + 1} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 18, normalized size = 0.75 \begin {gather*} \frac {3\,\ln \left (x-1\right )}{2}-\frac {\ln \left (x+1\right )}{2}+\frac {1}{x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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