Optimal. Leaf size=39 \[ \frac {3+x}{2 \left (1-x^2\right )}-\frac {3}{4} \log (1-x)+2 \log (x)-\frac {5}{4} \log (1+x) \]
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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1819, 815}
\begin {gather*} \frac {x+3}{2 \left (1-x^2\right )}-\frac {3}{4} \log (1-x)+2 \log (x)-\frac {5}{4} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 815
Rule 1819
Rubi steps
\begin {align*} \int \frac {2+x^2+x^3}{x \left (-1+x^2\right )^2} \, dx &=\frac {3+x}{2 \left (1-x^2\right )}+\frac {1}{2} \int \frac {-4+x}{x \left (-1+x^2\right )} \, dx\\ &=\frac {3+x}{2 \left (1-x^2\right )}+\frac {1}{2} \int \left (-\frac {3}{2 (-1+x)}+\frac {4}{x}-\frac {5}{2 (1+x)}\right ) \, dx\\ &=\frac {3+x}{2 \left (1-x^2\right )}-\frac {3}{4} \log (1-x)+2 \log (x)-\frac {5}{4} \log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.21 \begin {gather*} \frac {1}{4} \left (-\frac {2}{-1+x}-\frac {4}{-1+x^2}+\log (1-x)+8 \log (x)-\log (1+x)-4 \log \left (1-x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 32, normalized size = 0.82
method | result | size |
norman | \(\frac {-\frac {x}{2}-\frac {3}{2}}{x^{2}-1}+2 \ln \left (x \right )-\frac {3 \ln \left (-1+x \right )}{4}-\frac {5 \ln \left (1+x \right )}{4}\) | \(31\) |
risch | \(\frac {-\frac {x}{2}-\frac {3}{2}}{x^{2}-1}+2 \ln \left (x \right )-\frac {3 \ln \left (-1+x \right )}{4}-\frac {5 \ln \left (1+x \right )}{4}\) | \(31\) |
default | \(2 \ln \left (x \right )-\frac {1}{-1+x}-\frac {3 \ln \left (-1+x \right )}{4}+\frac {1}{2+2 x}-\frac {5 \ln \left (1+x \right )}{4}\) | \(32\) |
meijerg | \(\frac {i \left (-\frac {i x}{-x^{2}+1}+i \arctanh \left (x \right )\right )}{2}+\frac {3 x^{2}}{-2 x^{2}+2}+1+2 \ln \left (x \right )+i \pi -\ln \left (-x^{2}+1\right )\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.33, size = 29, normalized size = 0.74 \begin {gather*} -\frac {x + 3}{2 \, {\left (x^{2} - 1\right )}} - \frac {5}{4} \, \log \left (x + 1\right ) - \frac {3}{4} \, \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.53, size = 45, normalized size = 1.15 \begin {gather*} -\frac {5 \, {\left (x^{2} - 1\right )} \log \left (x + 1\right ) + 3 \, {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 8 \, {\left (x^{2} - 1\right )} \log \left (x\right ) + 2 \, x + 6}{4 \, {\left (x^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 32, normalized size = 0.82 \begin {gather*} \frac {- x - 3}{2 x^{2} - 2} + 2 \log {\left (x \right )} - \frac {3 \log {\left (x - 1 \right )}}{4} - \frac {5 \log {\left (x + 1 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 35, normalized size = 0.90 \begin {gather*} -\frac {x + 3}{2 \, {\left (x + 1\right )} {\left (x - 1\right )}} - \frac {5}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {3}{4} \, \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 31, normalized size = 0.79 \begin {gather*} 2\,\ln \left (x\right )-\frac {5\,\ln \left (x+1\right )}{4}-\frac {3\,\ln \left (x-1\right )}{4}-\frac {\frac {x}{2}+\frac {3}{2}}{x^2-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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