Optimal. Leaf size=23 \[ \log \left (\frac {\log ^2\left (e^{2 x}\right )}{x \left (2+x-x^2\right )^2}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 0.74, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1608, 1642}
\begin {gather*} -2 \log (2-x)+\log (x)-2 \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 1608
Rule 1642
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+x-3 x^2}{x \left (-2-x+x^2\right )} \, dx\\ &=\int \left (-\frac {2}{-2+x}+\frac {1}{x}-\frac {2}{1+x}\right ) \, dx\\ &=-2 \log (2-x)+\log (x)-2 \log (1+x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.61 \begin {gather*} \log (x)-2 \log \left (2+x-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 16, normalized size = 0.70
method | result | size |
risch | \(\ln \left (x \right )-2 \ln \left (x^{2}-x -2\right )\) | \(15\) |
default | \(-2 \ln \left (x -2\right )-2 \ln \left (x +1\right )+\ln \left (x \right )\) | \(16\) |
norman | \(-2 \ln \left (x -2\right )-2 \ln \left (x +1\right )+\ln \left (x \right )\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.65 \begin {gather*} -2 \, \log \left (x + 1\right ) - 2 \, \log \left (x - 2\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 14, normalized size = 0.61 \begin {gather*} -2 \, \log \left (x^{2} - x - 2\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 12, normalized size = 0.52 \begin {gather*} \log {\left (x \right )} - 2 \log {\left (x^{2} - x - 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 0.78 \begin {gather*} -2 \, \log \left ({\left | x + 1 \right |}\right ) - 2 \, \log \left ({\left | x - 2 \right |}\right ) + \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 = 0.07, size = 14, normalized size = 0.61 \begin {gather*} \ln \left (x\right )-2\,\ln \left (x^2-x-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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