Optimal. Leaf size=27 \[ 5+e^5-\log \left ((5-x) x \left (\frac {1}{2}+x\right ) (-5-x+\log (x))\right ) \]
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Rubi [A] time = 0.58, antiderivative size = 31, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 4, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6741, 6728, 1612, 6684} \begin {gather*} -\log (5-x)-\log (x)-\log (2 x+1)-\log (x-\log (x)+5) \end {gather*}
Antiderivative was successfully verified.
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Rule 1612
Rule 6684
Rule 6728
Rule 6741
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-20-91 x+x^2+8 x^3+\left (5+18 x-6 x^2\right ) \log (x)}{x \left (5+9 x-2 x^2\right ) (5+x-\log (x))} \, dx\\ &=\int \left (\frac {5+18 x-6 x^2}{(-5+x) x (1+2 x)}+\frac {1-x}{x (5+x-\log (x))}\right ) \, dx\\ &=\int \frac {5+18 x-6 x^2}{(-5+x) x (1+2 x)} \, dx+\int \frac {1-x}{x (5+x-\log (x))} \, dx\\ &=-\log (5+x-\log (x))+\int \left (\frac {1}{5-x}-\frac {1}{x}-\frac {2}{1+2 x}\right ) \, dx\\ &=-\log (5-x)-\log (x)-\log (1+2 x)-\log (5+x-\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 28, normalized size = 1.04 \begin {gather*} -\log (x)-\log \left (5+9 x-2 x^2\right )-\log (5+x-\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 28, normalized size = 1.04 \begin {gather*} -\log \left (2 \, x^{3} - 9 \, x^{2} - 5 \, x\right ) - \log \left (-x + \log \relax (x) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 28, normalized size = 1.04 \begin {gather*} -\log \left (2 \, x^{2} - 9 \, x - 5\right ) - \log \relax (x) - \log \left (-x + \log \relax (x) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 29, normalized size = 1.07
method | result | size |
risch | \(-\ln \left (2 x^{3}-9 x^{2}-5 x \right )-\ln \left (\ln \relax (x )-5-x \right )\) | \(29\) |
norman | \(-\ln \relax (x )-\ln \left (x -5\right )-\ln \left (2 x +1\right )-\ln \left (x -\ln \relax (x )+5\right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (2 \, x + 1\right ) - \log \left (x - 5\right ) - \log \relax (x) - \log \left (-x + \log \relax (x) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 26, normalized size = 0.96 \begin {gather*} -\ln \left (x-\ln \relax (x)+5\right )-\ln \left (x\,\left (-2\,x^2+9\,x+5\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 24, normalized size = 0.89 \begin {gather*} - \log {\left (- x + \log {\relax (x )} - 5 \right )} - \log {\left (2 x^{3} - 9 x^{2} - 5 x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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