Optimal. Leaf size=14 \[ 4-\frac {x}{5-x+\log (x)} \]
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Rubi [F]
time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-4-\log (x)}{25-10 x+x^2+(10-2 x) \log (x)+\log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-\log (x)}{(5-x+\log (x))^2} \, dx\\ &=\int \left (\frac {1-x}{(-5+x-\log (x))^2}+\frac {1}{-5+x-\log (x)}\right ) \, dx\\ &=\int \frac {1-x}{(-5+x-\log (x))^2} \, dx+\int \frac {1}{-5+x-\log (x)} \, dx\\ &=\int \left (\frac {1}{(-5+x-\log (x))^2}-\frac {x}{(-5+x-\log (x))^2}\right ) \, dx+\int \frac {1}{-5+x-\log (x)} \, dx\\ &=\int \frac {1}{(-5+x-\log (x))^2} \, dx-\int \frac {x}{(-5+x-\log (x))^2} \, dx+\int \frac {1}{-5+x-\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} -\frac {x}{5-x+\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.05, size = 16, normalized size = 1.14
method | result | size |
risch | \(\frac {x}{-\ln \left (x \right )+x -5}\) | \(12\) |
norman | \(\frac {5+\ln \left (x \right )}{-\ln \left (x \right )+x -5}\) | \(15\) |
default | \(-\frac {5+\ln \left (x \right )}{\ln \left (x \right )-x +5}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 11, normalized size = 0.79 \begin {gather*} \frac {x}{x - \log \left (x\right ) - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 11, normalized size = 0.79 \begin {gather*} \frac {x}{x - \log \left (x\right ) - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.57 \begin {gather*} - \frac {x}{- x + \log {\left (x \right )} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 11, normalized size = 0.79 \begin {gather*} \frac {x}{x - \log \left (x\right ) - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.63, size = 12, normalized size = 0.86 \begin {gather*} -\frac {x}{\ln \left (x\right )-x+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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