Optimal. Leaf size=23 \[ 5+e^{20}+\log (x)-\frac {x \log ^4(x)}{-e^5+x} \]
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Rubi [A] time = 0.39, antiderivative size = 18, normalized size of antiderivative = 0.78, number of steps used = 13, number of rules used = 8, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1594, 27, 6688, 2317, 2374, 2383, 6589, 2318} \begin {gather*} \frac {x \log ^4(x)}{e^5-x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1594
Rule 2317
Rule 2318
Rule 2374
Rule 2383
Rule 6589
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{10}-2 e^5 x+x^2+\left (4 e^5 x-4 x^2\right ) \log ^3(x)+e^5 x \log ^4(x)}{x \left (e^{10}-2 e^5 x+x^2\right )} \, dx\\ &=\int \frac {e^{10}-2 e^5 x+x^2+\left (4 e^5 x-4 x^2\right ) \log ^3(x)+e^5 x \log ^4(x)}{x \left (-e^5+x\right )^2} \, dx\\ &=\int \left (\frac {1}{x}+\frac {4 \log ^3(x)}{e^5-x}+\frac {e^5 \log ^4(x)}{\left (e^5-x\right )^2}\right ) \, dx\\ &=\log (x)+4 \int \frac {\log ^3(x)}{e^5-x} \, dx+e^5 \int \frac {\log ^4(x)}{\left (e^5-x\right )^2} \, dx\\ &=\log (x)+\frac {x \log ^4(x)}{e^5-x}-4 \log ^3(x) \log \left (1-\frac {x}{e^5}\right )-4 \int \frac {\log ^3(x)}{e^5-x} \, dx+12 \int \frac {\log ^2(x) \log \left (1-\frac {x}{e^5}\right )}{x} \, dx\\ &=\log (x)+\frac {x \log ^4(x)}{e^5-x}-12 \log ^2(x) \text {Li}_2\left (\frac {x}{e^5}\right )-12 \int \frac {\log ^2(x) \log \left (1-\frac {x}{e^5}\right )}{x} \, dx+24 \int \frac {\log (x) \text {Li}_2\left (\frac {x}{e^5}\right )}{x} \, dx\\ &=\log (x)+\frac {x \log ^4(x)}{e^5-x}+24 \log (x) \text {Li}_3\left (\frac {x}{e^5}\right )-24 \int \frac {\log (x) \text {Li}_2\left (\frac {x}{e^5}\right )}{x} \, dx-24 \int \frac {\text {Li}_3\left (\frac {x}{e^5}\right )}{x} \, dx\\ &=\log (x)+\frac {x \log ^4(x)}{e^5-x}-24 \text {Li}_4\left (\frac {x}{e^5}\right )+24 \int \frac {\text {Li}_3\left (\frac {x}{e^5}\right )}{x} \, dx\\ &=\log (x)+\frac {x \log ^4(x)}{e^5-x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 25, normalized size = 1.09 \begin {gather*} \frac {\log (x) \left (e^5-x+x \log ^3(x)\right )}{e^5-x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 27, normalized size = 1.17 \begin {gather*} -\frac {x \log \relax (x)^{4} - {\left (x - e^{5}\right )} \log \relax (x)}{x - e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 27, normalized size = 1.17 \begin {gather*} -\frac {x \log \relax (x)^{4} - x \log \relax (x) + e^{5} \log \relax (x)}{x - e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 18, normalized size = 0.78
method | result | size |
risch | \(\frac {x \ln \relax (x )^{4}}{{\mathrm e}^{5}-x}+\ln \relax (x )\) | \(18\) |
norman | \(\frac {x \ln \relax (x )^{4}+{\mathrm e}^{5} \ln \relax (x )-x \ln \relax (x )}{{\mathrm e}^{5}-x}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 66, normalized size = 2.87 \begin {gather*} -\frac {x \log \relax (x)^{4}}{x - e^{5}} - {\left (e^{\left (-10\right )} \log \left (x - e^{5}\right ) - e^{\left (-10\right )} \log \relax (x) + \frac {1}{x e^{5} - e^{10}}\right )} e^{10} + \frac {e^{5}}{x - e^{5}} + \log \left (x - e^{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 22, normalized size = 0.96 \begin {gather*} \ln \relax (x)-{\ln \relax (x)}^4\,\left (\frac {{\mathrm {e}}^5}{x-{\mathrm {e}}^5}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 14, normalized size = 0.61 \begin {gather*} - \frac {x \log {\relax (x )}^{4}}{x - e^{5}} + \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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