Optimal. Leaf size=20 \[ \log \left (-95 \left (-2+\frac {2 e^x}{x (\log (5)+\log (x))}\right )\right ) \]
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Rubi [F]
time = 3.04, 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 {e^x (1+(1-x) \log (5))+e^x (1-x) \log (x)}{-e^x x \log (5)+x^2 \log ^2(5)+\left (-e^x x+2 x^2 \log (5)\right ) \log (x)+x^2 \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 {e^x (-1+x \log (5)+x \log (x)-\log (5 x))}{x \left (e^x-x \log (5)-x \log (x)\right ) \log (5 x)} \, dx\\ &=\int \left (\frac {e^x}{x \left (-e^x+x \log (5)+x \log (x)\right )}+\frac {e^x}{x \left (-e^x+x \log (5)+x \log (x)\right ) \log (5 x)}-\frac {e^x \log (5)}{\left (-e^x+x \log (5)+x \log (x)\right ) \log (5 x)}-\frac {e^x \log (x)}{\left (-e^x+x \log (5)+x \log (x)\right ) \log (5 x)}\right ) \, dx\\ &=-\left (\log (5) \int \frac {e^x}{\left (-e^x+x \log (5)+x \log (x)\right ) \log (5 x)} \, dx\right )+\int \frac {e^x}{x \left (-e^x+x \log (5)+x \log (x)\right )} \, dx+\int \frac {e^x}{x \left (-e^x+x \log (5)+x \log (x)\right ) \log (5 x)} \, dx-\int \frac {e^x \log (x)}{\left (-e^x+x \log (5)+x \log (x)\right ) \log (5 x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^x (1+(1-x) \log (5))+e^x (1-x) \log (x)}{-e^x x \log (5)+x^2 \log ^2(5)+\left (-e^x x+2 x^2 \log (5)\right ) \log (x)+x^2 \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.71, size = 27, normalized size = 1.35
method | result | size |
risch | \(\ln \left (\ln \left (x \right )+\frac {x \ln \left (5\right )-{\mathrm e}^{x}}{x}\right )-\ln \left (\ln \left (5\right )+\ln \left (x \right )\right )\) | \(27\) |
norman | \(-\ln \left (x \right )-\ln \left (\ln \left (5\right )+\ln \left (x \right )\right )+\ln \left (x \ln \left (5\right )+x \ln \left (x \right )-{\mathrm e}^{x}\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 27, normalized size = 1.35 \begin {gather*} \log \left (-x \log \left (5\right ) - x \log \left (x\right ) + e^{x}\right ) - \log \left (x\right ) - \log \left (\log \left (5\right ) + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 27, normalized size = 1.35 \begin {gather*} \log \left (\frac {x \log \left (5\right ) + x \log \left (x\right ) - e^{x}}{x}\right ) - \log \left (\log \left (5\right ) + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 26, normalized size = 1.30 \begin {gather*} - \log {\left (x \right )} - \log {\left (\log {\left (x \right )} + \log {\left (5 \right )} \right )} + \log {\left (- x \log {\left (x \right )} - x \log {\left (5 \right )} + e^{x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 27, normalized size = 1.35 \begin {gather*} \log \left (-x \log \left (5\right ) - x \log \left (x\right ) + e^{x}\right ) - \log \left (x\right ) - \log \left (\log \left (5\right ) + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\int \frac {{\mathrm {e}}^x\,\left (\ln \left (5\right )\,\left (x-1\right )-1\right )+{\mathrm {e}}^x\,\ln \left (x\right )\,\left (x-1\right )}{x^2\,{\ln \left (5\right )}^2+x^2\,{\ln \left (x\right )}^2+\ln \left (x\right )\,\left (2\,x^2\,\ln \left (5\right )-x\,{\mathrm {e}}^x\right )-x\,{\mathrm {e}}^x\,\ln \left (5\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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