Optimal. Leaf size=33 \[ \frac {e^{-e^{e^5 x}+x}-x+x^2 \log (5)}{\log ((-1+x) \log (x))} \]
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Rubi [F]
time = 6.84, 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 {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (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 {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{(-1+x) x \log (x) \log ^2((-1+x) \log (x))} \, dx\\ &=\int \left (-\frac {x (-1+x \log (5))}{(-1+x) \log ^2((-1+x) \log (x))}-\frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}+\frac {x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}-\frac {x \log (5)}{\log (x) \log ^2((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) x \log (x) \log ^2((-1+x) \log (x))}-\frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ((-1+x) \log (x))}+\frac {-1+x \log (25)}{\log ((-1+x) \log (x))}\right ) \, dx\\ &=-\left (\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx\right )-\int \frac {x (-1+x \log (5))}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ((-1+x) \log (x))} \, dx+\int \frac {-1+x \log (25)}{\log ((-1+x) \log (x))} \, dx\\ &=-\left (\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx\right )-\int \left (-\frac {1-\log (5)}{\log ^2((-1+x) \log (x))}+\frac {-1+\log (5)}{(-1+x) \log ^2((-1+x) \log (x))}+\frac {x \log (5)}{\log ^2((-1+x) \log (x))}\right ) \, dx+\int \left (\frac {1}{\log (x) \log ^2((-1+x) \log (x))}+\frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx-\int \left (\frac {e^{-e^{e^5 x}+x} (1-x-x \log (x))}{x \log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx+\int \left (\frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))}\right ) \, dx+\int \left (-\frac {1}{\log ((-1+x) \log (x))}+\frac {x \log (25)}{\log ((-1+x) \log (x))}\right ) \, dx-\int \frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))} \, dx\\ &=-\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} (1-x-x \log (x))}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx\\ &=-\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx-\int \left (-\frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx-\int \left (\frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ^2((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx\\ &=-\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx\\ &=-\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx-\int \left (\frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ^2((-1+x) \log (x))}\right ) \, dx-\int \left (\frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx\\ &=-\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.40, size = 33, normalized size = 1.00 \begin {gather*} \frac {e^{-e^{e^5 x}+x}-x+x^2 \log (5)}{\log ((-1+x) \log (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 20.20, size = 118, normalized size = 3.58
method | result | size |
risch | \(\frac {2 i \left (x^{2} \ln \left (5\right )+{\mathrm e}^{x -{\mathrm e}^{x \,{\mathrm e}^{5}}}-x \right )}{\pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \left (x -1\right )\right )-\pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \left (x -1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \left (x -1\right )\right )^{2}+\pi \mathrm {csgn}\left (i \ln \left (x \right ) \left (x -1\right )\right )^{3}+2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (x -1\right )}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 31, normalized size = 0.94 \begin {gather*} \frac {x^{2} \log \left (5\right ) - x + e^{\left (x - e^{\left (x e^{5}\right )}\right )}}{\log \left (x - 1\right ) + \log \left (\log \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 38, normalized size = 1.15 \begin {gather*} \frac {x^{2} \log \left (5\right ) - x + e^{\left ({\left (x e^{5} - e^{\left (x e^{5} + 5\right )}\right )} e^{\left (-5\right )}\right )}}{\log \left ({\left (x - 1\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 = 1.05, size = 34, normalized size = 1.03 \begin {gather*} \frac {x^{2} \log {\left (5 \right )} - x}{\log {\left (\left (x - 1\right ) \log {\left (x \right )} \right )}} + \frac {e^{x - e^{x e^{5}}}}{\log {\left (\left (x - 1\right ) \log {\left (x \right )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 33, normalized size = 1.00 \begin {gather*} \frac {x^{2} \log \left (5\right ) - x + e^{\left (x - e^{\left (x e^{5}\right )}\right )}}{\log \left (x \log \left (x\right ) - \log \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.89, size = 138, normalized size = 4.18 \begin {gather*} \frac {{\mathrm {e}}^{x-{\mathrm {e}}^{x\,{\mathrm {e}}^5}}}{\ln \left (\ln \left (x\right )\,\left (x-1\right )\right )}-x\,\left (\ln \left (25\right )+1\right )+2\,x^2\,\ln \left (5\right )+\frac {x\,\left (x\,\ln \left (5\right )-1\right )-\frac {x\,\ln \left (\ln \left (x\right )\,\left (x-1\right )\right )\,\ln \left (x\right )\,\left (2\,x\,\ln \left (5\right )-1\right )\,\left (x-1\right )}{x+x\,\ln \left (x\right )-1}}{\ln \left (\ln \left (x\right )\,\left (x-1\right )\right )}-\frac {x+2\,x\,\ln \left (5\right )-2\,x^2\,\ln \left (5\right )-2\,x^3\,\ln \left (5\right )+2\,x^4\,\ln \left (5\right )+x^2-x^3-1}{\left (x+1\right )\,\left (x+x\,\ln \left (x\right )-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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