Optimal. Leaf size=29 \[ 1+\frac {\left (-1+e^{5 x}\right ) x}{\log \left (\frac {x}{-2-\log (-x+\log (x))}\right )} \]
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Rubi [F]
time = 9.27, 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 {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+x-e^{5 x} (1+x)+2 \left (-1+e^{5 x}\right ) \log (x)-\left (-1+e^{5 x}\right ) (x-\log (x)) \log (-x+\log (x))+\left (-1+e^{5 x} (1+5 x)\right ) (x-\log (x)) (2+\log (-x+\log (x))) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx\\ &=\int \left (\frac {1}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )}+\frac {x}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )}-\frac {2 \log (x)}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )}+\frac {\log (-x+\log (x))}{(2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )}-\frac {1}{\log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}+\frac {e^{5 x} \left (-1-x+2 \log (x)-x \log (-x+\log (x))+\log (x) \log (-x+\log (x))+2 x \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )+10 x^2 \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-2 \log (x) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-10 x \log (x) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )+x \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )+5 x^2 \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-\log (x) \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-5 x \log (x) \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )\right )}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {\log (x)}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx\right )+\int \frac {1}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx+\int \frac {x}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx+\int \frac {\log (-x+\log (x))}{(2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx-\int \frac {1}{\log \left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx+\int \frac {e^{5 x} \left (-1-x+2 \log (x)-x \log (-x+\log (x))+\log (x) \log (-x+\log (x))+2 x \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )+10 x^2 \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-2 \log (x) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-10 x \log (x) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )+x \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )+5 x^2 \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-\log (x) \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-5 x \log (x) \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )\right )}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx\\ &=\frac {e^{5 x} \left (2 x^2 \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-2 x \log (x) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )+x^2 \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )-x \log (x) \log (-x+\log (x)) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )\right )}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )}-2 \int \frac {\log (x)}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx+\int \frac {1}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx+\int \frac {x}{(x-\log (x)) (2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx+\int \frac {\log (-x+\log (x))}{(2+\log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx-\int \frac {1}{\log \left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.24, size = 26, normalized size = 0.90 \begin {gather*} \frac {\left (-1+e^{5 x}\right ) x}{\log \left (-\frac {x}{2+\log (-x+\log (x))}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (\left (1+5 x \right ) {\mathrm e}^{5 x}-1\right ) \ln \left (x \right )+\left (-5 x^{2}-x \right ) {\mathrm e}^{5 x}+x \right ) \ln \left (\ln \left (x \right )-x \right )+\left (\left (10 x +2\right ) {\mathrm e}^{5 x}-2\right ) \ln \left (x \right )+\left (-10 x^{2}-2 x \right ) {\mathrm e}^{5 x}+2 x \right ) \ln \left (-\frac {x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )+\left (\left (-{\mathrm e}^{5 x}+1\right ) \ln \left (x \right )+x \,{\mathrm e}^{5 x}-x \right ) \ln \left (\ln \left (x \right )-x \right )+\left (-2 \,{\mathrm e}^{5 x}+2\right ) \ln \left (x \right )+\left (x +1\right ) {\mathrm e}^{5 x}-x -1}{\left (\left (\ln \left (x \right )-x \right ) \ln \left (\ln \left (x \right )-x \right )+2 \ln \left (x \right )-2 x \right ) \ln \left (-\frac {x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 30, normalized size = 1.03 \begin {gather*} \frac {x e^{\left (5 \, x\right )} - x}{\log \left (x\right ) - \log \left (-\log \left (-x + \log \left (x\right )\right ) - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 28, normalized size = 0.97 \begin {gather*} \frac {x e^{\left (5 \, x\right )} - x}{\log \left (-\frac {x}{\log \left (-x + \log \left (x\right )\right ) + 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.31, size = 34, normalized size = 1.17 \begin {gather*} \frac {x e^{5 x}}{\log {\left (- \frac {x}{\log {\left (- x + \log {\left (x \right )} \right )} + 2} \right )}} - \frac {x}{\log {\left (- \frac {x}{\log {\left (- x + \log {\left (x \right )} \right )} + 2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1672 vs.
\(2 (27) = 54\).
time = 1.29, size = 1672, normalized size = 57.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.57, size = 266, normalized size = 9.17 \begin {gather*} {\mathrm {e}}^{5\,x}\,\left (5\,x^2+x\right )-x+\frac {x\,\left ({\mathrm {e}}^{5\,x}-1\right )-\frac {x\,\ln \left (-\frac {x}{\ln \left (\ln \left (x\right )-x\right )+2}\right )\,\left (x-\ln \left (x\right )\right )\,\left (\ln \left (\ln \left (x\right )-x\right )+2\right )\,\left ({\mathrm {e}}^{5\,x}+5\,x\,{\mathrm {e}}^{5\,x}-1\right )}{x-2\,\ln \left (x\right )+x\,\ln \left (\ln \left (x\right )-x\right )-\ln \left (\ln \left (x\right )-x\right )\,\ln \left (x\right )+1}}{\ln \left (-\frac {x}{\ln \left (\ln \left (x\right )-x\right )+2}\right )}-\frac {x^3\,\ln \left (x\right )-x^2\,\ln \left (x\right )+x^2\,{\mathrm {e}}^{5\,x}+x^3\,{\mathrm {e}}^{5\,x}-16\,x^4\,{\mathrm {e}}^{5\,x}+19\,x^5\,{\mathrm {e}}^{5\,x}-5\,x^6\,{\mathrm {e}}^{5\,x}-x^2+4\,x^3-4\,x^4+x^5+x^2\,{\mathrm {e}}^{5\,x}\,\ln \left (x\right )+4\,x^3\,{\mathrm {e}}^{5\,x}\,\ln \left (x\right )-5\,x^4\,{\mathrm {e}}^{5\,x}\,\ln \left (x\right )}{\left (x-2\,\ln \left (x\right )+\ln \left (\ln \left (x\right )-x\right )\,\left (x-\ln \left (x\right )\right )+1\right )\,\left (x+x\,\ln \left (x\right )-3\,x^2+x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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