Optimal. Leaf size=33 \[ 5-\log ^2\left (-x+\left (\frac {\log (x)}{x}+\log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )^2\right ) \]
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Rubi [F]
time = 26.66, 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 {\left (\left (16 x+e^x \left (-4 x-4 x^2\right )\right ) \log (x)+\left (-8 x^3+2 e^x x^3+\left (16-4 e^x\right ) \log (x)+\left (-16+4 e^x\right ) \log ^2(x)\right ) \log \left (-4 x+e^x x\right )+\left (16 x^2+e^x \left (-4 x^2-4 x^3\right )+\left (16 x-4 e^x x+\left (-16 x+4 e^x x\right ) \log (x)\right ) \log \left (-4 x+e^x x\right )\right ) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )\right ) \log \left (\frac {-x^3+\log ^2(x)+2 x \log (x) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )+x^2 \log ^2\left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )}{x^2}\right )}{\left (4 x^4-e^x x^4+\left (-4 x+e^x x\right ) \log ^2(x)\right ) \log \left (-4 x+e^x x\right )+\left (-8 x^2+2 e^x x^2\right ) \log (x) \log \left (-4 x+e^x x\right ) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )+\left (-4 x^3+e^x x^3\right ) \log \left (-4 x+e^x x\right ) \log ^2\left (\frac {1}{3} \log \left (-4 x+e^x x\right )\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 {\left (\left (16 x+e^x \left (-4 x-4 x^2\right )\right ) \log (x)+\left (-8 x^3+2 e^x x^3+\left (16-4 e^x\right ) \log (x)+\left (-16+4 e^x\right ) \log ^2(x)\right ) \log \left (-4 x+e^x x\right )+\left (16 x^2+e^x \left (-4 x^2-4 x^3\right )+\left (16 x-4 e^x x+\left (-16 x+4 e^x x\right ) \log (x)\right ) \log \left (-4 x+e^x x\right )\right ) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )\right ) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )\right )}{\left (4-e^x\right ) x \log \left (-4 x+e^x x\right ) \left (x^3-\log ^2(x)-2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-x^2 \log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )} \, dx\\ &=\int \left (\frac {16 x \left (-\log (x)-x \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right ) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}{\left (4-e^x\right ) \log \left (-4 x+e^x x\right ) \left (x^3-\log ^2(x)-2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-x^2 \log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}+\frac {2 \left (2 x \log (x)+2 x^2 \log (x)-x^3 \log \left (\left (-4+e^x\right ) x\right )-x \log (9) \log \left (\left (-4+e^x\right ) x\right )+2 \log (x) \log \left (\left (-4+e^x\right ) x\right )+2 x \log (3) \log (x) \log \left (\left (-4+e^x\right ) x\right )-2 \log ^2(x) \log \left (\left (-4+e^x\right ) x\right )+2 x^2 \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )+2 x^3 \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )+2 x \log \left (\left (-4+e^x\right ) x\right ) \log \left (\log \left (\left (-4+e^x\right ) x\right )\right )-2 x \log (x) \log \left (\left (-4+e^x\right ) x\right ) \log \left (\log \left (\left (-4+e^x\right ) x\right )\right )\right ) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}{x \log \left (-4 x+e^x x\right ) \left (x^3-\log ^2(x)-2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-x^2 \log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}\right ) \, dx\\ &=2 \int \frac {\left (2 x \log (x)+2 x^2 \log (x)-x^3 \log \left (\left (-4+e^x\right ) x\right )-x \log (9) \log \left (\left (-4+e^x\right ) x\right )+2 \log (x) \log \left (\left (-4+e^x\right ) x\right )+2 x \log (3) \log (x) \log \left (\left (-4+e^x\right ) x\right )-2 \log ^2(x) \log \left (\left (-4+e^x\right ) x\right )+2 x^2 \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )+2 x^3 \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )+2 x \log \left (\left (-4+e^x\right ) x\right ) \log \left (\log \left (\left (-4+e^x\right ) x\right )\right )-2 x \log (x) \log \left (\left (-4+e^x\right ) x\right ) \log \left (\log \left (\left (-4+e^x\right ) x\right )\right )\right ) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}{x \log \left (-4 x+e^x x\right ) \left (x^3-\log ^2(x)-2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-x^2 \log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )} \, dx+16 \int \frac {x \left (-\log (x)-x \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right ) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}{\left (4-e^x\right ) \log \left (-4 x+e^x x\right ) \left (x^3-\log ^2(x)-2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-x^2 \log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )} \, dx\\ &=2 \int \frac {\left (2 \log ^2(x) \log \left (\left (-4+e^x\right ) x\right )-x \left (2 x (1+x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-\log \left (\left (-4+e^x\right ) x\right ) \left (x^2+\log (9)-2 \log \left (\log \left (\left (-4+e^x\right ) x\right )\right )\right )\right )-\log (x) \left (2 x (1+x)+\log \left (\left (-4+e^x\right ) x\right ) \left (2+x \log (9)-2 x \log \left (\log \left (\left (-4+e^x\right ) x\right )\right )\right )\right )\right ) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}{x \log \left (-4 x+e^x x\right ) \left (\log ^2(x)+2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )+x^2 \left (-x+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )\right )} \, dx+16 \int \left (\frac {x \log (x) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}{\left (-4+e^x\right ) \log \left (-4 x+e^x x\right ) \left (x^3-\log ^2(x)-2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-x^2 \log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}+\frac {x^2 \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right ) \log \left (-x+\frac {\log ^2(x)}{x^2}+\frac {2 \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )}{x}+\log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}{\left (-4+e^x\right ) \log \left (-4 x+e^x x\right ) \left (x^3-\log ^2(x)-2 x \log (x) \log \left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )-x^2 \log ^2\left (\frac {1}{3} \log \left (\left (-4+e^x\right ) x\right )\right )\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (16 x+e^x \left (-4 x-4 x^2\right )\right ) \log (x)+\left (-8 x^3+2 e^x x^3+\left (16-4 e^x\right ) \log (x)+\left (-16+4 e^x\right ) \log ^2(x)\right ) \log \left (-4 x+e^x x\right )+\left (16 x^2+e^x \left (-4 x^2-4 x^3\right )+\left (16 x-4 e^x x+\left (-16 x+4 e^x x\right ) \log (x)\right ) \log \left (-4 x+e^x x\right )\right ) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )\right ) \log \left (\frac {-x^3+\log ^2(x)+2 x \log (x) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )+x^2 \log ^2\left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )}{x^2}\right )}{\left (4 x^4-e^x x^4+\left (-4 x+e^x x\right ) \log ^2(x)\right ) \log \left (-4 x+e^x x\right )+\left (-8 x^2+2 e^x x^2\right ) \log (x) \log \left (-4 x+e^x x\right ) \log \left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )+\left (-4 x^3+e^x x^3\right ) \log \left (-4 x+e^x x\right ) \log ^2\left (\frac {1}{3} \log \left (-4 x+e^x x\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (\left (4 \,{\mathrm e}^{x} x -16 x \right ) \ln \left (x \right )-4 \,{\mathrm e}^{x} x +16 x \right ) \ln \left ({\mathrm e}^{x} x -4 x \right )+\left (-4 x^{3}-4 x^{2}\right ) {\mathrm e}^{x}+16 x^{2}\right ) \ln \left (\frac {\ln \left ({\mathrm e}^{x} x -4 x \right )}{3}\right )+\left (\left (4 \,{\mathrm e}^{x}-16\right ) \ln \left (x \right )^{2}+\left (-4 \,{\mathrm e}^{x}+16\right ) \ln \left (x \right )+2 \,{\mathrm e}^{x} x^{3}-8 x^{3}\right ) \ln \left ({\mathrm e}^{x} x -4 x \right )+\left (\left (-4 x^{2}-4 x \right ) {\mathrm e}^{x}+16 x \right ) \ln \left (x \right )\right ) \ln \left (\frac {x^{2} \ln \left (\frac {\ln \left ({\mathrm e}^{x} x -4 x \right )}{3}\right )^{2}+2 x \ln \left (x \right ) \ln \left (\frac {\ln \left ({\mathrm e}^{x} x -4 x \right )}{3}\right )+\ln \left (x \right )^{2}-x^{3}}{x^{2}}\right )}{\left ({\mathrm e}^{x} x^{3}-4 x^{3}\right ) \ln \left ({\mathrm e}^{x} x -4 x \right ) \ln \left (\frac {\ln \left ({\mathrm e}^{x} x -4 x \right )}{3}\right )^{2}+\left (2 \,{\mathrm e}^{x} x^{2}-8 x^{2}\right ) \ln \left (x \right ) \ln \left ({\mathrm e}^{x} x -4 x \right ) \ln \left (\frac {\ln \left ({\mathrm e}^{x} x -4 x \right )}{3}\right )+\left (\left ({\mathrm e}^{x} x -4 x \right ) \ln \left (x \right )^{2}-{\mathrm e}^{x} x^{4}+4 x^{4}\right ) \ln \left ({\mathrm e}^{x} x -4 x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 191 vs.
\(2 (30) = 60\).
time = 0.76, size = 191, normalized size = 5.79 \begin {gather*} \log \left (\frac {x^{2} \log \left (3\right )^{2} + x^{2} \log \left (\log \left (x\right ) + \log \left (e^{x} - 4\right )\right )^{2} - x^{3} - 2 \, x \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, {\left (x^{2} \log \left (3\right ) - x \log \left (x\right )\right )} \log \left (\log \left (x\right ) + \log \left (e^{x} - 4\right )\right )}{x^{2}}\right )^{2} - 2 \, \log \left (\frac {x^{2} \log \left (3\right )^{2} + x^{2} \log \left (\log \left (x\right ) + \log \left (e^{x} - 4\right )\right )^{2} - x^{3} - 2 \, x \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, {\left (x^{2} \log \left (3\right ) - x \log \left (x\right )\right )} \log \left (\log \left (x\right ) + \log \left (e^{x} - 4\right )\right )}{x^{2}}\right ) \log \left (\frac {x^{2} \log \left (\frac {1}{3} \, \log \left (x e^{x} - 4 \, x\right )\right )^{2} - x^{3} + 2 \, x \log \left (x\right ) \log \left (\frac {1}{3} \, \log \left (x e^{x} - 4 \, x\right )\right ) + \log \left (x\right )^{2}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 54, normalized size = 1.64 \begin {gather*} -\log \left (\frac {x^{2} \log \left (\frac {1}{3} \, \log \left (x e^{x} - 4 \, x\right )\right )^{2} - x^{3} + 2 \, x \log \left (x\right ) \log \left (\frac {1}{3} \, \log \left (x e^{x} - 4 \, x\right )\right ) + \log \left (x\right )^{2}}{x^{2}}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.65, size = 54, normalized size = 1.64 \begin {gather*} -{\ln \left (\frac {-x^3+x^2\,{\ln \left (\frac {\ln \left (x\,{\mathrm {e}}^x-4\,x\right )}{3}\right )}^2+2\,x\,\ln \left (\frac {\ln \left (x\,{\mathrm {e}}^x-4\,x\right )}{3}\right )\,\ln \left (x\right )+{\ln \left (x\right )}^2}{x^2}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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