Optimal. Leaf size=27 \[ x+\log (2)-(1-x+\log (x)) \log \left (\frac {\log (3)}{-1-2 e^3+x}\right ) \]
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Rubi [B] time = 0.40, antiderivative size = 91, normalized size of antiderivative = 3.37, number of steps used = 12, number of rules used = 10, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6, 1593, 6742, 2316, 2315, 43, 2416, 2389, 2295, 2394} \begin {gather*} x-\left (2 e^3-\log \left (1+2 e^3\right )\right ) \log \left (-x+2 e^3+1\right )-\left (-x+2 e^3+1\right ) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right )-\log \left (\frac {x}{1+2 e^3}\right ) \log \left (-\frac {\log (3)}{-x+2 e^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 43
Rule 1593
Rule 2295
Rule 2315
Rule 2316
Rule 2389
Rule 2394
Rule 2416
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{\left (1+2 e^3\right ) x-x^2} \, dx\\ &=\int \frac {2 e^3 x-x \log (x)+\left (-1+2 x-x^2+e^3 (-2+2 x)\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{\left (1+2 e^3-x\right ) x} \, dx\\ &=\int \left (\frac {2 e^3-\log (x)}{1+2 e^3-x}+\frac {(-1+x) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x}\right ) \, dx\\ &=\int \frac {2 e^3-\log (x)}{1+2 e^3-x} \, dx+\int \frac {(-1+x) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x} \, dx\\ &=-\left (\left (2 e^3-\log \left (1+2 e^3\right )\right ) \log \left (1+2 e^3-x\right )\right )-\int \frac {\log \left (\frac {x}{1+2 e^3}\right )}{1+2 e^3-x} \, dx+\int \left (\log \left (-\frac {\log (3)}{1+2 e^3-x}\right )-\frac {\log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x}\right ) \, dx\\ &=-\left (\left (2 e^3-\log \left (1+2 e^3\right )\right ) \log \left (1+2 e^3-x\right )\right )-\text {Li}_2\left (1-\frac {x}{1+2 e^3}\right )+\int \log \left (-\frac {\log (3)}{1+2 e^3-x}\right ) \, dx-\int \frac {\log \left (-\frac {\log (3)}{1+2 e^3-x}\right )}{x} \, dx\\ &=-\left (\left (2 e^3-\log \left (1+2 e^3\right )\right ) \log \left (1+2 e^3-x\right )\right )-\log \left (\frac {x}{1+2 e^3}\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )-\text {Li}_2\left (1-\frac {x}{1+2 e^3}\right )+\int \frac {\log \left (-\frac {x}{-1-2 e^3}\right )}{1+2 e^3-x} \, dx-\operatorname {Subst}\left (\int \log \left (-\frac {\log (3)}{x}\right ) \, dx,x,1+2 e^3-x\right )\\ &=x-\left (2 e^3-\log \left (1+2 e^3\right )\right ) \log \left (1+2 e^3-x\right )-\left (1+2 e^3-x\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )-\log \left (\frac {x}{1+2 e^3}\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.08, size = 69, normalized size = 2.56 \begin {gather*} x+\left (-2 e^3+\log \left (1+2 e^3\right )\right ) \log \left (1+2 e^3-x\right )-\left (1+2 e^3-x+\log \left (\frac {x}{1+2 e^3}\right )\right ) \log \left (-\frac {\log (3)}{1+2 e^3-x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 36, normalized size = 1.33 \begin {gather*} {\left (x - 1\right )} \log \left (\frac {\log \relax (3)}{x - 2 \, e^{3} - 1}\right ) - \log \relax (x) \log \left (\frac {\log \relax (3)}{x - 2 \, e^{3} - 1}\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 279, normalized size = 10.33 \begin {gather*} -\frac {2 \, \pi ^{2} e^{3} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) \mathrm {sgn}\relax (x) - 8 \, \pi ^{2} e^{6} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) - 2 \, \pi ^{2} e^{3} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) + 6 \, \pi ^{2} e^{3} \mathrm {sgn}\relax (x) + \pi ^{2} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) \mathrm {sgn}\relax (x) + 8 \, \pi ^{2} e^{6} - 6 \, \pi ^{2} e^{3} + 8 \, x e^{3} \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) - 16 \, e^{6} \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right )^{2} - 8 \, e^{3} \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) \log \left ({\left | x \right |}\right ) - 8 \, x e^{3} \log \left (\log \relax (3)\right ) + 8 \, e^{3} \log \left ({\left | x \right |}\right ) \log \left (\log \relax (3)\right ) - \pi ^{2} \mathrm {sgn}\left (x - 2 \, e^{3} - 1\right ) + 3 \, \pi ^{2} \mathrm {sgn}\relax (x) - 3 \, \pi ^{2} - 8 \, x e^{3} + 4 \, x \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) - 4 \, \log \left ({\left | x - 2 \, e^{3} - 1 \right |}\right ) \log \left ({\left | x \right |}\right ) - 4 \, x \log \left (\log \relax (3)\right ) + 4 \, \log \left ({\left | x \right |}\right ) \log \left (\log \relax (3)\right ) - 4 \, x}{4 \, {\left (2 \, e^{3} + 1\right )}} - \frac {4 \, e^{6} \log \left (x - 2 \, e^{3} - 1\right )^{2} - 2 \, e^{3} \log \left (x - 2 \, e^{3} - 1\right ) - \log \left (x - 2 \, e^{3} - 1\right )}{2 \, e^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.55, size = 59, normalized size = 2.19
method | result | size |
norman | \(x -\ln \left (-\frac {\ln \relax (3)}{2 \,{\mathrm e}^{3}-x +1}\right )+\ln \left (-\frac {\ln \relax (3)}{2 \,{\mathrm e}^{3}-x +1}\right ) x -\ln \relax (x ) \ln \left (-\frac {\ln \relax (3)}{2 \,{\mathrm e}^{3}-x +1}\right )\) | \(59\) |
default | \(\left (x -1-2 \,{\mathrm e}^{3}\right ) \ln \left (\frac {1}{x -1-2 \,{\mathrm e}^{3}}\right )+x -1-2 \,{\mathrm e}^{3}-\frac {2 \,{\mathrm e}^{3} \dilog \left (1+\frac {2 \,{\mathrm e}^{3}+1}{x -1-2 \,{\mathrm e}^{3}}\right )}{2 \,{\mathrm e}^{3}+1}-\frac {2 \,{\mathrm e}^{3} \ln \left (\frac {1}{x -1-2 \,{\mathrm e}^{3}}\right ) \ln \left (1+\frac {2 \,{\mathrm e}^{3}+1}{x -1-2 \,{\mathrm e}^{3}}\right )}{2 \,{\mathrm e}^{3}+1}-\frac {\dilog \left (1+\frac {2 \,{\mathrm e}^{3}+1}{x -1-2 \,{\mathrm e}^{3}}\right )}{2 \,{\mathrm e}^{3}+1}-\frac {\ln \left (\frac {1}{x -1-2 \,{\mathrm e}^{3}}\right ) \ln \left (1+\frac {2 \,{\mathrm e}^{3}+1}{x -1-2 \,{\mathrm e}^{3}}\right )}{2 \,{\mathrm e}^{3}+1}+\frac {\ln \left (\frac {1}{x -1-2 \,{\mathrm e}^{3}}\right )^{2}}{2}-\ln \left (\ln \relax (3)\right ) \ln \relax (x )+\ln \left (\ln \relax (3)\right ) x -2 \,{\mathrm e}^{3} \ln \left (2 \,{\mathrm e}^{3}-x +1\right )+\left (\ln \relax (x )-\ln \left (\frac {x}{2 \,{\mathrm e}^{3}+1}\right )\right ) \ln \left (\frac {2 \,{\mathrm e}^{3}-x +1}{2 \,{\mathrm e}^{3}+1}\right )-\dilog \left (\frac {x}{2 \,{\mathrm e}^{3}+1}\right )\) | \(255\) |
risch | \(\left (\ln \relax (x )-x \right ) \ln \left ({\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}\right )-i \pi x \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+i \pi \ln \left (\left (-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \relax (3)\right )+2 i \ln \relax (2)-2 i\right ) x \right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+i \pi x -i \pi \ln \left (\left (-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \relax (3)\right )+2 i \ln \relax (2)-2 i\right ) x \right )-i \pi \ln \left (\left (-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \relax (3)\right )+2 i \ln \relax (2)-2 i\right ) x \right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+\ln \left (\ln \relax (3)\right ) x -x \ln \relax (2)+x +i \pi x \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}-\ln \left (\ln \relax (3)\right ) \ln \left (\left (-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \relax (3)\right )+2 i \ln \relax (2)-2 i\right ) x \right )+\ln \relax (2) \ln \left (\left (-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3}-\frac {x}{2}+\frac {1}{2}}\right )^{3}+2 \pi -2 i \ln \left (\ln \relax (3)\right )+2 i \ln \relax (2)-2 i\right ) x \right )+\ln \left (2 \,{\mathrm e}^{3}-x +1\right )\) | \(413\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 48, normalized size = 1.78 \begin {gather*} x {\left (\log \left (\log \relax (3)\right ) + 1\right )} - {\left (x - 2 \, e^{3} - \log \relax (x) - 1\right )} \log \left (x - 2 \, e^{3} - 1\right ) - 2 \, e^{3} \log \left (x - 2 \, e^{3} - 1\right ) - \log \relax (x) \log \left (\log \relax (3)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.66, size = 33, normalized size = 1.22 \begin {gather*} x+\ln \left (x-2\,{\mathrm {e}}^3-1\right )+\ln \left (-\frac {\ln \relax (3)}{2\,{\mathrm {e}}^3-x+1}\right )\,\left (x-\ln \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 31, normalized size = 1.15 \begin {gather*} x + \left (x - \log {\relax (x )}\right ) \log {\left (- \frac {\log {\relax (3 )}}{- x + 1 + 2 e^{3}} \right )} + \log {\left (x - 2 e^{3} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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