Optimal. Leaf size=31 \[ 1+\left (3-\frac {\log \left (\log \left (\left (4+2 \left (3+e^x-x\right )\right ) x+\log (x)\right )\right )}{\log (x)}\right )^2 \]
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Rubi [A] time = 6.86, antiderivative size = 46, normalized size of antiderivative = 1.48, number of steps used = 4, number of rules used = 3, integrand size = 233, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6688, 12, 6712} \begin {gather*} \frac {\log ^2\left (\log \left (2 \left (-x+e^x+5\right ) x+\log (x)\right )\right )}{\log ^2(x)}-\frac {6 \log \left (\log \left (2 \left (-x+e^x+5\right ) x+\log (x)\right )\right )}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6688
Rule 6712
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (3 \log (x)-\log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right ) \left (2 x \left (-5-e^x+x\right ) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )+\log (x) \left (1+2 \left (5+e^x\right ) x+2 \left (-2+e^x\right ) x^2-\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right )\right )}{x \left (2 x \left (-5-e^x+x\right )-\log (x)\right ) \log ^3(x) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )} \, dx\\ &=2 \int \frac {\left (3 \log (x)-\log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right ) \left (2 x \left (-5-e^x+x\right ) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )+\log (x) \left (1+2 \left (5+e^x\right ) x+2 \left (-2+e^x\right ) x^2-\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right ) \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )\right )\right )}{x \left (2 x \left (-5-e^x+x\right )-\log (x)\right ) \log ^3(x) \log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int (3-x) \, dx,x,\frac {\log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )}{\log (x)}\right )\right )\\ &=-\frac {6 \log \left (\log \left (2 \left (5+e^x-x\right ) x+\log (x)\right )\right )}{\log (x)}+\frac {\log ^2\left (\log \left (2 \left (5+e^x-x\right ) x+\log (x)\right )\right )}{\log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.20, size = 51, normalized size = 1.65 \begin {gather*} 2 \left (-\frac {3 \log \left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )}{\log (x)}+\frac {\log ^2\left (\log \left (-2 x \left (-5-e^x+x\right )+\log (x)\right )\right )}{2 \log ^2(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 51, normalized size = 1.65 \begin {gather*} -\frac {6 \, \log \relax (x) \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \relax (x)\right )\right ) - \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \relax (x)\right )\right )^{2}}{\log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 1.65
method | result | size |
risch | \(\frac {\ln \left (\ln \left (\ln \relax (x )+2 \,{\mathrm e}^{x} x -2 x^{2}+10 x \right )\right )^{2}}{\ln \relax (x )^{2}}-\frac {6 \ln \left (\ln \left (\ln \relax (x )+2 \,{\mathrm e}^{x} x -2 x^{2}+10 x \right )\right )}{\ln \relax (x )}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 51, normalized size = 1.65 \begin {gather*} -\frac {6 \, \log \relax (x) \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \relax (x)\right )\right ) - \log \left (\log \left (-2 \, x^{2} + 2 \, x e^{x} + 10 \, x + \log \relax (x)\right )\right )^{2}}{\log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 46, normalized size = 1.48 \begin {gather*} \frac {\ln \left (\ln \left (10\,x+\ln \relax (x)+2\,x\,{\mathrm {e}}^x-2\,x^2\right )\right )\,\left (\ln \left (\ln \left (10\,x+\ln \relax (x)+2\,x\,{\mathrm {e}}^x-2\,x^2\right )\right )-6\,\ln \relax (x)\right )}{{\ln \relax (x)}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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