Optimal. Leaf size=35 \[ \log \left (\frac {x}{x-\frac {3}{\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}}\right ) \]
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Rubi [F] time = 19.48, 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 {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (e^{e^x}+e^{3+2 x}+e^{e^x+x} x+2 e^{3+2 x} x-3 e^3 x^2+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log (x)+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}{x \left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\\ &=3 \int \frac {e^{e^x}+e^{3+2 x}+e^{e^x+x} x+2 e^{3+2 x} x-3 e^3 x^2+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log (x)+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{x \left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\\ &=3 \int \left (-\frac {-2 e^{e^x}+e^{e^x+x}-2 e^3 x+2 e^3 x^2}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {-1-2 x-\log (x)-\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx\\ &=-\left (3 \int \frac {-2 e^{e^x}+e^{e^x+x}-2 e^3 x+2 e^3 x^2}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\right )+3 \int \frac {-1-2 x-\log (x)-\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\\ &=3 \int \left (\frac {1+2 x}{3 x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {-3-x-2 x^2}{3 x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx-3 \int \left (-\frac {2 e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {2 e^3 x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}-\frac {2 e^3 x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx\\ &=-\left (3 \int \frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\right )+6 \int \frac {e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-\left (6 e^3\right ) \int \frac {x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\left (6 e^3\right ) \int \frac {x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {1+2 x}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {-3-x-2 x^2}{x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\\ &=-\left (3 \int \frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\right )+6 \int \frac {e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-\left (6 e^3\right ) \int \frac {x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\left (6 e^3\right ) \int \frac {x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \left (\frac {2}{\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}+\frac {1}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx+\int \left (\frac {1}{3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}-\frac {3}{x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}-\frac {2 x}{-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}\right ) \, dx\\ &=2 \int \frac {1}{\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx-2 \int \frac {x}{-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx-3 \int \frac {1}{x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-3 \int \frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+6 \int \frac {e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-\left (6 e^3\right ) \int \frac {x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\left (6 e^3\right ) \int \frac {x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {1}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {1}{3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.27, size = 74, normalized size = 2.11 \begin {gather*} -3 \left (-\frac {\log (x)}{3}-\frac {1}{3} \log \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )+\frac {1}{3} \log \left (3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 56, normalized size = 1.60 \begin {gather*} -\log \left (\frac {x \log \left (x^{2} - e^{\left (2 \, x\right )} - e^{\left (e^{x} - 3\right )}\right ) + x \log \relax (x) - 3}{x}\right ) + \log \left (\log \left (x^{2} - e^{\left (2 \, x\right )} - e^{\left (e^{x} - 3\right )}\right ) + \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.46, size = 82, normalized size = 2.34 \begin {gather*} -\log \left (x \log \left ({\left (x^{2} e^{\left (x + 3\right )} - e^{\left (3 \, x + 3\right )} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right ) + x \log \relax (x) - 3 \, x - 3\right ) + \log \relax (x) + \log \left (\log \left ({\left (x^{2} e^{\left (x + 3\right )} - e^{\left (3 \, x + 3\right )} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right ) + \log \relax (x) - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 56, normalized size = 1.60
method | result | size |
risch | \(\ln \left (\ln \left (-{\mathrm e}^{{\mathrm e}^{x}-3}-{\mathrm e}^{2 x}+x^{2}\right )+\ln \relax (x )\right )-\ln \left (\ln \left (-{\mathrm e}^{{\mathrm e}^{x}-3}-{\mathrm e}^{2 x}+x^{2}\right )+\frac {x \ln \relax (x )-3}{x}\right )\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.75, size = 66, normalized size = 1.89 \begin {gather*} -\log \left (\frac {x \log \left (x^{2} e^{3} - e^{\left (2 \, x + 3\right )} - e^{\left (e^{x}\right )}\right ) + x \log \relax (x) - 3 \, x - 3}{x}\right ) + \log \left (\log \left (x^{2} e^{3} - e^{\left (2 \, x + 3\right )} - e^{\left (e^{x}\right )}\right ) + \log \relax (x) - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{{\mathrm {e}}^x-3}\,\left (3\,\ln \relax (x)+3\,x\,{\mathrm {e}}^x+3\right )+\ln \relax (x)\,\left (3\,{\mathrm {e}}^{2\,x}-3\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (6\,x+3\right )+\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^x-3}-{\mathrm {e}}^{2\,x}\right )\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{{\mathrm {e}}^x-3}-3\,x^2\right )-9\,x^2}{{\mathrm {e}}^{{\mathrm {e}}^x-3}\,\left (x^2\,{\ln \relax (x)}^2-3\,x\,\ln \relax (x)\right )+{\ln \relax (x)}^2\,\left (x^2\,{\mathrm {e}}^{2\,x}-x^4\right )+{\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^x-3}-{\mathrm {e}}^{2\,x}\right )}^2\,\left (x^2\,{\mathrm {e}}^{2\,x}-x^4+x^2\,{\mathrm {e}}^{{\mathrm {e}}^x-3}\right )-\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^x-3}-{\mathrm {e}}^{2\,x}\right )\,\left (3\,x\,{\mathrm {e}}^{2\,x}-\ln \relax (x)\,\left (2\,x^2\,{\mathrm {e}}^{2\,x}-2\,x^4\right )+{\mathrm {e}}^{{\mathrm {e}}^x-3}\,\left (3\,x-2\,x^2\,\ln \relax (x)\right )-3\,x^3\right )-\ln \relax (x)\,\left (3\,x\,{\mathrm {e}}^{2\,x}-3\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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