Optimal. Leaf size=27 \[ \log \left (-1+\frac {x^2}{-5+\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )}\right ) \]
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Rubi [A] time = 4.69, antiderivative size = 48, normalized size of antiderivative = 1.78, number of steps used = 5, number of rules used = 3, integrand size = 181, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6688, 6742, 6684} \begin {gather*} \log \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right )-\log \left (5-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6684
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (11-e^{\sqrt [4]{e}} (11+12 x)+2 \left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )}{\left (1-e^{\sqrt [4]{e}}-e^{\sqrt [4]{e}} x\right ) \left (5-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right ) \left (5+x^2-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )} \, dx\\ &=\int \left (\frac {-1+e^{\sqrt [4]{e}}+2 e^{\sqrt [4]{e}} x+2 \left (1-e^{\sqrt [4]{e}}\right ) x^2-2 e^{\sqrt [4]{e}} x^3}{x \left (1-e^{\sqrt [4]{e}}-e^{\sqrt [4]{e}} x\right ) \left (5+x^2-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )}-\frac {-1+e^{\sqrt [4]{e}}+2 e^{\sqrt [4]{e}} x}{x \left (-1+e^{\sqrt [4]{e}}+e^{\sqrt [4]{e}} x\right ) \left (-5+\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )}\right ) \, dx\\ &=\int \frac {-1+e^{\sqrt [4]{e}}+2 e^{\sqrt [4]{e}} x+2 \left (1-e^{\sqrt [4]{e}}\right ) x^2-2 e^{\sqrt [4]{e}} x^3}{x \left (1-e^{\sqrt [4]{e}}-e^{\sqrt [4]{e}} x\right ) \left (5+x^2-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )} \, dx-\int \frac {-1+e^{\sqrt [4]{e}}+2 e^{\sqrt [4]{e}} x}{x \left (-1+e^{\sqrt [4]{e}}+e^{\sqrt [4]{e}} x\right ) \left (-5+\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )} \, dx\\ &=-\log \left (5-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )+\log \left (5+x^2-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 48, normalized size = 1.78 \begin {gather*} -\log \left (5-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )+\log \left (5+x^2-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 40, normalized size = 1.48 \begin {gather*} \log \left (-x^{2} + \log \left (-{\left (x^{2} + x\right )} e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) - \log \left (\log \left (-{\left (x^{2} + x\right )} e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 48, normalized size = 1.78 \begin {gather*} \log \left (-x^{2} + \log \left (-x^{2} e^{\left (e^{\frac {1}{4}}\right )} - x e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) - \log \left (\log \left (-x^{2} e^{\left (e^{\frac {1}{4}}\right )} - x e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 47, normalized size = 1.74
method | result | size |
norman | \(-\ln \left (\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )+\ln \left (x^{2}-\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )+5\right )\) | \(47\) |
risch | \(\ln \left (-x^{2}+\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )-\ln \left (\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 46, normalized size = 1.70 \begin {gather*} \log \left (-x^{2} + \log \left (-x e^{\left (e^{\frac {1}{4}}\right )} - e^{\left (e^{\frac {1}{4}}\right )} + 1\right ) + \log \relax (x) - 5\right ) - \log \left (\log \left (-x e^{\left (e^{\frac {1}{4}}\right )} - e^{\left (e^{\frac {1}{4}}\right )} + 1\right ) + \log \relax (x) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.49, size = 250, normalized size = 9.26 \begin {gather*} \ln \left (\frac {{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}}{2}-x-x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}+x^2+x^3-\frac {1}{2}\right )+\ln \left (\left (x^2-\ln \left (-x\,\left ({\mathrm {e}}^{{\mathrm {e}}^{1/4}}+x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-1\right )\right )+5\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{1/4}}+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-1\right )\right )-\ln \left (20\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-2\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-10\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}+20\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}+2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )+4\,x^2\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-20\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-20\,x^2+4\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-4\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-4\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )+10\right )-\ln \left (x-\frac {{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}}{2}+\frac {1}{2}\right ) \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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