Integrand size = 181, antiderivative size = 27 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-1+\frac {x^2}{-5+\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )}\right ) \]
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Time = 2.94 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6820, 6874, 6816} \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right )-\log \left (5-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )\right ) \]
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Rule 6816
Rule 6820
Rule 6874
Rubi steps \begin{align*} \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{align*}
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=-\log \left (5-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )+\log \left (5+x^2-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right ) \]
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Time = 5.57 (sec) , antiderivative size = 47, normalized size of antiderivative = 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 | \(\ln \left (-x^{2}+\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )-\ln \left (\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )\) | \(47\) |
parallelrisch | \(\left (-\ln \left (\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right ) {\mathrm e}^{2 \,{\mathrm e}^{\frac {1}{4}}}+\ln \left (x^{2}-\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )+5\right ) {\mathrm e}^{2 \,{\mathrm e}^{\frac {1}{4}}}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{\frac {1}{4}}}\) | \(64\) |
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Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-x^{2} + \log \left (-{\left (x^{2} + x\right )} e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) - \log \left (\log \left (-{\left (x^{2} + x\right )} e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) \]
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Exception generated. \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\text {Exception raised: PolynomialError} \]
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-x^{2} + \log \left (-x e^{\left (e^{\frac {1}{4}}\right )} - e^{\left (e^{\frac {1}{4}}\right )} + 1\right ) + \log \left (x\right ) - 5\right ) - \log \left (\log \left (-x e^{\left (e^{\frac {1}{4}}\right )} - e^{\left (e^{\frac {1}{4}}\right )} + 1\right ) + \log \left (x\right ) - 5\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.43 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-x^{2} + \log \left (-x^{2} e^{\left (e^{\frac {1}{4}}\right )} - x e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) - \log \left (\log \left (-x^{2} e^{\left (e^{\frac {1}{4}}\right )} - x e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) \]
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Time = 15.10 (sec) , antiderivative size = 250, normalized size of antiderivative = 9.26 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\ln \left (\frac {{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}}{2}-x-x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}+x^2+x^3-\frac {1}{2}\right )+\ln \left (\left (x^2-\ln \left (-x\,\left ({\mathrm {e}}^{{\mathrm {e}}^{1/4}}+x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-1\right )\right )+5\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{1/4}}+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-1\right )\right )-\ln \left (20\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-2\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-10\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}+20\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}+2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )+4\,x^2\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-20\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-20\,x^2+4\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-4\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-4\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )+10\right )-\ln \left (x-\frac {{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}}{2}+\frac {1}{2}\right ) \]
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