Integrand size = 98, antiderivative size = 27 \[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\frac {\left (5+e^{e^x}\right ) x \log \left (x^2\right )}{3+\frac {3 x^2}{1+x}} \]
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\[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {10}{3 \left (1+x+x^2\right )^2}+\frac {2 e^{e^x}}{3 \left (1+x+x^2\right )^2}+\frac {20 x}{3 \left (1+x+x^2\right )^2}+\frac {4 e^{e^x} x}{3 \left (1+x+x^2\right )^2}+\frac {20 x^2}{3 \left (1+x+x^2\right )^2}+\frac {4 e^{e^x} x^2}{3 \left (1+x+x^2\right )^2}+\frac {10 x^3}{3 \left (1+x+x^2\right )^2}+\frac {2 e^{e^x} x^3}{3 \left (1+x+x^2\right )^2}+\frac {e^{e^x} \log \left (x^2\right )}{3 \left (1+x+x^2\right )^2}+\frac {2 e^{e^x} x \log \left (x^2\right )}{3 \left (1+x+x^2\right )^2}+\frac {5 (1+2 x) \log \left (x^2\right )}{3 \left (1+x+x^2\right )^2}+\frac {e^{e^x+x} x (1+x) \log \left (x^2\right )}{3 \left (1+x+x^2\right )}\right ) \, dx \\ & = \frac {1}{3} \int \frac {e^{e^x} \log \left (x^2\right )}{\left (1+x+x^2\right )^2} \, dx+\frac {1}{3} \int \frac {e^{e^x+x} x (1+x) \log \left (x^2\right )}{1+x+x^2} \, dx+\frac {2}{3} \int \frac {e^{e^x}}{\left (1+x+x^2\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^x} x^3}{\left (1+x+x^2\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^x} x \log \left (x^2\right )}{\left (1+x+x^2\right )^2} \, dx+\frac {4}{3} \int \frac {e^{e^x} x}{\left (1+x+x^2\right )^2} \, dx+\frac {4}{3} \int \frac {e^{e^x} x^2}{\left (1+x+x^2\right )^2} \, dx+\frac {5}{3} \int \frac {(1+2 x) \log \left (x^2\right )}{\left (1+x+x^2\right )^2} \, dx+\frac {10}{3} \int \frac {1}{\left (1+x+x^2\right )^2} \, dx+\frac {10}{3} \int \frac {x^3}{\left (1+x+x^2\right )^2} \, dx+\frac {20}{3} \int \frac {x}{\left (1+x+x^2\right )^2} \, dx+\frac {20}{3} \int \frac {x^2}{\left (1+x+x^2\right )^2} \, dx \\ & = -\frac {20 (2+x)}{9 \left (1+x+x^2\right )}-\frac {20 x (2+x)}{9 \left (1+x+x^2\right )}-\frac {10 x^2 (2+x)}{9 \left (1+x+x^2\right )}+\frac {10 (1+2 x)}{9 \left (1+x+x^2\right )}+\frac {1}{3} e^{e^x} \log \left (x^2\right )-\frac {1}{3} \int \frac {8 i \left (\sqrt {3} \int \frac {e^{e^x}}{-1+i \sqrt {3}-2 x} \, dx+3 i \int -\frac {e^{e^x}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx+3 i \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\sqrt {3} \int \frac {e^{e^x}}{1+i \sqrt {3}+2 x} \, dx\right )}{9 x} \, dx-\frac {1}{3} \int \frac {2 \left (e^{e^x}-\frac {2 i \int \frac {e^{e^x+x}}{-1+i \sqrt {3}-2 x} \, dx}{\sqrt {3}}-\frac {2 i \int \frac {e^{e^x+x}}{1+i \sqrt {3}+2 x} \, dx}{\sqrt {3}}\right )}{x} \, dx+\frac {2}{3} \int \left (-\frac {4 e^{e^x}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i e^{e^x}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 e^{e^x}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i e^{e^x}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx+\frac {2}{3} \int \left (\frac {e^{e^x}}{\left (1+x+x^2\right )^2}+\frac {e^{e^x} (-1+x)}{1+x+x^2}\right ) \, dx-\frac {2}{3} \int \frac {4 \left (-i \sqrt {3} \int \frac {e^{e^x}}{-1+i \sqrt {3}-2 x} \, dx+3 \left (1-i \sqrt {3}\right ) \int -\frac {e^{e^x}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx+3 \left (1+i \sqrt {3}\right ) \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-i \sqrt {3} \int \frac {e^{e^x}}{1+i \sqrt {3}+2 x} \, dx\right )}{9 x} \, dx+\frac {10}{9} \int \frac {x (4+x)}{1+x+x^2} \, dx+\frac {4}{3} \int \left (-\frac {2 \left (-1+i \sqrt {3}\right ) e^{e^x}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}-\frac {2 i e^{e^x}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {2 \left (-1-i \sqrt {3}\right ) e^{e^x}}{3 \left (1+i \sqrt {3}+2 x\right )^2}-\frac {2 i e^{e^x}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx+\frac {4}{3} \int \left (\frac {e^{e^x} (-1-x)}{\left (1+x+x^2\right )^2}+\frac {e^{e^x}}{1+x+x^2}\right ) \, dx+\frac {5}{3} \int \left (\frac {\log \left (x^2\right )}{\left (1+x+x^2\right )^2}+\frac {2 x \log \left (x^2\right )}{\left (1+x+x^2\right )^2}\right ) \, dx+\frac {40}{9} \int \frac {1}{1+x+x^2} \, dx-\frac {1}{9} \left (4 \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx-\frac {1}{9} \left (4 \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {\left (2 i \log \left (x^2\right )\right ) \int \frac {e^{e^x+x}}{-1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}-\frac {\left (2 i \log \left (x^2\right )\right ) \int \frac {e^{e^x+x}}{1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}+\frac {1}{9} \left (4 \left (1-i \sqrt {3}\right ) \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{9} \left (4 \left (1+i \sqrt {3}\right ) \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx \\ & = \frac {10 x}{9}-\frac {20 (2+x)}{9 \left (1+x+x^2\right )}-\frac {20 x (2+x)}{9 \left (1+x+x^2\right )}-\frac {10 x^2 (2+x)}{9 \left (1+x+x^2\right )}+\frac {10 (1+2 x)}{9 \left (1+x+x^2\right )}+\frac {1}{3} e^{e^x} \log \left (x^2\right )-\frac {8}{27} i \int \frac {\sqrt {3} \int \frac {e^{e^x}}{-1+i \sqrt {3}-2 x} \, dx-3 i \int \frac {e^{e^x}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx+3 i \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\sqrt {3} \int \frac {e^{e^x}}{1+i \sqrt {3}+2 x} \, dx}{x} \, dx-\frac {8}{27} \int \frac {-i \sqrt {3} \int \frac {e^{e^x}}{-1+i \sqrt {3}-2 x} \, dx-3 \left (1-i \sqrt {3}\right ) \int \frac {e^{e^x}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx+3 \left (1+i \sqrt {3}\right ) \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-i \sqrt {3} \int \frac {e^{e^x}}{1+i \sqrt {3}+2 x} \, dx}{x} \, dx+\frac {2}{3} \int \frac {e^{e^x}}{\left (1+x+x^2\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^x} (-1+x)}{1+x+x^2} \, dx-\frac {2}{3} \int \frac {e^{e^x}-\frac {2 i \int \frac {e^{e^x+x}}{-1+i \sqrt {3}-2 x} \, dx}{\sqrt {3}}-\frac {2 i \int \frac {e^{e^x+x}}{1+i \sqrt {3}+2 x} \, dx}{\sqrt {3}}}{x} \, dx-\frac {8}{9} \int \frac {e^{e^x}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx-\frac {8}{9} \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {10}{9} \int \frac {-1+3 x}{1+x+x^2} \, dx+\frac {4}{3} \int \frac {e^{e^x} (-1-x)}{\left (1+x+x^2\right )^2} \, dx+\frac {4}{3} \int \frac {e^{e^x}}{1+x+x^2} \, dx+\frac {5}{3} \int \frac {\log \left (x^2\right )}{\left (1+x+x^2\right )^2} \, dx+\frac {10}{3} \int \frac {x \log \left (x^2\right )}{\left (1+x+x^2\right )^2} \, dx-\frac {80}{9} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+\frac {1}{9} \left (8 \left (1-i \sqrt {3}\right )\right ) \int \frac {e^{e^x}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{9} \left (8 \left (1+i \sqrt {3}\right )\right ) \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{9} \left (4 \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx-\frac {1}{9} \left (4 \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {\left (2 i \log \left (x^2\right )\right ) \int \frac {e^{e^x+x}}{-1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}-\frac {\left (2 i \log \left (x^2\right )\right ) \int \frac {e^{e^x+x}}{1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}+\frac {1}{9} \left (4 \left (1-i \sqrt {3}\right ) \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{9} \left (4 \left (1+i \sqrt {3}\right ) \log \left (x^2\right )\right ) \int \frac {e^{e^x}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\frac {1}{3} \left (10 \log (x)+\frac {\left (-5+e^{e^x} x (1+x)\right ) \log \left (x^2\right )}{1+x+x^2}\right ) \]
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Time = 2.95 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85
method | result | size |
parallelrisch | \(\frac {2 \ln \left (x^{2}\right ) {\mathrm e}^{{\mathrm e}^{x}} x^{2}+10 x^{2} \ln \left (x^{2}\right )+2 \ln \left (x^{2}\right ) {\mathrm e}^{{\mathrm e}^{x}} x +10 x \ln \left (x^{2}\right )}{6 x^{2}+6 x +6}\) | \(50\) |
risch | \(-\frac {10 \ln \left (x \right )}{3 \left (x^{2}+x +1\right )}+\frac {\frac {5 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}}{6}-\frac {5 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )}{3}+\frac {5 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{6}+\frac {10 x^{2} \ln \left (x \right )}{3}+\frac {10 x \ln \left (x \right )}{3}+\frac {10 \ln \left (x \right )}{3}}{x^{2}+x +1}+\frac {x \left (-i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x \ln \left (x \right )+4 \ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{x}}}{6 x^{2}+6 x +6}\) | \(215\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\frac {{\left (x^{2} + x\right )} e^{\left (e^{x}\right )} \log \left (x^{2}\right ) + 5 \, {\left (x^{2} + x\right )} \log \left (x^{2}\right )}{3 \, {\left (x^{2} + x + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\frac {\left (x^{2} \log {\left (x^{2} \right )} + x \log {\left (x^{2} \right )}\right ) e^{e^{x}}}{3 x^{2} + 3 x + 3} + \frac {10 \log {\left (x \right )}}{3} - \frac {5 \log {\left (x^{2} \right )}}{3 x^{2} + 3 x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (25) = 50\).
Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\frac {2 \, {\left ({\left (x^{2} + x\right )} e^{\left (e^{x}\right )} \log \left (x\right ) + 5 \, {\left (x^{2} + x\right )} \log \left (x\right )\right )}}{3 \, {\left (x^{2} + x + 1\right )}} + \frac {20 \, {\left (2 \, x + 1\right )}}{9 \, {\left (x^{2} + x + 1\right )}} - \frac {20 \, {\left (x + 2\right )}}{9 \, {\left (x^{2} + x + 1\right )}} - \frac {20 \, {\left (x - 1\right )}}{9 \, {\left (x^{2} + x + 1\right )}} \]
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\[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\int { \frac {10 \, x^{3} + 20 \, x^{2} + {\left (2 \, x^{3} + 4 \, x^{2} + {\left ({\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + x\right )} e^{x} + 2 \, x + 1\right )} \log \left (x^{2}\right ) + 4 \, x + 2\right )} e^{\left (e^{x}\right )} + 5 \, {\left (2 \, x + 1\right )} \log \left (x^{2}\right ) + 20 \, x + 10}{3 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {10+20 x+20 x^2+10 x^3+(5+10 x) \log \left (x^2\right )+e^{e^x} \left (2+4 x+4 x^2+2 x^3+\left (1+2 x+e^x \left (x+2 x^2+2 x^3+x^4\right )\right ) \log \left (x^2\right )\right )}{3+6 x+9 x^2+6 x^3+3 x^4} \, dx=\int \frac {20\,x+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (4\,x+\ln \left (x^2\right )\,\left (2\,x+{\mathrm {e}}^x\,\left (x^4+2\,x^3+2\,x^2+x\right )+1\right )+4\,x^2+2\,x^3+2\right )+20\,x^2+10\,x^3+\ln \left (x^2\right )\,\left (10\,x+5\right )+10}{3\,x^4+6\,x^3+9\,x^2+6\,x+3} \,d x \]
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