Integrand size = 167, antiderivative size = 26 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {e^4 \left (x^2+\log (3 x) \log (1+x)\right )}{e^{x^2}+x} \]
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\[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^4 \left (-\left ((1+x) \left (x^2 \left (-x+2 e^{x^2} \left (-1+x^2\right )\right )-\left (e^{x^2}+x\right ) \log (1+x)\right )\right )-x \log (3 x) \left (-e^{x^2}-x+(1+x) \left (1+2 e^{x^2} x\right ) \log (1+x)\right )\right )}{x (1+x) \left (e^{x^2}+x\right )^2} \, dx \\ & = e^4 \int \frac {-\left ((1+x) \left (x^2 \left (-x+2 e^{x^2} \left (-1+x^2\right )\right )-\left (e^{x^2}+x\right ) \log (1+x)\right )\right )-x \log (3 x) \left (-e^{x^2}-x+(1+x) \left (1+2 e^{x^2} x\right ) \log (1+x)\right )}{x (1+x) \left (e^{x^2}+x\right )^2} \, dx \\ & = e^4 \int \left (\frac {\left (-1+2 x^2\right ) \left (x^2+\log (3 x) \log (1+x)\right )}{\left (e^{x^2}+x\right )^2}-\frac {-2 x^2-2 x^3+2 x^4+2 x^5-x \log (3 x)-\log (1+x)-x \log (1+x)+2 x^2 \log (3 x) \log (1+x)+2 x^3 \log (3 x) \log (1+x)}{x (1+x) \left (e^{x^2}+x\right )}\right ) \, dx \\ & = e^4 \int \frac {\left (-1+2 x^2\right ) \left (x^2+\log (3 x) \log (1+x)\right )}{\left (e^{x^2}+x\right )^2} \, dx-e^4 \int \frac {-2 x^2-2 x^3+2 x^4+2 x^5-x \log (3 x)-\log (1+x)-x \log (1+x)+2 x^2 \log (3 x) \log (1+x)+2 x^3 \log (3 x) \log (1+x)}{x (1+x) \left (e^{x^2}+x\right )} \, dx \\ & = -\left (e^4 \int \frac {(1+x) \left (2 x^2 \left (-1+x^2\right )-\log (1+x)\right )+x \log (3 x) (-1+2 x (1+x) \log (1+x))}{x (1+x) \left (e^{x^2}+x\right )} \, dx\right )+e^4 \int \left (-\frac {x^2+\log (3 x) \log (1+x)}{\left (e^{x^2}+x\right )^2}+\frac {2 x^2 \left (x^2+\log (3 x) \log (1+x)\right )}{\left (e^{x^2}+x\right )^2}\right ) \, dx \\ & = -\left (e^4 \int \frac {x^2+\log (3 x) \log (1+x)}{\left (e^{x^2}+x\right )^2} \, dx\right )-e^4 \int \left (\frac {-2 x^2-2 x^3+2 x^4+2 x^5-x \log (3 x)-\log (1+x)-x \log (1+x)+2 x^2 \log (3 x) \log (1+x)+2 x^3 \log (3 x) \log (1+x)}{x \left (e^{x^2}+x\right )}-\frac {-2 x^2-2 x^3+2 x^4+2 x^5-x \log (3 x)-\log (1+x)-x \log (1+x)+2 x^2 \log (3 x) \log (1+x)+2 x^3 \log (3 x) \log (1+x)}{(1+x) \left (e^{x^2}+x\right )}\right ) \, dx+\left (2 e^4\right ) \int \frac {x^2 \left (x^2+\log (3 x) \log (1+x)\right )}{\left (e^{x^2}+x\right )^2} \, dx \\ & = -\left (e^4 \int \frac {-2 x^2-2 x^3+2 x^4+2 x^5-x \log (3 x)-\log (1+x)-x \log (1+x)+2 x^2 \log (3 x) \log (1+x)+2 x^3 \log (3 x) \log (1+x)}{x \left (e^{x^2}+x\right )} \, dx\right )+e^4 \int \frac {-2 x^2-2 x^3+2 x^4+2 x^5-x \log (3 x)-\log (1+x)-x \log (1+x)+2 x^2 \log (3 x) \log (1+x)+2 x^3 \log (3 x) \log (1+x)}{(1+x) \left (e^{x^2}+x\right )} \, dx-e^4 \int \left (\frac {x^2}{\left (e^{x^2}+x\right )^2}+\frac {\log (3 x) \log (1+x)}{\left (e^{x^2}+x\right )^2}\right ) \, dx+\left (2 e^4\right ) \int \left (\frac {x^4}{\left (e^{x^2}+x\right )^2}+\frac {x^2 \log (3 x) \log (1+x)}{\left (e^{x^2}+x\right )^2}\right ) \, dx \\ & = -\left (e^4 \int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx\right )-e^4 \int \frac {\log (3 x) \log (1+x)}{\left (e^{x^2}+x\right )^2} \, dx-e^4 \int \frac {(1+x) \left (2 x^2 \left (-1+x^2\right )-\log (1+x)\right )+x \log (3 x) (-1+2 x (1+x) \log (1+x))}{x \left (e^{x^2}+x\right )} \, dx+e^4 \int \frac {(1+x) \left (2 x^2 \left (-1+x^2\right )-\log (1+x)\right )+x \log (3 x) (-1+2 x (1+x) \log (1+x))}{(1+x) \left (e^{x^2}+x\right )} \, dx+\left (2 e^4\right ) \int \frac {x^4}{\left (e^{x^2}+x\right )^2} \, dx+\left (2 e^4\right ) \int \frac {x^2 \log (3 x) \log (1+x)}{\left (e^{x^2}+x\right )^2} \, dx \\ & = -\left (e^4 \int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx\right )-e^4 \int \left (-\frac {2 x}{e^{x^2}+x}-\frac {2 x^2}{e^{x^2}+x}+\frac {2 x^3}{e^{x^2}+x}+\frac {2 x^4}{e^{x^2}+x}-\frac {\log (3 x)}{e^{x^2}+x}-\frac {\log (1+x)}{e^{x^2}+x}-\frac {\log (1+x)}{x \left (e^{x^2}+x\right )}+\frac {2 x \log (3 x) \log (1+x)}{e^{x^2}+x}+\frac {2 x^2 \log (3 x) \log (1+x)}{e^{x^2}+x}\right ) \, dx+e^4 \int \left (-\frac {2 x^2}{(1+x) \left (e^{x^2}+x\right )}-\frac {2 x^3}{(1+x) \left (e^{x^2}+x\right )}+\frac {2 x^4}{(1+x) \left (e^{x^2}+x\right )}+\frac {2 x^5}{(1+x) \left (e^{x^2}+x\right )}-\frac {x \log (3 x)}{(1+x) \left (e^{x^2}+x\right )}-\frac {\log (1+x)}{(1+x) \left (e^{x^2}+x\right )}-\frac {x \log (1+x)}{(1+x) \left (e^{x^2}+x\right )}+\frac {2 x^2 \log (3 x) \log (1+x)}{(1+x) \left (e^{x^2}+x\right )}+\frac {2 x^3 \log (3 x) \log (1+x)}{(1+x) \left (e^{x^2}+x\right )}\right ) \, dx+e^4 \int \frac {\log (3 x) \int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx}{1+x} \, dx+e^4 \int \frac {\log (1+x) \int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx}{x} \, dx+\left (2 e^4\right ) \int \frac {x^4}{\left (e^{x^2}+x\right )^2} \, dx-\left (2 e^4\right ) \int \frac {\log (3 x) \int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx}{1+x} \, dx-\left (2 e^4\right ) \int \frac {\log (1+x) \int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx}{x} \, dx-\left (e^4 \log (3 x) \log (1+x)\right ) \int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx+\left (2 e^4 \log (3 x) \log (1+x)\right ) \int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx \\ & = -\left (e^4 \int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx\right )+e^4 \int \frac {\log (3 x)}{e^{x^2}+x} \, dx-e^4 \int \frac {x \log (3 x)}{(1+x) \left (e^{x^2}+x\right )} \, dx+e^4 \int \frac {\log (1+x)}{e^{x^2}+x} \, dx+e^4 \int \frac {\log (1+x)}{x \left (e^{x^2}+x\right )} \, dx-e^4 \int \frac {\log (1+x)}{(1+x) \left (e^{x^2}+x\right )} \, dx-e^4 \int \frac {x \log (1+x)}{(1+x) \left (e^{x^2}+x\right )} \, dx-e^4 \int \frac {\int \frac {\int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx}{x} \, dx}{1+x} \, dx-e^4 \int \frac {\int \frac {\int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx}{1+x} \, dx}{x} \, dx+\left (2 e^4\right ) \int \frac {x^4}{\left (e^{x^2}+x\right )^2} \, dx+\left (2 e^4\right ) \int \frac {x}{e^{x^2}+x} \, dx+\left (2 e^4\right ) \int \frac {x^2}{e^{x^2}+x} \, dx-\left (2 e^4\right ) \int \frac {x^3}{e^{x^2}+x} \, dx-\left (2 e^4\right ) \int \frac {x^4}{e^{x^2}+x} \, dx-\left (2 e^4\right ) \int \frac {x^2}{(1+x) \left (e^{x^2}+x\right )} \, dx-\left (2 e^4\right ) \int \frac {x^3}{(1+x) \left (e^{x^2}+x\right )} \, dx+\left (2 e^4\right ) \int \frac {x^4}{(1+x) \left (e^{x^2}+x\right )} \, dx+\left (2 e^4\right ) \int \frac {x^5}{(1+x) \left (e^{x^2}+x\right )} \, dx-\left (2 e^4\right ) \int \frac {x \log (3 x) \log (1+x)}{e^{x^2}+x} \, dx-\left (2 e^4\right ) \int \frac {x^2 \log (3 x) \log (1+x)}{e^{x^2}+x} \, dx+\left (2 e^4\right ) \int \frac {x^2 \log (3 x) \log (1+x)}{(1+x) \left (e^{x^2}+x\right )} \, dx+\left (2 e^4\right ) \int \frac {x^3 \log (3 x) \log (1+x)}{(1+x) \left (e^{x^2}+x\right )} \, dx+\left (2 e^4\right ) \int \frac {\int \frac {\int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx}{x} \, dx}{1+x} \, dx+\left (2 e^4\right ) \int \frac {\int \frac {\int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx}{1+x} \, dx}{x} \, dx+\left (e^4 \log (3 x)\right ) \int \frac {\int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx}{1+x} \, dx-\left (2 e^4 \log (3 x)\right ) \int \frac {\int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx}{1+x} \, dx+\left (e^4 \log (1+x)\right ) \int \frac {\int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx}{x} \, dx-\left (2 e^4 \log (1+x)\right ) \int \frac {\int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx}{x} \, dx-\left (e^4 \log (3 x) \log (1+x)\right ) \int \frac {1}{\left (e^{x^2}+x\right )^2} \, dx+\left (2 e^4 \log (3 x) \log (1+x)\right ) \int \frac {x^2}{\left (e^{x^2}+x\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {e^4 \left (x^2+\log (3 x) \log (1+x)\right )}{e^{x^2}+x} \]
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Time = 36.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08
method | result | size |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{4}+{\mathrm e}^{4} \ln \left (1+x \right ) \ln \left (3 x \right )}{{\mathrm e}^{x^{2}}+x}\) | \(28\) |
risch | \(\frac {{\mathrm e}^{4} \ln \left (3 x \right ) \ln \left (1+x \right )}{{\mathrm e}^{x^{2}}+x}+\frac {x^{2} {\mathrm e}^{4}}{{\mathrm e}^{x^{2}}+x}\) | \(35\) |
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Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{8} + e^{8} \log \left (3 \, x\right ) \log \left (x + 1\right )}{x e^{4} + e^{\left (x^{2} + 4\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{4} + e^{4} \log {\left (3 x \right )} \log {\left (x + 1 \right )}}{x + e^{x^{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{4} + {\left (e^{4} \log \left (3\right ) + e^{4} \log \left (x\right )\right )} \log \left (x + 1\right )}{x + e^{\left (x^{2}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {x^{2} e^{4} + e^{4} \log \left (3\right ) \log \left (x + 1\right ) + e^{4} \log \left (x + 1\right ) \log \left (x\right )}{x + e^{\left (x^{2}\right )}} \]
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Time = 9.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^4 \left (x^3+x^4\right )+e^{4+x^2} \left (2 x^2+2 x^3-2 x^4-2 x^5\right )+\left (e^{4+x^2} (1+x)+e^4 \left (x+x^2\right )\right ) \log (1+x)+\log (3 x) \left (e^{4+x^2} x+e^4 x^2+\left (e^4 \left (-x-x^2\right )+e^{4+x^2} \left (-2 x^2-2 x^3\right )\right ) \log (1+x)\right )}{x^3+x^4+e^{2 x^2} \left (x+x^2\right )+e^{x^2} \left (2 x^2+2 x^3\right )} \, dx=\frac {{\mathrm {e}}^4\,\left (x^2+\ln \left (3\,x\right )\,\ln \left (x+1\right )\right )}{x+{\mathrm {e}}^{x^2}} \]
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