Integrand size = 68, antiderivative size = 29 \[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\frac {1}{4} \log \left (\left (-e^{\frac {-1+x+x^2+\log ^2(x)}{x}}-2 x\right )^2\right ) \]
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\[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1+x^2+2 \log (x)-\log ^2(x)}{2 x^2}-\frac {e^{\frac {1}{x}} \left (1-x+x^2+2 \log (x)-\log ^2(x)\right )}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1+x^2+2 \log (x)-\log ^2(x)}{x^2} \, dx-\int \frac {e^{\frac {1}{x}} \left (1-x+x^2+2 \log (x)-\log ^2(x)\right )}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx \\ & = \frac {1}{2} \int \left (\frac {1+x^2}{x^2}+\frac {2 \log (x)}{x^2}-\frac {\log ^2(x)}{x^2}\right ) \, dx-\int \left (-\frac {e^{\frac {1}{x}}}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x}+\frac {e^{\frac {1}{x}}}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )}+\frac {e^{\frac {1}{x}} x}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x}+\frac {2 e^{\frac {1}{x}} \log (x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )}-\frac {e^{\frac {1}{x}} \log ^2(x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1+x^2}{x^2} \, dx-\frac {1}{2} \int \frac {\log ^2(x)}{x^2} \, dx-2 \int \frac {e^{\frac {1}{x}} \log (x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx+\int \frac {e^{\frac {1}{x}}}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x} \, dx-\int \frac {e^{\frac {1}{x}}}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx-\int \frac {e^{\frac {1}{x}} x}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x} \, dx+\int \frac {\log (x)}{x^2} \, dx+\int \frac {e^{\frac {1}{x}} \log ^2(x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx \\ & = -\frac {1}{x}-\frac {\log (x)}{x}+\frac {\log ^2(x)}{2 x}+\frac {1}{2} \int \left (1+\frac {1}{x^2}\right ) \, dx-2 \int \frac {e^{\frac {1}{x}} \log (x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx+e^{-1-x-\frac {\log ^2(x)}{x}} \text {Subst}\left (\int \frac {\exp \left (-2 e^{-1+\frac {1}{x}-x-\frac {\log ^2(x)}{x}}+e^{-1-x-\frac {\log ^2(x)}{x}} x\right )}{x} \, dx,x,\frac {e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x}{x}\right )+\int \frac {e^{\frac {1}{x}}}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x} \, dx-\int \frac {e^{\frac {1}{x}} x}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x} \, dx-\int \frac {\log (x)}{x^2} \, dx+\int \frac {e^{\frac {1}{x}} \log ^2(x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx \\ & = -\frac {1}{2 x}+\frac {x}{2}+\frac {\log ^2(x)}{2 x}-2 \int \frac {e^{\frac {1}{x}} \log (x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx+e^{-1-x-\frac {\log ^2(x)}{x}} \text {Subst}\left (\int \frac {e^{e^{-1-x-\frac {\log ^2(x)}{x}} \left (-2 e^{\frac {1}{x}}+x\right )}}{x} \, dx,x,\frac {e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x}{x}\right )+\int \frac {e^{\frac {1}{x}}}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x} \, dx-\int \frac {e^{\frac {1}{x}} x}{e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x} \, dx+\int \frac {e^{\frac {1}{x}} \log ^2(x)}{x \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )} \, dx \\ \end{align*}
Time = 2.73 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\frac {-1+x \log \left (e^{1+x+\frac {\log ^2(x)}{x}}+2 e^{\frac {1}{x}} x\right )}{2 x} \]
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Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {\ln \left (x +\frac {{\mathrm e}^{\frac {\ln \left (x \right )^{2}+x^{2}+x -1}{x}}}{2}\right )}{2}\) | \(23\) |
risch | \(\frac {\ln \left (x \right )^{2}}{2 x}+\frac {x^{2}-1}{2 x}-\frac {\ln \left (x \right )^{2}+x^{2}+x -1}{2 x}+\frac {\ln \left (2 x +{\mathrm e}^{\frac {\ln \left (x \right )^{2}+x^{2}+x -1}{x}}\right )}{2}\) | \(58\) |
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\frac {1}{2} \, \log \left (2 \, x + e^{\left (\frac {x^{2} + \log \left (x\right )^{2} + x - 1}{x}\right )}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\frac {\log {\left (2 x + e^{\frac {x^{2} + x + \log {\left (x \right )}^{2} - 1}{x}} \right )}}{2} \]
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Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\frac {x^{2} - 1}{2 \, x} + \frac {1}{2} \, \log \left ({\left (2 \, x e^{\frac {1}{x}} + e^{\left (x + \frac {\log \left (x\right )^{2}}{x} + 1\right )}\right )} e^{\left (-x - 1\right )}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\frac {1}{2} \, \log \left (2 \, x + e^{\left (\frac {x^{2} + \log \left (x\right )^{2} + x - 1}{x}\right )}\right ) \]
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Time = 11.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {2 x^2+e^{\frac {-1+x+x^2+\log ^2(x)}{x}} \left (1+x^2+2 \log (x)-\log ^2(x)\right )}{2 e^{\frac {-1+x+x^2+\log ^2(x)}{x}} x^2+4 x^3} \, dx=\frac {\ln \left (x+\frac {\mathrm {e}\,{\mathrm {e}}^{-\frac {1}{x}}\,{\mathrm {e}}^x\,{\mathrm {e}}^{\frac {{\ln \left (x\right )}^2}{x}}}{2}\right )}{2} \]
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