Integrand size = 113, antiderivative size = 31 \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=e^x \left (5-\log \left (e^{-1/e} \left (x+\frac {3 e^{x^2} x}{\log (x)}\right )\right )\right ) \]
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Time = 3.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10, number of steps used = 27, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6873, 6874, 2230, 2225, 2209, 2207, 2635} \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=(1+5 e) e^{x-1}-e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right ) \]
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Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2635
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{x \log (x) \left (3 e^{x^2}+\log (x)\right )} \, dx \\ & = \int \left (\frac {e^x \left (-1+2 x^2 \log (x)\right )}{x \left (3 e^{x^2}+\log (x)\right )}+\frac {e^{-1+x} \left (e-e \log (x)+(1+5 e) x \log (x)-2 e x^2 \log (x)-e x \log (x) \log \left (x+\frac {3 e^{x^2} x}{\log (x)}\right )\right )}{x \log (x)}\right ) \, dx \\ & = \int \frac {e^x \left (-1+2 x^2 \log (x)\right )}{x \left (3 e^{x^2}+\log (x)\right )} \, dx+\int \frac {e^{-1+x} \left (e-e \log (x)+(1+5 e) x \log (x)-2 e x^2 \log (x)-e x \log (x) \log \left (x+\frac {3 e^{x^2} x}{\log (x)}\right )\right )}{x \log (x)} \, dx \\ & = \int \left (-\frac {e^x}{x \left (3 e^{x^2}+\log (x)\right )}+\frac {2 e^x x \log (x)}{3 e^{x^2}+\log (x)}\right ) \, dx+\int \left (\frac {e^{-1+x} \left (e-e \log (x)+(1+5 e) x \log (x)-2 e x^2 \log (x)\right )}{x \log (x)}-e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )\right ) \, dx \\ & = 2 \int \frac {e^x x \log (x)}{3 e^{x^2}+\log (x)} \, dx-\int \frac {e^x}{x \left (3 e^{x^2}+\log (x)\right )} \, dx+\int \frac {e^{-1+x} \left (e-e \log (x)+(1+5 e) x \log (x)-2 e x^2 \log (x)\right )}{x \log (x)} \, dx-\int e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right ) \, dx \\ & = -e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )+2 \int \frac {e^x x \log (x)}{3 e^{x^2}+\log (x)} \, dx+\int \left (\frac {e^{-1+x} \left (-e+(1+5 e) x-2 e x^2\right )}{x}+\frac {e^x}{x \log (x)}\right ) \, dx-\int \frac {e^x}{x \left (3 e^{x^2}+\log (x)\right )} \, dx+\int \frac {e^x \left (-3 e^{x^2}+3 e^{x^2} \left (1+2 x^2\right ) \log (x)+\log ^2(x)\right )}{x \log (x) \left (3 e^{x^2}+\log (x)\right )} \, dx \\ & = -e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )+2 \int \frac {e^x x \log (x)}{3 e^{x^2}+\log (x)} \, dx+\int \frac {e^{-1+x} \left (-e+(1+5 e) x-2 e x^2\right )}{x} \, dx+\int \frac {e^x}{x \log (x)} \, dx-\int \frac {e^x}{x \left (3 e^{x^2}+\log (x)\right )} \, dx+\int \left (-\frac {e^x \left (-1+2 x^2 \log (x)\right )}{x \left (3 e^{x^2}+\log (x)\right )}+\frac {e^x \left (-1+\log (x)+2 x^2 \log (x)\right )}{x \log (x)}\right ) \, dx \\ & = -e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )+2 \int \frac {e^x x \log (x)}{3 e^{x^2}+\log (x)} \, dx+\int \left (e^{-1+x} (1+5 e)-\frac {e^x}{x}-2 e^x x\right ) \, dx+\int \frac {e^x}{x \log (x)} \, dx-\int \frac {e^x}{x \left (3 e^{x^2}+\log (x)\right )} \, dx-\int \frac {e^x \left (-1+2 x^2 \log (x)\right )}{x \left (3 e^{x^2}+\log (x)\right )} \, dx+\int \frac {e^x \left (-1+\log (x)+2 x^2 \log (x)\right )}{x \log (x)} \, dx \\ & = -e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )-2 \int e^x x \, dx+2 \int \frac {e^x x \log (x)}{3 e^{x^2}+\log (x)} \, dx+(1+5 e) \int e^{-1+x} \, dx-\int \frac {e^x}{x} \, dx+\int \left (\frac {e^x \left (1+2 x^2\right )}{x}-\frac {e^x}{x \log (x)}\right ) \, dx+\int \frac {e^x}{x \log (x)} \, dx-\int \frac {e^x}{x \left (3 e^{x^2}+\log (x)\right )} \, dx-\int \left (-\frac {e^x}{x \left (3 e^{x^2}+\log (x)\right )}+\frac {2 e^x x \log (x)}{3 e^{x^2}+\log (x)}\right ) \, dx \\ & = e^{-1+x} (1+5 e)-2 e^x x-\text {Ei}(x)-e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )+2 \int e^x \, dx+\int \frac {e^x \left (1+2 x^2\right )}{x} \, dx \\ & = 2 e^x+e^{-1+x} (1+5 e)-2 e^x x-\text {Ei}(x)-e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )+\int \left (\frac {e^x}{x}+2 e^x x\right ) \, dx \\ & = 2 e^x+e^{-1+x} (1+5 e)-2 e^x x-\text {Ei}(x)-e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )+2 \int e^x x \, dx+\int \frac {e^x}{x} \, dx \\ & = 2 e^x+e^{-1+x} (1+5 e)-e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right )-2 \int e^x \, dx \\ & = e^{-1+x} (1+5 e)-e^x \log \left (\frac {x \left (3 e^{x^2}+\log (x)\right )}{\log (x)}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=e^{-1+x} \left (1+5 e-e \log \left (x+\frac {3 e^{x^2} x}{\log (x)}\right )\right ) \]
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Time = 6.43 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \(-{\mathrm e}^{x} \ln \left (\frac {x \left (3 \,{\mathrm e}^{x^{2}}+\ln \left (x \right )\right ) {\mathrm e}^{-{\mathrm e}^{-1}}}{\ln \left (x \right )}\right )+5 \,{\mathrm e}^{x}\) | \(33\) |
risch | \(-{\mathrm e}^{x} \ln \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )+\frac {\left (i \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right )}^{3} {\mathrm e}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right ) {\mathrm e}+i \pi {\operatorname {csgn}\left (\frac {i x \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right )}^{3} {\mathrm e}-i \pi \,\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right ) {\operatorname {csgn}\left (\frac {i x \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right )}^{2} {\mathrm e}-i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right )}^{2} {\mathrm e}+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right ) {\mathrm e}-i \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right )}^{2} {\mathrm e}-i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{2}}+\frac {\ln \left (x \right )}{3}\right )}{\ln \left (x \right )}\right )}^{2} {\mathrm e}+2-2 \ln \left (3\right ) {\mathrm e}-2 \,{\mathrm e} \ln \left (x \right )+2 \ln \left (\ln \left (x \right )\right ) {\mathrm e}+10 \,{\mathrm e}\right ) {\mathrm e}^{-1+x}}{2}\) | \(333\) |
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=-{\left (e^{\left (x^{2} + x\right )} \log \left (\frac {{\left (3 \, x e^{\left (x^{2}\right )} + x \log \left (x\right )\right )} e^{\left (-e^{\left (-1\right )}\right )}}{\log \left (x\right )}\right ) - 5 \, e^{\left (x^{2} + x\right )}\right )} e^{\left (-x^{2}\right )} \]
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Timed out. \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=-{\left ({\left (e \log \left (x\right ) - 5 \, e - 1\right )} e^{x} + e^{\left (x + 1\right )} \log \left (3 \, e^{\left (x^{2}\right )} + \log \left (x\right )\right ) - e^{\left (x + 1\right )} \log \left (\log \left (x\right )\right )\right )} e^{\left (-1\right )} \]
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Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=-{\left (e^{\left (x + 1\right )} \log \left (x\right ) + e^{\left (x + 1\right )} \log \left (3 \, e^{\left (x^{2}\right )} + \log \left (x\right )\right ) - e^{\left (x + 1\right )} \log \left (\log \left (x\right )\right ) - 5 \, e^{\left (x + 1\right )} - e^{x}\right )} e^{\left (-1\right )} \]
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Time = 11.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {3 e^{x+x^2}+e^{x+x^2} \left (-3+15 x-6 x^2\right ) \log (x)+e^x (-1+5 x) \log ^2(x)+\left (-3 e^{x+x^2} x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{-1/e} \left (3 e^{x^2} x+x \log (x)\right )}{\log (x)}\right )}{3 e^{x^2} x \log (x)+x \log ^2(x)} \, dx=-{\mathrm {e}}^x\,\left (\ln \left (\frac {{\mathrm {e}}^{-{\mathrm {e}}^{-1}}\,\left (3\,x\,{\mathrm {e}}^{x^2}+x\,\ln \left (x\right )\right )}{\ln \left (x\right )}\right )-5\right ) \]
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