Integrand size = 91, antiderivative size = 23 \[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=5 e^{x^2} \left (20+x-x \left (-2-\log ^2(x)\right )\right )^2 \]
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\[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=\int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = 20 \int e^{x^2} x \log ^3(x) \, dx+\int e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right ) \, dx+\int e^{x^2} (400+60 x) \log (x) \, dx+\int e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x) \, dx+\int e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x) \, dx \\ & = 30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)+20 \int e^{x^2} x \log ^3(x) \, dx+\int \left (600 e^{x^2}+4090 e^{x^2} x+1200 e^{x^2} x^2+90 e^{x^2} x^3\right ) \, dx-\int \frac {10 \left (3 e^{x^2}+20 \sqrt {\pi } \text {erfi}(x)\right )}{x} \, dx+\int e^{x^2} x \left (10+10 x^2\right ) \log ^4(x) \, dx+\int \left (200 e^{x^2} \log ^2(x)+60 e^{x^2} x \log ^2(x)+400 e^{x^2} x^2 \log ^2(x)+60 e^{x^2} x^3 \log ^2(x)\right ) \, dx \\ & = 30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)-10 \int \frac {3 e^{x^2}+20 \sqrt {\pi } \text {erfi}(x)}{x} \, dx+20 \int e^{x^2} x \log ^3(x) \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx+90 \int e^{x^2} x^3 \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx+600 \int e^{x^2} \, dx+1200 \int e^{x^2} x^2 \, dx+4090 \int e^{x^2} x \, dx+\int \left (10 e^{x^2} x \log ^4(x)+10 e^{x^2} x^3 \log ^4(x)\right ) \, dx \\ & = 2045 e^{x^2}+600 e^{x^2} x+45 e^{x^2} x^2+300 \sqrt {\pi } \text {erfi}(x)+30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)-10 \int \left (\frac {3 e^{x^2}}{x}+\frac {20 \sqrt {\pi } \text {erfi}(x)}{x}\right ) \, dx+10 \int e^{x^2} x \log ^4(x) \, dx+10 \int e^{x^2} x^3 \log ^4(x) \, dx+20 \int e^{x^2} x \log ^3(x) \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx-90 \int e^{x^2} x \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx-600 \int e^{x^2} \, dx \\ & = 2000 e^{x^2}+600 e^{x^2} x+45 e^{x^2} x^2+30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)+10 \int e^{x^2} x \log ^4(x) \, dx+10 \int e^{x^2} x^3 \log ^4(x) \, dx+20 \int e^{x^2} x \log ^3(x) \, dx-30 \int \frac {e^{x^2}}{x} \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx-\left (200 \sqrt {\pi }\right ) \int \frac {\text {erfi}(x)}{x} \, dx \\ & = 2000 e^{x^2}+600 e^{x^2} x+45 e^{x^2} x^2-15 \operatorname {ExpIntegralEi}\left (x^2\right )-400 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )+30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)+10 \int e^{x^2} x \log ^4(x) \, dx+10 \int e^{x^2} x^3 \log ^4(x) \, dx+20 \int e^{x^2} x \log ^3(x) \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx \\ \end{align*}
Time = 5.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=5 e^{x^2} \left (20+3 x+x \log ^2(x)\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(22)=44\).
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\left (45 x^{2}+600 x +2000\right ) {\mathrm e}^{x^{2}}+5 x^{2} {\mathrm e}^{x^{2}} \ln \left (x \right )^{4}+10 x \left (3 x +20\right ) {\mathrm e}^{x^{2}} \ln \left (x \right )^{2}\) | \(46\) |
parallelrisch | \(5 x^{2} {\mathrm e}^{x^{2}} \ln \left (x \right )^{4}+30 \ln \left (x \right )^{2} {\mathrm e}^{x^{2}} x^{2}+200 \,{\mathrm e}^{x^{2}} \ln \left (x \right )^{2} x +45 x^{2} {\mathrm e}^{x^{2}}+600 \,{\mathrm e}^{x^{2}} x +2000 \,{\mathrm e}^{x^{2}}\) | \(61\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=5 \, x^{2} e^{\left (x^{2}\right )} \log \left (x\right )^{4} + 10 \, {\left (3 \, x^{2} + 20 \, x\right )} e^{\left (x^{2}\right )} \log \left (x\right )^{2} + 5 \, {\left (9 \, x^{2} + 120 \, x + 400\right )} e^{\left (x^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=\left (5 x^{2} \log {\left (x \right )}^{4} + 30 x^{2} \log {\left (x \right )}^{2} + 45 x^{2} + 200 x \log {\left (x \right )}^{2} + 600 x + 2000\right ) e^{x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (19) = 38\).
Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=5 \, {\left (x^{2} \log \left (x\right )^{4} + 2 \, {\left (3 \, x^{2} + 20 \, x\right )} \log \left (x\right )^{2}\right )} e^{\left (x^{2}\right )} + 45 \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} + 600 \, x e^{\left (x^{2}\right )} + 2045 \, e^{\left (x^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (19) = 38\).
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.48 \[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=5 \, {\left ({\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )}\right )} \log \left (x\right )^{4} + 10 \, {\left (3 \, x^{2} + 20 \, x\right )} e^{\left (x^{2}\right )} \log \left (x\right )^{2} + 5 \, {\left (9 \, x^{2} + 120 \, x + 400\right )} e^{\left (x^{2}\right )} \]
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Timed out. \[ \int \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx=\int {\mathrm {e}}^{x^2}\,\left (10\,x^3+10\,x\right )\,{\ln \left (x\right )}^4+20\,x\,{\mathrm {e}}^{x^2}\,{\ln \left (x\right )}^3+{\mathrm {e}}^{x^2}\,\left (60\,x^3+400\,x^2+60\,x+200\right )\,{\ln \left (x\right )}^2+{\mathrm {e}}^{x^2}\,\left (60\,x+400\right )\,\ln \left (x\right )+{\mathrm {e}}^{x^2}\,\left (90\,x^3+1200\,x^2+4090\,x+600\right ) \,d x \]
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