Integrand size = 232, antiderivative size = 24 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {e^{8+x^2} \left (e^4+x+\frac {x}{\log (4)}\right )^4}{x^2} \]
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Time = 0.57 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {12, 6820, 2326} \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {e^{x^2+8} \left (x (1+\log (4))+e^4 \log (4)\right )^3 \left (x^3 (1+\log (4))+e^4 x^2 \log (4)\right )}{x^4 \log ^4(4)} \]
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Rule 12
Rule 2326
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3} \, dx}{\log ^4(4)} \\ & = \frac {\int \frac {2 e^{8+x^2} \left (e^4 \log (4)+x (1+\log (4))\right )^3 \left (-e^4 \log (4)+e^4 x^2 \log (4)+x (1+\log (4))+x^3 (1+\log (4))\right )}{x^3} \, dx}{\log ^4(4)} \\ & = \frac {2 \int \frac {e^{8+x^2} \left (e^4 \log (4)+x (1+\log (4))\right )^3 \left (-e^4 \log (4)+e^4 x^2 \log (4)+x (1+\log (4))+x^3 (1+\log (4))\right )}{x^3} \, dx}{\log ^4(4)} \\ & = \frac {e^{8+x^2} \left (e^4 \log (4)+x (1+\log (4))\right )^3 \left (e^4 x^2 \log (4)+x^3 (1+\log (4))\right )}{x^4 \log ^4(4)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {e^{8+x^2} \left (x+e^4 \log (4)+x \log (4)\right )^4}{x^2 \log ^4(4)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(25)=50\).
Time = 1.80 (sec) , antiderivative size = 155, normalized size of antiderivative = 6.46
method | result | size |
risch | \(\frac {\left (16 \ln \left (2\right )^{4} {\mathrm e}^{16}+64 \,{\mathrm e}^{12} \ln \left (2\right )^{4} x +32 \,{\mathrm e}^{12} \ln \left (2\right )^{3} x +96 \,{\mathrm e}^{8} \ln \left (2\right )^{4} x^{2}+96 \,{\mathrm e}^{8} \ln \left (2\right )^{3} x^{2}+64 \,{\mathrm e}^{4} \ln \left (2\right )^{4} x^{3}+24 \ln \left (2\right )^{2} {\mathrm e}^{8} x^{2}+96 \,{\mathrm e}^{4} \ln \left (2\right )^{3} x^{3}+16 \ln \left (2\right )^{4} x^{4}+48 \,{\mathrm e}^{4} \ln \left (2\right )^{2} x^{3}+32 x^{4} \ln \left (2\right )^{3}+8 x^{3} {\mathrm e}^{4} \ln \left (2\right )+24 x^{4} \ln \left (2\right )^{2}+8 x^{4} \ln \left (2\right )+x^{4}\right ) {\mathrm e}^{x^{2}+8}}{16 x^{2} \ln \left (2\right )^{4}}\) | \(155\) |
norman | \(\frac {{\mathrm e}^{8} {\mathrm e}^{16} \ln \left (2\right )^{3} {\mathrm e}^{x^{2}}+\frac {{\mathrm e}^{8} {\mathrm e}^{4} \left (8 \ln \left (2\right )^{3}+12 \ln \left (2\right )^{2}+6 \ln \left (2\right )+1\right ) x^{3} {\mathrm e}^{x^{2}}}{2}+\frac {{\mathrm e}^{8} \left (16 \ln \left (2\right )^{4}+32 \ln \left (2\right )^{3}+24 \ln \left (2\right )^{2}+8 \ln \left (2\right )+1\right ) x^{4} {\mathrm e}^{x^{2}}}{16 \ln \left (2\right )}+2 \,{\mathrm e}^{8} {\mathrm e}^{12} \ln \left (2\right )^{2} \left (1+2 \ln \left (2\right )\right ) x \,{\mathrm e}^{x^{2}}+\frac {3 \ln \left (2\right ) \left ({\mathrm e}^{8}\right )^{2} \left (4 \ln \left (2\right )^{2}+4 \ln \left (2\right )+1\right ) x^{2} {\mathrm e}^{x^{2}}}{2}}{x^{2} \ln \left (2\right )^{3}}\) | \(157\) |
gosper | \(\frac {{\mathrm e}^{8} \left (16 \ln \left (2\right )^{4} {\mathrm e}^{16}+64 \,{\mathrm e}^{12} \ln \left (2\right )^{4} x +32 \,{\mathrm e}^{12} \ln \left (2\right )^{3} x +96 \,{\mathrm e}^{8} \ln \left (2\right )^{4} x^{2}+96 \,{\mathrm e}^{8} \ln \left (2\right )^{3} x^{2}+64 \,{\mathrm e}^{4} \ln \left (2\right )^{4} x^{3}+24 \ln \left (2\right )^{2} {\mathrm e}^{8} x^{2}+96 \,{\mathrm e}^{4} \ln \left (2\right )^{3} x^{3}+16 \ln \left (2\right )^{4} x^{4}+48 \,{\mathrm e}^{4} \ln \left (2\right )^{2} x^{3}+32 x^{4} \ln \left (2\right )^{3}+8 x^{3} {\mathrm e}^{4} \ln \left (2\right )+24 x^{4} \ln \left (2\right )^{2}+8 x^{4} \ln \left (2\right )+x^{4}\right ) {\mathrm e}^{x^{2}}}{16 \ln \left (2\right )^{4} x^{2}}\) | \(169\) |
parallelrisch | \(\frac {16 \,{\mathrm e}^{8} {\mathrm e}^{16} \ln \left (2\right )^{4} {\mathrm e}^{x^{2}}+64 \,{\mathrm e}^{8} {\mathrm e}^{12} \ln \left (2\right )^{4} x \,{\mathrm e}^{x^{2}}+96 \left ({\mathrm e}^{8}\right )^{2} \ln \left (2\right )^{4} x^{2} {\mathrm e}^{x^{2}}+64 \,{\mathrm e}^{8} {\mathrm e}^{4} \ln \left (2\right )^{4} x^{3} {\mathrm e}^{x^{2}}+16 \,{\mathrm e}^{8} \ln \left (2\right )^{4} x^{4} {\mathrm e}^{x^{2}}+32 \,{\mathrm e}^{8} {\mathrm e}^{12} \ln \left (2\right )^{3} x \,{\mathrm e}^{x^{2}}+96 \left ({\mathrm e}^{8}\right )^{2} \ln \left (2\right )^{3} x^{2} {\mathrm e}^{x^{2}}+96 \,{\mathrm e}^{8} {\mathrm e}^{4} \ln \left (2\right )^{3} x^{3} {\mathrm e}^{x^{2}}+32 \,{\mathrm e}^{8} \ln \left (2\right )^{3} x^{4} {\mathrm e}^{x^{2}}+24 \left ({\mathrm e}^{8}\right )^{2} \ln \left (2\right )^{2} x^{2} {\mathrm e}^{x^{2}}+48 \,{\mathrm e}^{8} {\mathrm e}^{4} \ln \left (2\right )^{2} x^{3} {\mathrm e}^{x^{2}}+24 \,{\mathrm e}^{8} \ln \left (2\right )^{2} x^{4} {\mathrm e}^{x^{2}}+8 \,{\mathrm e}^{8} {\mathrm e}^{4} \ln \left (2\right ) x^{3} {\mathrm e}^{x^{2}}+8 \,{\mathrm e}^{8} \ln \left (2\right ) x^{4} {\mathrm e}^{x^{2}}+{\mathrm e}^{8} x^{4} {\mathrm e}^{x^{2}}}{16 \ln \left (2\right )^{4} x^{2}}\) | \(282\) |
meijerg | \({\mathrm e}^{24} \left (\frac {1}{x^{2}}+1-2 \ln \left (x \right )-i \pi -\frac {2 x^{2}+2}{2 x^{2}}+\frac {{\mathrm e}^{x^{2}}}{x^{2}}+\ln \left (-x^{2}\right )+\operatorname {Ei}_{1}\left (-x^{2}\right )\right )+{\mathrm e}^{24} \left (2 \ln \left (x \right )+i \pi -\ln \left (-x^{2}\right )-\operatorname {Ei}_{1}\left (-x^{2}\right )\right )+\frac {\left (2 \ln \left (2\right )^{4} {\mathrm e}^{8}+4 \ln \left (2\right )^{3} {\mathrm e}^{8}+3 \,{\mathrm e}^{8} \ln \left (2\right )^{2}+{\mathrm e}^{8} \ln \left (2\right )+\frac {{\mathrm e}^{8}}{8}\right ) \left (1-\frac {\left (-2 x^{2}+2\right ) {\mathrm e}^{x^{2}}}{2}\right )}{2 \ln \left (2\right )^{4}}-\frac {\left (12 \ln \left (2\right )^{4} {\mathrm e}^{16}+12 \,{\mathrm e}^{16} \ln \left (2\right )^{3}+3 \,{\mathrm e}^{16} \ln \left (2\right )^{2}+2 \ln \left (2\right )^{4} {\mathrm e}^{8}+4 \ln \left (2\right )^{3} {\mathrm e}^{8}+3 \,{\mathrm e}^{8} \ln \left (2\right )^{2}+{\mathrm e}^{8} \ln \left (2\right )+\frac {{\mathrm e}^{8}}{8}\right ) \left (1-{\mathrm e}^{x^{2}}\right )}{2 \ln \left (2\right )^{4}}+\frac {i \left (8 \ln \left (2\right )^{4} {\mathrm e}^{12}+12 \ln \left (2\right )^{3} {\mathrm e}^{12}+6 \ln \left (2\right )^{2} {\mathrm e}^{12}+{\mathrm e}^{12} \ln \left (2\right )\right ) \left (-i {\mathrm e}^{x^{2}} x +\frac {i \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2}\right )}{2 \ln \left (2\right )^{4}}+\frac {i \left (-4 \,{\mathrm e}^{20} \ln \left (2\right )^{4}-2 \,{\mathrm e}^{20} \ln \left (2\right )^{3}\right ) \left (\frac {2 i {\mathrm e}^{x^{2}}}{x}-2 i \sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )}{2 \ln \left (2\right )^{4}}+\frac {\left (8 \,{\mathrm e}^{20} \ln \left (2\right )^{4}+4 \,{\mathrm e}^{20} \ln \left (2\right )^{3}+4 \ln \left (2\right )^{4} {\mathrm e}^{12}+6 \ln \left (2\right )^{3} {\mathrm e}^{12}+3 \ln \left (2\right )^{2} {\mathrm e}^{12}+\frac {{\mathrm e}^{12} \ln \left (2\right )}{2}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2 \ln \left (2\right )^{4}}\) | \(362\) |
default | \(\frac {128 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{4} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+16 \ln \left (2\right ) {\mathrm e}^{8} {\mathrm e}^{4} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+24 \ln \left (2\right )^{2} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}+96 \ln \left (2\right )^{3} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}+96 \ln \left (2\right )^{4} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}-16 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{16} \operatorname {Ei}_{1}\left (-x^{2}\right )-64 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{12} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )-32 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{12} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )-32 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{16} \left (-\frac {{\mathrm e}^{x^{2}}}{2 x^{2}}-\frac {\operatorname {Ei}_{1}\left (-x^{2}\right )}{2}\right )+96 \ln \left (2\right )^{2} {\mathrm e}^{8} {\mathrm e}^{4} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+192 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{4} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+{\mathrm e}^{x^{2}} {\mathrm e}^{8}+16 \,{\mathrm e}^{8} \ln \left (2\right ) \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+24 \ln \left (2\right )^{2} {\mathrm e}^{8} {\mathrm e}^{x^{2}}+8 \,{\mathrm e}^{8} \ln \left (2\right ) {\mathrm e}^{x^{2}}+64 \ln \left (2\right )^{3} {\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+16 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{x^{2}}+32 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{12} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+32 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{4} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+4 \ln \left (2\right ) {\mathrm e}^{8} {\mathrm e}^{4} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+48 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{4} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+64 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{12} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+24 \ln \left (2\right )^{2} {\mathrm e}^{8} {\mathrm e}^{4} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+48 \ln \left (2\right )^{2} {\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+32 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{x^{2}}+32 \ln \left (2\right )^{4} {\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+2 \,{\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )}{16 \ln \left (2\right )^{4}}\) | \(573\) |
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 5.33 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {{\left (x^{4} e^{8} + 16 \, {\left (x^{4} e^{8} + 4 \, x^{3} e^{12} + 6 \, x^{2} e^{16} + 4 \, x e^{20} + e^{24}\right )} \log \left (2\right )^{4} + 32 \, {\left (x^{4} e^{8} + 3 \, x^{3} e^{12} + 3 \, x^{2} e^{16} + x e^{20}\right )} \log \left (2\right )^{3} + 24 \, {\left (x^{4} e^{8} + 2 \, x^{3} e^{12} + x^{2} e^{16}\right )} \log \left (2\right )^{2} + 8 \, {\left (x^{4} e^{8} + x^{3} e^{12}\right )} \log \left (2\right )\right )} e^{\left (x^{2}\right )}}{16 \, x^{2} \log \left (2\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 201, normalized size of antiderivative = 8.38 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {\left (x^{4} e^{8} + 16 x^{4} e^{8} \log {\left (2 \right )}^{4} + 8 x^{4} e^{8} \log {\left (2 \right )} + 32 x^{4} e^{8} \log {\left (2 \right )}^{3} + 24 x^{4} e^{8} \log {\left (2 \right )}^{2} + 8 x^{3} e^{12} \log {\left (2 \right )} + 64 x^{3} e^{12} \log {\left (2 \right )}^{4} + 48 x^{3} e^{12} \log {\left (2 \right )}^{2} + 96 x^{3} e^{12} \log {\left (2 \right )}^{3} + 24 x^{2} e^{16} \log {\left (2 \right )}^{2} + 96 x^{2} e^{16} \log {\left (2 \right )}^{4} + 96 x^{2} e^{16} \log {\left (2 \right )}^{3} + 32 x e^{20} \log {\left (2 \right )}^{3} + 64 x e^{20} \log {\left (2 \right )}^{4} + 16 e^{24} \log {\left (2 \right )}^{4}\right ) e^{x^{2}}}{16 x^{2} \log {\left (2 \right )}^{4}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 467, normalized size of antiderivative = 19.46 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=-\frac {64 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{20} \log \left (2\right )^{4} + 32 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \left (2\right )^{4} - 16 \, {\rm Ei}\left (x^{2}\right ) e^{24} \log \left (2\right )^{4} - 16 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \left (2\right )^{4} + 16 \, e^{24} \Gamma \left (-1, -x^{2}\right ) \log \left (2\right )^{4} + 32 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{20} \log \left (2\right )^{3} + 48 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \left (2\right )^{3} - \frac {32 \, \sqrt {-x^{2}} e^{20} \Gamma \left (-\frac {1}{2}, -x^{2}\right ) \log \left (2\right )^{4}}{x} - 32 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \left (2\right )^{3} + 32 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \left (2\right )^{4} - 96 \, e^{\left (x^{2} + 16\right )} \log \left (2\right )^{4} - 16 \, e^{\left (x^{2} + 8\right )} \log \left (2\right )^{4} + 24 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \left (2\right )^{2} - \frac {16 \, \sqrt {-x^{2}} e^{20} \Gamma \left (-\frac {1}{2}, -x^{2}\right ) \log \left (2\right )^{3}}{x} - 24 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \left (2\right )^{2} + 48 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \left (2\right )^{3} - 96 \, e^{\left (x^{2} + 16\right )} \log \left (2\right )^{3} - 32 \, e^{\left (x^{2} + 8\right )} \log \left (2\right )^{3} + 4 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} \log \left (2\right ) - 8 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} \log \left (2\right ) + 24 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \left (2\right )^{2} - 24 \, e^{\left (x^{2} + 16\right )} \log \left (2\right )^{2} - 24 \, e^{\left (x^{2} + 8\right )} \log \left (2\right )^{2} - {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} + 4 \, {\left (-i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) e^{12} - 2 \, x e^{\left (x^{2} + 12\right )}\right )} \log \left (2\right ) - 8 \, e^{\left (x^{2} + 8\right )} \log \left (2\right ) - e^{\left (x^{2} + 8\right )}}{16 \, \log \left (2\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 9.12 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {16 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right )^{4} + 32 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right )^{3} + 64 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right )^{4} + 24 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right )^{2} + 96 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right )^{3} + 96 \, x^{2} e^{\left (x^{2} + 16\right )} \log \left (2\right )^{4} + 8 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right ) + 48 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right )^{2} + 96 \, x^{2} e^{\left (x^{2} + 16\right )} \log \left (2\right )^{3} + 64 \, x e^{\left (x^{2} + 20\right )} \log \left (2\right )^{4} + x^{4} e^{\left (x^{2} + 8\right )} + 8 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right ) + 24 \, x^{2} e^{\left (x^{2} + 16\right )} \log \left (2\right )^{2} + 32 \, x e^{\left (x^{2} + 20\right )} \log \left (2\right )^{3} + 16 \, e^{\left (x^{2} + 24\right )} \log \left (2\right )^{4}}{16 \, x^{2} \log \left (2\right )^{4}} \]
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Time = 9.98 (sec) , antiderivative size = 159, normalized size of antiderivative = 6.62 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {{\mathrm {e}}^{x^2}\,\left (3\,{\mathrm {e}}^{16}+12\,{\mathrm {e}}^{16}\,\ln \left (2\right )+12\,{\mathrm {e}}^{16}\,{\ln \left (2\right )}^2\right )}{2\,{\ln \left (2\right )}^2}+\frac {{\mathrm {e}}^{x^2+24}\,{\ln \left (2\right )}^4+2\,x\,{\mathrm {e}}^{x^2+20}\,{\ln \left (2\right )}^3\,\left (2\,\ln \left (2\right )+1\right )}{x^2\,{\ln \left (2\right )}^4}+\frac {x\,{\mathrm {e}}^{x^2}\,\left (4\,{\mathrm {e}}^{12}\,\ln \left (2\right )+24\,{\mathrm {e}}^{12}\,{\ln \left (2\right )}^2+48\,{\mathrm {e}}^{12}\,{\ln \left (2\right )}^3+32\,{\mathrm {e}}^{12}\,{\ln \left (2\right )}^4\right )}{8\,{\ln \left (2\right )}^4}+\frac {x^2\,{\mathrm {e}}^{x^2}\,\left (\frac {{\mathrm {e}}^8}{2}+4\,{\mathrm {e}}^8\,\ln \left (2\right )+12\,{\mathrm {e}}^8\,{\ln \left (2\right )}^2+16\,{\mathrm {e}}^8\,{\ln \left (2\right )}^3+8\,{\mathrm {e}}^8\,{\ln \left (2\right )}^4\right )}{8\,{\ln \left (2\right )}^4} \]
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