Integrand size = 124, antiderivative size = 29 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {1}{16} (-3+x)^2 x^2+\frac {1}{x^4 \left (5+2 e^{-e}+x\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(29)=58\).
Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.24, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2099} \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {x^4}{16}+\frac {e^e}{\left (2+5 e^e\right ) x^4}-\frac {3 x^3}{8}-\frac {e^{2 e}}{\left (2+5 e^e\right )^2 x^3}+\frac {9 x^2}{16}+\frac {e^{3 e}}{\left (2+5 e^e\right )^3 x^2}+\frac {e^{5 e}}{\left (2+5 e^e\right )^4 \left (e^e x+5 e^e+2\right )}-\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x} \]
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Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {4 e^e}{\left (2+5 e^e\right ) x^5}+\frac {3 e^{2 e}}{\left (2+5 e^e\right )^2 x^4}-\frac {2 e^{3 e}}{\left (2+5 e^e\right )^3 x^3}+\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x^2}+\frac {9 x}{8}-\frac {9 x^2}{8}+\frac {x^3}{4}-\frac {e^{6 e}}{\left (2+5 e^e\right )^4 \left (2+5 e^e+e^e x\right )^2}\right ) \, dx \\ & = \frac {e^e}{\left (2+5 e^e\right ) x^4}-\frac {e^{2 e}}{\left (2+5 e^e\right )^2 x^3}+\frac {e^{3 e}}{\left (2+5 e^e\right )^3 x^2}-\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x}+\frac {9 x^2}{16}-\frac {3 x^3}{8}+\frac {x^4}{16}+\frac {e^{5 e}}{\left (2+5 e^e\right )^4 \left (2+5 e^e+e^e x\right )} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {2 (-3+x)^2 x^6+e^e \left (16+45 x^6-21 x^7-x^8+x^9\right )}{16 x^4 \left (2+e^e (5+x)\right )} \]
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Time = 0.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
method | result | size |
risch | \(\frac {x^{4}}{16}-\frac {3 x^{3}}{8}+\frac {9 x^{2}}{16}+\frac {{\mathrm e}^{{\mathrm e}}}{x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(38\) |
norman | \(\frac {\left (-\frac {21 \,{\mathrm e}^{{\mathrm e}}}{16}-\frac {3}{4}\right ) x^{7}+\left (-\frac {{\mathrm e}^{{\mathrm e}}}{16}+\frac {1}{8}\right ) x^{8}+\left (\frac {45 \,{\mathrm e}^{{\mathrm e}}}{16}+\frac {9}{8}\right ) x^{6}+\frac {{\mathrm e}^{{\mathrm e}} x^{9}}{16}+{\mathrm e}^{{\mathrm e}}}{x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(64\) |
gosper | \(\frac {{\mathrm e}^{{\mathrm e}} x^{9}-{\mathrm e}^{{\mathrm e}} x^{8}-21 \,{\mathrm e}^{{\mathrm e}} x^{7}+2 x^{8}+45 \,{\mathrm e}^{{\mathrm e}} x^{6}-12 x^{7}+18 x^{6}+16 \,{\mathrm e}^{{\mathrm e}}}{16 x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(72\) |
parallelrisch | \(\frac {\left ({\mathrm e}^{2 \,{\mathrm e}} x^{9}-{\mathrm e}^{2 \,{\mathrm e}} x^{8}-21 \,{\mathrm e}^{2 \,{\mathrm e}} x^{7}+2 \,{\mathrm e}^{{\mathrm e}} x^{8}+45 \,{\mathrm e}^{2 \,{\mathrm e}} x^{6}-12 \,{\mathrm e}^{{\mathrm e}} x^{7}+18 \,{\mathrm e}^{{\mathrm e}} x^{6}+16 \,{\mathrm e}^{2 \,{\mathrm e}}\right ) {\mathrm e}^{-{\mathrm e}}}{16 x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(96\) |
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {2 \, x^{8} - 12 \, x^{7} + 18 \, x^{6} + {\left (x^{9} - x^{8} - 21 \, x^{7} + 45 \, x^{6} + 16\right )} e^{e}}{16 \, {\left (2 \, x^{4} + {\left (x^{5} + 5 \, x^{4}\right )} e^{e}\right )}} \]
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Time = 0.59 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {x^{4}}{16} - \frac {3 x^{3}}{8} + \frac {9 x^{2}}{16} + \frac {e^{e}}{x^{5} e^{e} + x^{4} \cdot \left (2 + 5 e^{e}\right )} \]
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Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {1}{16} \, x^{4} - \frac {3}{8} \, x^{3} + \frac {9}{16} \, x^{2} + \frac {e^{e}}{x^{5} e^{e} + x^{4} {\left (5 \, e^{e} + 2\right )}} \]
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Exception generated. \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\text {Exception raised: NotImplementedError} \]
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Time = 0.27 (sec) , antiderivative size = 314, normalized size of antiderivative = 10.83 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=x^2\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (4\,{\mathrm {e}}^{\mathrm {e}}-31\,{\mathrm {e}}^{2\,\mathrm {e}}+8\right )}{16}-\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2}{8}+{\mathrm {e}}^{-\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\right )+\frac {{\mathrm {e}}^{\mathrm {e}}}{{\mathrm {e}}^{\mathrm {e}}\,x^5+\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\,x^4}-x^3\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{6}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{24}\right )+\frac {x^4}{16}-x\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (135\,{\mathrm {e}}^{2\,\mathrm {e}}+144\,{\mathrm {e}}^{\mathrm {e}}+36\right )}{8}+2\,{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (4\,{\mathrm {e}}^{\mathrm {e}}-31\,{\mathrm {e}}^{2\,\mathrm {e}}+8\right )}{8}-\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2}{4}+2\,{\mathrm {e}}^{-\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\right )-{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2\right ) \]
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