Integrand size = 163, antiderivative size = 24 \[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx=e^{6 x \left (6+\left (1+4 e^4+x+\frac {x}{3+x}\right )^2\right )} \]
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\[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx=\int \frac {\exp \left (\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}\right ) \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx \\ & = \int \left (24 \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \left (1+2 e^4\right )^2+48 \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \left (1+2 e^4\right ) x+18 \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) x^2-\frac {54 \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \left (-3 \left (3-8 e^4\right )-\left (1-8 e^4\right ) x\right )}{27+27 x+9 x^2+x^3}\right ) \, dx \\ & = 18 \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) x^2 \, dx-54 \int \frac {\exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \left (-3 \left (3-8 e^4\right )-\left (1-8 e^4\right ) x\right )}{27+27 x+9 x^2+x^3} \, dx+\left (48 \left (1+2 e^4\right )\right ) \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) x \, dx+\left (24 \left (1+2 e^4\right )^2\right ) \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \, dx \\ & = 18 \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) x^2 \, dx-54 \int \left (-\frac {6 \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right )}{(3+x)^3}+\frac {\exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \left (-1+8 e^4\right )}{(3+x)^2}\right ) \, dx+\left (48 \left (1+2 e^4\right )\right ) \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) x \, dx+\left (24 \left (1+2 e^4\right )^2\right ) \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \, dx \\ & = 18 \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) x^2 \, dx+324 \int \frac {\exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right )}{(3+x)^3} \, dx+\left (54 \left (1-8 e^4\right )\right ) \int \frac {\exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right )}{(3+x)^2} \, dx+\left (48 \left (1+2 e^4\right )\right ) \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) x \, dx+\left (24 \left (1+2 e^4\right )^2\right ) \int \exp \left (\frac {6 x \left (9 \left (7+8 e^4+16 e^8\right )+6 \left (11+24 e^4+16 e^8\right ) x+\left (37+64 e^4+16 e^8\right ) x^2+2 \left (5+4 e^4\right ) x^3+x^4\right )}{9+6 x+x^2}\right ) \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(24)=48\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.33 \[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx=e^{\frac {6 x \left (63+66 x+37 x^2+10 x^3+x^4+16 e^8 (3+x)^2+8 e^4 \left (9+18 x+8 x^2+x^3\right )\right )}{(3+x)^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(24)=48\).
Time = 142.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79
method | result | size |
risch | \({\mathrm e}^{\frac {6 x \left (8 x^{3} {\mathrm e}^{4}+x^{4}+64 x^{2} {\mathrm e}^{4}+16 x^{2} {\mathrm e}^{8}+10 x^{3}+144 x \,{\mathrm e}^{4}+96 x \,{\mathrm e}^{8}+37 x^{2}+72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{8}+66 x +63\right )}{\left (3+x \right )^{2}}}\) | \(67\) |
gosper | \({\mathrm e}^{\frac {6 x \left (8 x^{3} {\mathrm e}^{4}+x^{4}+64 x^{2} {\mathrm e}^{4}+16 x^{2} {\mathrm e}^{8}+10 x^{3}+144 x \,{\mathrm e}^{4}+96 x \,{\mathrm e}^{8}+37 x^{2}+72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{8}+66 x +63\right )}{x^{2}+6 x +9}}\) | \(80\) |
parallelrisch | \({\mathrm e}^{\frac {2 \left (48 x^{3}+288 x^{2}+432 x \right ) {\mathrm e}^{8}+2 \left (24 x^{4}+192 x^{3}+432 x^{2}+216 x \right ) {\mathrm e}^{4}+6 x^{5}+60 x^{4}+222 x^{3}+396 x^{2}+378 x}{x^{2}+6 x +9}}\) | \(80\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {2 \left (48 x^{3}+288 x^{2}+432 x \right ) {\mathrm e}^{8}+2 \left (24 x^{4}+192 x^{3}+432 x^{2}+216 x \right ) {\mathrm e}^{4}+6 x^{5}+60 x^{4}+222 x^{3}+396 x^{2}+378 x}{x^{2}+6 x +9}}+9 \,{\mathrm e}^{\frac {2 \left (48 x^{3}+288 x^{2}+432 x \right ) {\mathrm e}^{8}+2 \left (24 x^{4}+192 x^{3}+432 x^{2}+216 x \right ) {\mathrm e}^{4}+6 x^{5}+60 x^{4}+222 x^{3}+396 x^{2}+378 x}{x^{2}+6 x +9}}+6 x \,{\mathrm e}^{\frac {2 \left (48 x^{3}+288 x^{2}+432 x \right ) {\mathrm e}^{8}+2 \left (24 x^{4}+192 x^{3}+432 x^{2}+216 x \right ) {\mathrm e}^{4}+6 x^{5}+60 x^{4}+222 x^{3}+396 x^{2}+378 x}{x^{2}+6 x +9}}}{\left (3+x \right )^{2}}\) | \(254\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.00 \[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx=e^{\left (\frac {6 \, {\left (x^{5} + 10 \, x^{4} + 37 \, x^{3} + 66 \, x^{2} + 16 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} e^{8} + 8 \, {\left (x^{4} + 8 \, x^{3} + 18 \, x^{2} + 9 \, x\right )} e^{4} + 63 \, x\right )}}{x^{2} + 6 \, x + 9}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx=e^{\frac {2 \cdot \left (3 x^{5} + 30 x^{4} + 111 x^{3} + 198 x^{2} + 189 x + \left (48 x^{3} + 288 x^{2} + 432 x\right ) e^{8} + \left (24 x^{4} + 192 x^{3} + 432 x^{2} + 216 x\right ) e^{4}\right )}{x^{2} + 6 x + 9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (22) = 44\).
Time = 0.82 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx=e^{\left (6 \, x^{3} + 48 \, x^{2} e^{4} + 24 \, x^{2} + 96 \, x e^{8} + 96 \, x e^{4} + 24 \, x + \frac {432 \, e^{4}}{x + 3} - \frac {162}{x^{2} + 6 \, x + 9} - \frac {54}{x + 3} - 144 \, e^{4} + 36\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.55 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx=e^{\left (\frac {6 \, {\left (x^{5} + 8 \, x^{4} e^{4} + 10 \, x^{4} + 16 \, x^{3} e^{8} + 64 \, x^{3} e^{4} + 37 \, x^{3} + 96 \, x^{2} e^{8} + 144 \, x^{2} e^{4} + 66 \, x^{2} + 144 \, x e^{8} + 72 \, x e^{4} + 63 \, x\right )}}{x^{2} + 6 \, x + 9}\right )} \]
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Time = 8.57 (sec) , antiderivative size = 201, normalized size of antiderivative = 8.38 \[ \int \frac {e^{\frac {2 \left (189 x+198 x^2+111 x^3+30 x^4+3 x^5+e^8 \left (432 x+288 x^2+48 x^3\right )+e^4 \left (216 x+432 x^2+192 x^3+24 x^4\right )\right )}{9+6 x+x^2}} \left (1134+1998 x+1998 x^2+942 x^3+210 x^4+18 x^5+e^8 \left (2592+2592 x+864 x^2+96 x^3\right )+e^4 \left (1296+4752 x+3456 x^2+960 x^3+96 x^4\right )\right )}{27+27 x+9 x^2+x^3} \, dx={\mathrm {e}}^{\frac {6\,x^5}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {60\,x^4}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {222\,x^3}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {396\,x^2}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {432\,x\,{\mathrm {e}}^4}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {864\,x\,{\mathrm {e}}^8}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {48\,x^4\,{\mathrm {e}}^4}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {96\,x^3\,{\mathrm {e}}^8}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {384\,x^3\,{\mathrm {e}}^4}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {576\,x^2\,{\mathrm {e}}^8}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {864\,x^2\,{\mathrm {e}}^4}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {378\,x}{x^2+6\,x+9}} \]
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