Integrand size = 332, antiderivative size = 29 \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx=e^{2 \left (\frac {x}{-2+e^2-e^4}-\frac {6 x}{-1+x}\right )^2} \]
[Out]
Time = 2.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6, 6820, 12, 6838} \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx=\exp \left (\frac {2 x^2 \left (x+6 e^4-6 e^2+11\right )^2}{\left (2-e^2+e^4\right )^2 (1-x)^2}\right ) \]
[In]
[Out]
Rule 6
Rule 12
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}\right ) \left (-144 e^8 x+\left (-484-144 e^4\right ) x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx \\ & = \int \frac {\exp \left (\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}\right ) \left (\left (-484-144 e^4-144 e^8\right ) x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx \\ & = \int \frac {\exp \left (\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}\right ) \left (\left (-484-144 e^4-144 e^8\right ) x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+\left (e^4+e^8\right ) \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx \\ & = \int \frac {4 \exp \left (\frac {2 x^2 \left (11-6 e^2+6 e^4+x\right )^2}{\left (2-e^2+e^4\right )^2 (-1+x)^2}\right ) x \left (\left (11-6 e^2+6 e^4\right )^2+3 \left (11-6 e^2+6 e^4\right ) x-3 \left (3-2 e^2+2 e^4\right ) x^2-x^3\right )}{\left (2-e^2+e^4\right )^2 (1-x)^3} \, dx \\ & = \frac {4 \int \frac {\exp \left (\frac {2 x^2 \left (11-6 e^2+6 e^4+x\right )^2}{\left (2-e^2+e^4\right )^2 (-1+x)^2}\right ) x \left (\left (11-6 e^2+6 e^4\right )^2+3 \left (11-6 e^2+6 e^4\right ) x-3 \left (3-2 e^2+2 e^4\right ) x^2-x^3\right )}{(1-x)^3} \, dx}{\left (2-e^2+e^4\right )^2} \\ & = \exp \left (\frac {2 x^2 \left (11-6 e^2+6 e^4+x\right )^2}{\left (2-e^2+e^4\right )^2 (1-x)^2}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx=e^{\frac {2 x^2 \left (11-6 e^2+6 e^4+x\right )^2}{\left (2-e^2+e^4\right )^2 (-1+x)^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(26)=52\).
Time = 109.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.34
method | result | size |
risch | \({\mathrm e}^{\frac {2 x^{2} \left (12 \,{\mathrm e}^{2} x -12 x \,{\mathrm e}^{4}-x^{2}+132 \,{\mathrm e}^{2}-168 \,{\mathrm e}^{4}-36 \,{\mathrm e}^{8}+72 \,{\mathrm e}^{6}-22 x -121\right )}{\left (-1+x \right )^{2} \left (4 \,{\mathrm e}^{2}-5 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{6}-{\mathrm e}^{8}-4\right )}}\) | \(68\) |
gosper | \({\mathrm e}^{\frac {2 x^{2} \left (168 \,{\mathrm e}^{4}-72 \,{\mathrm e}^{6}-12 \,{\mathrm e}^{2} x +36 \,{\mathrm e}^{8}+12 x \,{\mathrm e}^{4}+x^{2}-132 \,{\mathrm e}^{2}+22 x +121\right )}{-2 x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{2}+5 x^{2} {\mathrm e}^{4}+x^{2} {\mathrm e}^{8}-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{2} x -10 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{8}+4 x^{2}-4 \,{\mathrm e}^{2}+5 \,{\mathrm e}^{4}+{\mathrm e}^{8}-8 x +4}}\) | \(156\) |
norman | \(\frac {\left ({\mathrm e}^{2}-2-{\mathrm e}^{4}\right ) {\mathrm e}^{\frac {72 x^{2} {\mathrm e}^{8}+2 \left (-72 x^{2} {\mathrm e}^{2}+12 x^{3}+132 x^{2}\right ) {\mathrm e}^{4}+72 x^{2} {\mathrm e}^{4}+2 \left (-12 x^{3}-132 x^{2}\right ) {\mathrm e}^{2}+2 x^{4}+44 x^{3}+242 x^{2}}{\left (x^{2}-2 x +1\right ) {\mathrm e}^{8}+\left (\left (-2 x^{2}+4 x -2\right ) {\mathrm e}^{2}+4 x^{2}-8 x +4\right ) {\mathrm e}^{4}+\left (x^{2}-2 x +1\right ) {\mathrm e}^{4}+\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{2}+4 x^{2}-8 x +4}}+\left (-2 \,{\mathrm e}^{2}+2 \,{\mathrm e}^{4}+4\right ) x \,{\mathrm e}^{\frac {72 x^{2} {\mathrm e}^{8}+2 \left (-72 x^{2} {\mathrm e}^{2}+12 x^{3}+132 x^{2}\right ) {\mathrm e}^{4}+72 x^{2} {\mathrm e}^{4}+2 \left (-12 x^{3}-132 x^{2}\right ) {\mathrm e}^{2}+2 x^{4}+44 x^{3}+242 x^{2}}{\left (x^{2}-2 x +1\right ) {\mathrm e}^{8}+\left (\left (-2 x^{2}+4 x -2\right ) {\mathrm e}^{2}+4 x^{2}-8 x +4\right ) {\mathrm e}^{4}+\left (x^{2}-2 x +1\right ) {\mathrm e}^{4}+\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{2}+4 x^{2}-8 x +4}}+\left ({\mathrm e}^{2}-2-{\mathrm e}^{4}\right ) x^{2} {\mathrm e}^{\frac {72 x^{2} {\mathrm e}^{8}+2 \left (-72 x^{2} {\mathrm e}^{2}+12 x^{3}+132 x^{2}\right ) {\mathrm e}^{4}+72 x^{2} {\mathrm e}^{4}+2 \left (-12 x^{3}-132 x^{2}\right ) {\mathrm e}^{2}+2 x^{4}+44 x^{3}+242 x^{2}}{\left (x^{2}-2 x +1\right ) {\mathrm e}^{8}+\left (\left (-2 x^{2}+4 x -2\right ) {\mathrm e}^{2}+4 x^{2}-8 x +4\right ) {\mathrm e}^{4}+\left (x^{2}-2 x +1\right ) {\mathrm e}^{4}+\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{2}+4 x^{2}-8 x +4}}}{\left (-1+x \right )^{2} \left ({\mathrm e}^{2}-2-{\mathrm e}^{4}\right )}\) | \(495\) |
parallelrisch | \(\frac {5 \,{\mathrm e}^{4} {\mathrm e}^{\frac {2 x^{2} \left (168 \,{\mathrm e}^{4}-72 \,{\mathrm e}^{6}-12 \,{\mathrm e}^{2} x +36 \,{\mathrm e}^{8}+12 x \,{\mathrm e}^{4}+x^{2}-132 \,{\mathrm e}^{2}+22 x +121\right )}{-2 x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{2}+5 x^{2} {\mathrm e}^{4}+x^{2} {\mathrm e}^{8}-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{2} x -10 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{8}+4 x^{2}-4 \,{\mathrm e}^{2}+5 \,{\mathrm e}^{4}+{\mathrm e}^{8}-8 x +4}}-2 \,{\mathrm e}^{2} {\mathrm e}^{4} {\mathrm e}^{\frac {2 x^{2} \left (168 \,{\mathrm e}^{4}-72 \,{\mathrm e}^{6}-12 \,{\mathrm e}^{2} x +36 \,{\mathrm e}^{8}+12 x \,{\mathrm e}^{4}+x^{2}-132 \,{\mathrm e}^{2}+22 x +121\right )}{-2 x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{2}+5 x^{2} {\mathrm e}^{4}+x^{2} {\mathrm e}^{8}-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{2} x -10 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{8}+4 x^{2}-4 \,{\mathrm e}^{2}+5 \,{\mathrm e}^{4}+{\mathrm e}^{8}-8 x +4}}+{\mathrm e}^{8} {\mathrm e}^{\frac {2 x^{2} \left (168 \,{\mathrm e}^{4}-72 \,{\mathrm e}^{6}-12 \,{\mathrm e}^{2} x +36 \,{\mathrm e}^{8}+12 x \,{\mathrm e}^{4}+x^{2}-132 \,{\mathrm e}^{2}+22 x +121\right )}{-2 x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{2}+5 x^{2} {\mathrm e}^{4}+x^{2} {\mathrm e}^{8}-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{2} x -10 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{8}+4 x^{2}-4 \,{\mathrm e}^{2}+5 \,{\mathrm e}^{4}+{\mathrm e}^{8}-8 x +4}}-4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {2 x^{2} \left (168 \,{\mathrm e}^{4}-72 \,{\mathrm e}^{6}-12 \,{\mathrm e}^{2} x +36 \,{\mathrm e}^{8}+12 x \,{\mathrm e}^{4}+x^{2}-132 \,{\mathrm e}^{2}+22 x +121\right )}{-2 x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{2}+5 x^{2} {\mathrm e}^{4}+x^{2} {\mathrm e}^{8}-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{2} x -10 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{8}+4 x^{2}-4 \,{\mathrm e}^{2}+5 \,{\mathrm e}^{4}+{\mathrm e}^{8}-8 x +4}}+4 \,{\mathrm e}^{\frac {2 x^{2} \left (168 \,{\mathrm e}^{4}-72 \,{\mathrm e}^{6}-12 \,{\mathrm e}^{2} x +36 \,{\mathrm e}^{8}+12 x \,{\mathrm e}^{4}+x^{2}-132 \,{\mathrm e}^{2}+22 x +121\right )}{-2 x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{2}+5 x^{2} {\mathrm e}^{4}+x^{2} {\mathrm e}^{8}-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{2} x -10 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{8}+4 x^{2}-4 \,{\mathrm e}^{2}+5 \,{\mathrm e}^{4}+{\mathrm e}^{8}-8 x +4}}}{-2 \,{\mathrm e}^{2} {\mathrm e}^{4}-4 \,{\mathrm e}^{2}+5 \,{\mathrm e}^{4}+{\mathrm e}^{8}+4}\) | \(985\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.00 \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx=e^{\left (\frac {2 \, {\left (x^{4} + 22 \, x^{3} + 36 \, x^{2} e^{8} - 72 \, x^{2} e^{6} + 121 \, x^{2} + 12 \, {\left (x^{3} + 14 \, x^{2}\right )} e^{4} - 12 \, {\left (x^{3} + 11 \, x^{2}\right )} e^{2}\right )}}{4 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{8} - 2 \, {\left (x^{2} - 2 \, x + 1\right )} e^{6} + 5 \, {\left (x^{2} - 2 \, x + 1\right )} e^{4} - 4 \, {\left (x^{2} - 2 \, x + 1\right )} e^{2} - 8 \, x + 4}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (20) = 40\).
Time = 1.91 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.10 \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx=e^{\frac {2 \left (x^{4} + 22 x^{3} + 121 x^{2} + 36 x^{2} e^{4} + 36 x^{2} e^{8} + \left (- 12 x^{3} - 132 x^{2}\right ) e^{2} + \left (12 x^{3} - 72 x^{2} e^{2} + 132 x^{2}\right ) e^{4}\right )}{4 x^{2} - 8 x + \left (- 4 x^{2} + 8 x - 4\right ) e^{2} + \left (x^{2} - 2 x + 1\right ) e^{4} + \left (x^{2} - 2 x + 1\right ) e^{8} + \left (4 x^{2} - 8 x + \left (- 2 x^{2} + 4 x - 2\right ) e^{2} + 4\right ) e^{4} + 4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (26) = 52\).
Time = 3.35 (sec) , antiderivative size = 711, normalized size of antiderivative = 24.52 \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx=e^{\left (\frac {2 \, x^{2}}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {24 \, x e^{4}}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} - \frac {24 \, x e^{2}}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {48 \, x}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {72 \, e^{8}}{x^{2} {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - 2 \, x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} + e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {144 \, e^{8}}{x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - e^{8} + 2 \, e^{6} - 5 \, e^{4} + 4 \, e^{2} - 4} + \frac {72 \, e^{8}}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} - \frac {144 \, e^{6}}{x^{2} {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - 2 \, x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} + e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} - \frac {288 \, e^{6}}{x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - e^{8} + 2 \, e^{6} - 5 \, e^{4} + 4 \, e^{2} - 4} - \frac {144 \, e^{6}}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {360 \, e^{4}}{x^{2} {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - 2 \, x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} + e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {744 \, e^{4}}{x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - e^{8} + 2 \, e^{6} - 5 \, e^{4} + 4 \, e^{2} - 4} + \frac {384 \, e^{4}}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} - \frac {288 \, e^{2}}{x^{2} {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - 2 \, x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} + e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} - \frac {600 \, e^{2}}{x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - e^{8} + 2 \, e^{6} - 5 \, e^{4} + 4 \, e^{2} - 4} - \frac {312 \, e^{2}}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {288}{x^{2} {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - 2 \, x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} + e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4} + \frac {624}{x {\left (e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4\right )} - e^{8} + 2 \, e^{6} - 5 \, e^{4} + 4 \, e^{2} - 4} + \frac {336}{e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (26) = 52\).
Time = 4.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.55 \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx=e^{\left (\frac {2 \, {\left (x^{4} + 12 \, x^{3} e^{4} - 12 \, x^{3} e^{2} + 22 \, x^{3} + 36 \, x^{2} e^{8} - 72 \, x^{2} e^{6} + 168 \, x^{2} e^{4} - 132 \, x^{2} e^{2} + 121 \, x^{2}\right )}}{x^{2} e^{8} - 2 \, x^{2} e^{6} + 5 \, x^{2} e^{4} - 4 \, x^{2} e^{2} + 4 \, x^{2} - 2 \, x e^{8} + 4 \, x e^{6} - 10 \, x e^{4} + 8 \, x e^{2} - 8 \, x + e^{8} - 2 \, e^{6} + 5 \, e^{4} - 4 \, e^{2} + 4}\right )} \]
[In]
[Out]
Time = 10.77 (sec) , antiderivative size = 724, normalized size of antiderivative = 24.97 \[ \int \frac {e^{\frac {2 \left (121 x^2+36 e^4 x^2+36 e^8 x^2+22 x^3+x^4+e^2 \left (-132 x^2-12 x^3\right )+e^4 \left (132 x^2-72 e^2 x^2+12 x^3\right )\right )}{4-8 x+4 x^2+e^2 \left (-4+8 x-4 x^2\right )+e^4 \left (1-2 x+x^2\right )+e^8 \left (1-2 x+x^2\right )+e^4 \left (4-8 x+4 x^2+e^2 \left (-2+4 x-2 x^2\right )\right )}} \left (-484 x-144 e^4 x-144 e^8 x-132 x^2+36 x^3+4 x^4+e^2 \left (528 x+72 x^2-24 x^3\right )+e^4 \left (-528 x+288 e^2 x-72 x^2+24 x^3\right )\right )}{-4+12 x-12 x^2+4 x^3+e^2 \left (4-12 x+12 x^2-4 x^3\right )+e^4 \left (-1+3 x-3 x^2+x^3\right )+e^8 \left (-1+3 x-3 x^2+x^3\right )+e^4 \left (-4+12 x-12 x^2+4 x^3+e^2 \left (2-6 x+6 x^2-2 x^3\right )\right )} \, dx={\mathrm {e}}^{\frac {2\,x^4}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{\frac {44\,x^3}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{\frac {242\,x^2}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{-\frac {24\,x^3\,{\mathrm {e}}^2}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{\frac {24\,x^3\,{\mathrm {e}}^4}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{\frac {72\,x^2\,{\mathrm {e}}^8}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{-\frac {144\,x^2\,{\mathrm {e}}^6}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{-\frac {264\,x^2\,{\mathrm {e}}^2}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}}\,{\mathrm {e}}^{\frac {336\,x^2\,{\mathrm {e}}^4}{5\,{\mathrm {e}}^4-4\,{\mathrm {e}}^2-8\,x-2\,{\mathrm {e}}^6+{\mathrm {e}}^8+8\,x\,{\mathrm {e}}^2-10\,x\,{\mathrm {e}}^4+4\,x\,{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^8-4\,x^2\,{\mathrm {e}}^2+5\,x^2\,{\mathrm {e}}^4-2\,x^2\,{\mathrm {e}}^6+x^2\,{\mathrm {e}}^8+4\,x^2+4}} \]
[In]
[Out]