Integrand size = 247, antiderivative size = 29 \[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=\frac {e^{e^{-2 x} x}}{\left (-2+\left (1-e^2\right ) \left (e^x-x\right )\right )^2} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(232\) vs. \(2(29)=58\).
Time = 3.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 8.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.004, Rules used = {2326} \[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=-\frac {e^{e^{-2 x} x} \left (-2 x^2-e^2 \left (x-2 x^2\right )-3 x-e^x \left (-e^2 (1-2 x)-2 x+1\right )+2\right )}{\left (e^{-2 x}-2 e^{-2 x} x\right ) \left (3 e^{3 x} \left (-e^6 x^2+x^2+e^4 \left (3 x^2+4 x\right )-e^2 \left (3 x^2+8 x+4\right )+4 x+4\right )-e^{2 x} \left (-e^6 x^3+x^3+6 x^2+3 e^4 \left (x^3+2 x^2\right )-3 e^2 \left (x^3+4 x^2+4 x\right )+12 x+8\right )-3 e^{4 x} \left (-e^6 x+x+e^4 (3 x+2)-e^2 (3 x+4)+2\right )+\left (1-e^2\right )^3 e^{5 x}\right )} \]
[In]
[Out]
Rule 2326
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{e^{-2 x} x} \left (2-e^x \left (1-e^2 (1-2 x)-2 x\right )-3 x-2 x^2-e^2 \left (x-2 x^2\right )\right )}{\left (e^{-2 x}-2 e^{-2 x} x\right ) \left (e^{5 x} \left (1-e^2\right )^3-3 e^{4 x} \left (2+x-e^6 x+e^4 (2+3 x)-e^2 (4+3 x)\right )+3 e^{3 x} \left (4+4 x+x^2-e^6 x^2+e^4 \left (4 x+3 x^2\right )-e^2 \left (4+8 x+3 x^2\right )\right )-e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+3 e^4 \left (2 x^2+x^3\right )-3 e^2 \left (4 x+4 x^2+x^3\right )\right )\right )} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=\frac {e^{e^{-2 x} x}}{\left (2-e^x+e^{2+x}+x-e^2 x\right )^2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(25)=50\).
Time = 5.68 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{x \,{\mathrm e}^{-2 x}}}{x^{2} {\mathrm e}^{4}-2 x \,{\mathrm e}^{4} {\mathrm e}^{x}+{\mathrm e}^{4} {\mathrm e}^{2 x}-2 x^{2} {\mathrm e}^{2}+4 x \,{\mathrm e}^{2} {\mathrm e}^{x}-2 \,{\mathrm e}^{2 x} {\mathrm e}^{2}-4 \,{\mathrm e}^{2} x +4 \,{\mathrm e}^{2} {\mathrm e}^{x}+x^{2}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}+4 x -4 \,{\mathrm e}^{x}+4}\) | \(91\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=\frac {e^{\left (x e^{\left (-2 \, x\right )}\right )}}{x^{2} e^{4} + x^{2} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{2} + {\left (e^{4} - 2 \, e^{2} + 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x e^{4} - 2 \, {\left (x + 1\right )} e^{2} + x + 2\right )} e^{x} + 4 \, x + 4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (20) = 40\).
Time = 1.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.52 \[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=\frac {e^{x e^{- 2 x}}}{- 2 x^{2} e^{2} + x^{2} + x^{2} e^{4} - 2 x e^{4} e^{x} - 2 x e^{x} + 4 x e^{2} e^{x} - 4 x e^{2} + 4 x - 2 e^{2} e^{2 x} + e^{2 x} + e^{4} e^{2 x} - 4 e^{x} + 4 e^{2} e^{x} + 4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).
Time = 0.66 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=\frac {e^{\left (x e^{\left (-2 \, x\right )}\right )}}{x^{2} {\left (e^{4} - 2 \, e^{2} + 1\right )} - 4 \, x {\left (e^{2} - 1\right )} + {\left (e^{4} - 2 \, e^{2} + 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x {\left (e^{4} - 2 \, e^{2} + 1\right )} - 2 \, e^{2} + 2\right )} e^{x} + 4} \]
[In]
[Out]
\[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=\int { -\frac {{\left (2 \, x^{2} - {\left (2 \, x^{2} - x\right )} e^{2} + 2 \, {\left (e^{2} - 1\right )} e^{\left (3 \, x\right )} - 2 \, {\left (e^{2} - 1\right )} e^{\left (2 \, x\right )} + {\left ({\left (2 \, x - 1\right )} e^{2} - 2 \, x + 1\right )} e^{x} + 3 \, x - 2\right )} e^{\left (x e^{\left (-2 \, x\right )}\right )}}{{\left (e^{6} - 3 \, e^{4} + 3 \, e^{2} - 1\right )} e^{\left (5 \, x\right )} - 3 \, {\left (x e^{6} - {\left (3 \, x + 2\right )} e^{4} + {\left (3 \, x + 4\right )} e^{2} - x - 2\right )} e^{\left (4 \, x\right )} + 3 \, {\left (x^{2} e^{6} - x^{2} - {\left (3 \, x^{2} + 4 \, x\right )} e^{4} + {\left (3 \, x^{2} + 8 \, x + 4\right )} e^{2} - 4 \, x - 4\right )} e^{\left (3 \, x\right )} - {\left (x^{3} e^{6} - x^{3} - 6 \, x^{2} - 3 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{4} + 3 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{2} - 12 \, x - 8\right )} e^{\left (2 \, x\right )}} \,d x } \]
[In]
[Out]
Time = 9.60 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.31 \[ \int \frac {e^{e^{-2 x} x} \left (2+e^{3 x} \left (2-2 e^2\right )+e^{2 x} \left (-2+2 e^2\right )-3 x-2 x^2+e^x \left (-1+e^2 (1-2 x)+2 x\right )+e^2 \left (-x+2 x^2\right )\right )}{e^{5 x} \left (-1+3 e^2-3 e^4+e^6\right )+e^{4 x} \left (6+e^2 (-12-9 x)+3 x-3 e^6 x+e^4 (6+9 x)\right )+e^{3 x} \left (-12-12 x-3 x^2+3 e^6 x^2+e^4 \left (-12 x-9 x^2\right )+e^2 \left (12+24 x+9 x^2\right )\right )+e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+e^2 \left (-12 x-12 x^2-3 x^3\right )+e^4 \left (6 x^2+3 x^3\right )\right )} \, dx=\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^{-2\,x}}}{{\left ({\mathrm {e}}^2-1\right )}^2\,\left ({\mathrm {e}}^{2\,x}+\frac {4}{{\left ({\mathrm {e}}^2-1\right )}^2}-2\,x\,{\mathrm {e}}^x+x^2-\frac {x\,\left (4\,{\mathrm {e}}^2-4\right )}{{\left ({\mathrm {e}}^2-1\right )}^2}+\frac {{\mathrm {e}}^x\,\left (4\,{\mathrm {e}}^2-4\right )}{{\left ({\mathrm {e}}^2-1\right )}^2}\right )} \]
[In]
[Out]