Integrand size = 161, antiderivative size = 27 \[ \int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx=e^{\frac {\left (-2+x+x^2\right )^2}{\left (1-e^{e^8 x}\right )^2 x}} \]
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx=e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \]
Integrate[(E^((4 - 4*x - 3*x^2 + 2*x^3 + x^4)/(x - 2*E^(E^8*x)*x + E^(2*E^ 8*x)*x))*(4 + 3*x^2 - 4*x^3 - 3*x^4 + E^(E^8*x)*(-4 - 3*x^2 + 4*x^3 + 3*x^ 4 + E^8*(-8*x + 8*x^2 + 6*x^3 - 4*x^4 - 2*x^5))))/(-x^2 + 3*E^(E^8*x)*x^2 - 3*E^(2*E^8*x)*x^2 + E^(3*E^8*x)*x^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (-3 x^4-4 x^3+3 x^2+e^{e^8 x} \left (3 x^4+4 x^3-3 x^2+e^8 \left (-2 x^5-4 x^4+6 x^3+8 x^2-8 x\right )-4\right )+4\right ) \exp \left (\frac {x^4+2 x^3-3 x^2-4 x+4}{-2 e^{e^8 x} x+e^{2 e^8 x} x+x}\right )}{3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2-x^2} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (e^{e^8 x}-1\right )^2 x}} \left (-x^2-x+2\right ) \left (-3 x^2-2 e^{e^8 x+8} \left (x^2+x-2\right ) x+e^{e^8 x} \left (3 x^2+x+2\right )-x-2\right )}{\left (1-e^{e^8 x}\right )^3 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (e^{e^8 x}-1\right )^2 x}} \left (-x^2-x+2\right ) \left (2 e^8 x^3-\left (3-2 e^8\right ) x^2-\left (1+4 e^8\right ) x-2\right )}{\left (1-e^{e^8 x}\right )^2 x^2}-\frac {2 e^{\frac {\left (x^2+x-2\right )^2}{\left (e^{e^8 x}-1\right )^2 x}+8} \left (x^2+x-2\right )^2}{\left (e^{e^8 x}-1\right )^3 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}+8}}{\left (-1+e^{e^8 x}\right )^3}dx-\left (3-8 e^8\right ) \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2}dx-4 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x^2}dx-8 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}+8}}{\left (-1+e^{e^8 x}\right )^3 x}dx-8 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}+8}}{\left (-1+e^{e^8 x}\right )^2 x}dx+6 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}+8} x}{\left (-1+e^{e^8 x}\right )^3}dx+2 \left (2+3 e^8\right ) \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^2}dx-4 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}+8} x^2}{\left (-1+e^{e^8 x}\right )^3}dx+\left (3-4 e^8\right ) \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^2}dx-2 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}+8} x^3}{\left (-1+e^{e^8 x}\right )^3}dx-2 \int \frac {e^{\frac {\left (x^2+x-2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}+8} x^3}{\left (-1+e^{e^8 x}\right )^2}dx\) |
Int[(E^((4 - 4*x - 3*x^2 + 2*x^3 + x^4)/(x - 2*E^(E^8*x)*x + E^(2*E^8*x)*x ))*(4 + 3*x^2 - 4*x^3 - 3*x^4 + E^(E^8*x)*(-4 - 3*x^2 + 4*x^3 + 3*x^4 + E^ 8*(-8*x + 8*x^2 + 6*x^3 - 4*x^4 - 2*x^5))))/(-x^2 + 3*E^(E^8*x)*x^2 - 3*E^ (2*E^8*x)*x^2 + E^(3*E^8*x)*x^2),x]
3.30.12.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 8.72 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22
method | result | size |
risch | \({\mathrm e}^{\frac {\left (2+x \right )^{2} \left (-1+x \right )^{2}}{x \left ({\mathrm e}^{2 x \,{\mathrm e}^{8}}-2 \,{\mathrm e}^{x \,{\mathrm e}^{8}}+1\right )}}\) | \(33\) |
parallelrisch | \({\mathrm e}^{\frac {x^{4}+2 x^{3}-3 x^{2}-4 x +4}{x \left ({\mathrm e}^{2 x \,{\mathrm e}^{8}}-2 \,{\mathrm e}^{x \,{\mathrm e}^{8}}+1\right )}}\) | \(46\) |
int((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*exp(x*e xp(4)^2)-3*x^4-4*x^3+3*x^2+4)*exp((x^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*exp(4)^ 2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*exp(4)^2)^ 2+3*x^2*exp(x*exp(4)^2)-x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx=e^{\left (\frac {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 4 \, x + 4}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x}\right )} \]
integrate((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*e xp(x*exp(4)^2)-3*x^4-4*x^3+3*x^2+4)*exp((x^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*e xp(4)^2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*exp( 4)^2)^2+3*x^2*exp(x*exp(4)^2)-x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.42 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx=e^{\frac {x^{4} + 2 x^{3} - 3 x^{2} - 4 x + 4}{x e^{2 x e^{8}} - 2 x e^{x e^{8}} + x}} \]
integrate((((-2*x**5-4*x**4+6*x**3+8*x**2-8*x)*exp(4)**2+3*x**4+4*x**3-3*x **2-4)*exp(x*exp(4)**2)-3*x**4-4*x**3+3*x**2+4)*exp((x**4+2*x**3-3*x**2-4* x+4)/(x*exp(x*exp(4)**2)**2-2*x*exp(x*exp(4)**2)+x))/(x**2*exp(x*exp(4)**2 )**3-3*x**2*exp(x*exp(4)**2)**2+3*x**2*exp(x*exp(4)**2)-x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (22) = 44\).
Time = 0.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.93 \[ \int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx=e^{\left (\frac {x^{3}}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} + \frac {2 \, x^{2}}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} - \frac {3 \, x}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} + \frac {4}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x} - \frac {4}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1}\right )} \]
integrate((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*e xp(x*exp(4)^2)-3*x^4-4*x^3+3*x^2+4)*exp((x^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*e xp(4)^2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*exp( 4)^2)^2+3*x^2*exp(x*exp(4)^2)-x^2),x, algorithm=\
e^(x^3/(e^(2*x*e^8) - 2*e^(x*e^8) + 1) + 2*x^2/(e^(2*x*e^8) - 2*e^(x*e^8) + 1) - 3*x/(e^(2*x*e^8) - 2*e^(x*e^8) + 1) + 4/(x*e^(2*x*e^8) - 2*x*e^(x*e ^8) + x) - 4/(e^(2*x*e^8) - 2*e^(x*e^8) + 1))
Exception generated. \[ \int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx=\text {Exception raised: TypeError} \]
integrate((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*e xp(x*exp(4)^2)-3*x^4-4*x^3+3*x^2+4)*exp((x^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*e xp(4)^2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*exp( 4)^2)^2+3*x^2*exp(x*exp(4)^2)-x^2),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-96,[2,24,6]%%%}+%%%{-704,[2,23,6]%%%}+%%%{288,[2,23,5]%%% }+%%%{-96
Time = 9.49 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.07 \[ \int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx={\mathrm {e}}^{-\frac {4}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{-\frac {3\,x}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{\frac {4}{x-2\,x\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}}}\,{\mathrm {e}}^{\frac {x^3}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{\frac {2\,x^2}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}} \]
int((exp((2*x^3 - 3*x^2 - 4*x + x^4 + 4)/(x - 2*x*exp(x*exp(8)) + x*exp(2* x*exp(8))))*(4*x^3 - 3*x^2 + 3*x^4 + exp(x*exp(8))*(exp(8)*(8*x - 8*x^2 - 6*x^3 + 4*x^4 + 2*x^5) + 3*x^2 - 4*x^3 - 3*x^4 + 4) - 4))/(x^2 - 3*x^2*exp (x*exp(8)) + 3*x^2*exp(2*x*exp(8)) - x^2*exp(3*x*exp(8))),x)