Integrand size = 124, antiderivative size = 28 \[ \int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} \left (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} \left (-4 x^2+2 x^3\right )\right )\right )}{x^2} \, dx=\frac {e^{e^{4+x^2 \left (e^{e^4}+(4+x)^4\right )}} (-2+x)}{x} \]
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} \left (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} \left (-4 x^2+2 x^3\right )\right )\right )}{x^2} \, dx=e^{e^{4+\left (256+e^{e^4}\right ) x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (1-\frac {2}{x}\right ) \]
Integrate[(E^E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6) *(2 + E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(-1024 *x^2 - 1024*x^3 + 224*x^5 + 68*x^6 + 6*x^7 + E^E^4*(-4*x^2 + 2*x^3))))/x^2 ,x]
Leaf count is larger than twice the leaf count of optimal. \(119\) vs. \(2(28)=56\).
Time = 0.58 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.25, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2726}
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.
\(\displaystyle \int \frac {\exp \left (\exp \left (x^6+16 x^5+96 x^4+256 x^3+e^{e^4} x^2+256 x^2+4\right )\right ) \left (\left (6 x^7+68 x^6+224 x^5-1024 x^3-1024 x^2+e^{e^4} \left (2 x^3-4 x^2\right )\right ) \exp \left (x^6+16 x^5+96 x^4+256 x^3+e^{e^4} x^2+256 x^2+4\right )+2\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -\frac {\left (-3 x^7-34 x^6-112 x^5+512 x^3+512 x^2+e^{e^4} \left (2 x^2-x^3\right )\right ) \exp \left (\exp \left (x^6+16 x^5+96 x^4+256 x^3+e^{e^4} x^2+256 x^2+4\right )\right )}{x^2 \left (3 x^5+40 x^4+192 x^3+384 x^2+e^{e^4} x+256 x\right )}\) |
Int[(E^E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(2 + E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(-1024*x^2 - 1024*x^3 + 224*x^5 + 68*x^6 + 6*x^7 + E^E^4*(-4*x^2 + 2*x^3))))/x^2,x]
-((E^E^(4 + 256*x^2 + E^E^4*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6)*(512*x^ 2 + 512*x^3 - 112*x^5 - 34*x^6 - 3*x^7 + E^E^4*(2*x^2 - x^3)))/(x^2*(256*x + E^E^4*x + 384*x^2 + 192*x^3 + 40*x^4 + 3*x^5)))
3.5.50.3.1 Defintions of rubi rules used
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 6.53 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50
method | result | size |
risch | \(\frac {\left (-2+x \right ) {\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}}{x}\) | \(42\) |
norman | \(\frac {x \,{\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}-2 \,{\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}}{x}\) | \(78\) |
parallelrisch | \(\frac {x \,{\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}-2 \,{\mathrm e}^{{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+x^{6}+16 x^{5}+96 x^{4}+256 x^{3}+256 x^{2}+4}}}{x}\) | \(78\) |
int((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x^2)*ex p(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4)+2)*exp(exp(x^2*exp( exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} \left (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} \left (-4 x^2+2 x^3\right )\right )\right )}{x^2} \, dx=\text {Timed out} \]
integrate((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x ^2)*exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4)+2)*exp(exp(x^ 2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x, algorithm=\
Time = 0.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} \left (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} \left (-4 x^2+2 x^3\right )\right )\right )}{x^2} \, dx=\frac {\left (x - 2\right ) e^{e^{x^{6} + 16 x^{5} + 96 x^{4} + 256 x^{3} + 256 x^{2} + x^{2} e^{e^{4}} + 4}}}{x} \]
integrate((((2*x**3-4*x**2)*exp(exp(4))+6*x**7+68*x**6+224*x**5-1024*x**3- 1024*x**2)*exp(x**2*exp(exp(4))+x**6+16*x**5+96*x**4+256*x**3+256*x**2+4)+ 2)*exp(exp(x**2*exp(exp(4))+x**6+16*x**5+96*x**4+256*x**3+256*x**2+4))/x** 2,x)
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} \left (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} \left (-4 x^2+2 x^3\right )\right )\right )}{x^2} \, dx=\frac {{\left (x - 2\right )} e^{\left (e^{\left (x^{6} + 16 \, x^{5} + 96 \, x^{4} + 256 \, x^{3} + x^{2} e^{\left (e^{4}\right )} + 256 \, x^{2} + 4\right )}\right )}}{x} \]
integrate((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x ^2)*exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4)+2)*exp(exp(x^ 2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x, algorithm=\
\[ \int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} \left (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} \left (-4 x^2+2 x^3\right )\right )\right )}{x^2} \, dx=\int { \frac {2 \, {\left ({\left (3 \, x^{7} + 34 \, x^{6} + 112 \, x^{5} - 512 \, x^{3} - 512 \, x^{2} + {\left (x^{3} - 2 \, x^{2}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (x^{6} + 16 \, x^{5} + 96 \, x^{4} + 256 \, x^{3} + x^{2} e^{\left (e^{4}\right )} + 256 \, x^{2} + 4\right )} + 1\right )} e^{\left (e^{\left (x^{6} + 16 \, x^{5} + 96 \, x^{4} + 256 \, x^{3} + x^{2} e^{\left (e^{4}\right )} + 256 \, x^{2} + 4\right )}\right )}}{x^{2}} \,d x } \]
integrate((((2*x^3-4*x^2)*exp(exp(4))+6*x^7+68*x^6+224*x^5-1024*x^3-1024*x ^2)*exp(x^2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4)+2)*exp(exp(x^ 2*exp(exp(4))+x^6+16*x^5+96*x^4+256*x^3+256*x^2+4))/x^2,x, algorithm=\
integrate(2*((3*x^7 + 34*x^6 + 112*x^5 - 512*x^3 - 512*x^2 + (x^3 - 2*x^2) *e^(e^4))*e^(x^6 + 16*x^5 + 96*x^4 + 256*x^3 + x^2*e^(e^4) + 256*x^2 + 4) + 1)*e^(e^(x^6 + 16*x^5 + 96*x^4 + 256*x^3 + x^2*e^(e^4) + 256*x^2 + 4))/x ^2, x)
Time = 9.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {e^{e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6}} \left (2+e^{4+256 x^2+e^{e^4} x^2+256 x^3+96 x^4+16 x^5+x^6} \left (-1024 x^2-1024 x^3+224 x^5+68 x^6+6 x^7+e^{e^4} \left (-4 x^2+2 x^3\right )\right )\right )}{x^2} \, dx=\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^6}\,{\mathrm {e}}^4\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,{\mathrm {e}}^{16\,x^5}\,{\mathrm {e}}^{96\,x^4}\,{\mathrm {e}}^{256\,x^2}\,{\mathrm {e}}^{256\,x^3}}\,\left (x-2\right )}{x} \]
int(-(exp(exp(x^2*exp(exp(4)) + 256*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6 + 4))*(exp(x^2*exp(exp(4)) + 256*x^2 + 256*x^3 + 96*x^4 + 16*x^5 + x^6 + 4 )*(1024*x^2 + 1024*x^3 - 224*x^5 - 68*x^6 - 6*x^7 + exp(exp(4))*(4*x^2 - 2 *x^3)) - 2))/x^2,x)