Integrand size = 125, antiderivative size = 22 \[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=e^{\left (2 \left (-10+e^{3+e^{e^x} x}\right )+\log (x)\right )^2} \] Output:
exp((2*exp(x*exp(exp(x))+3)-20+ln(x))^2)
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=e^{4 \left (-10+e^{3+e^{e^x} x}\right )^2+\log ^2(x)} x^{-40+4 e^{3+e^{e^x} x}} \] Input:
Integrate[(E^(400 + 4*E^(6 + 2*E^E^x*x) + Log[x]^2 + E^(3 + E^E^x*x)*(-80 + 4*Log[x]))*(-40 + E^(6 + E^x + 2*E^E^x*x)*(8*x + 8*E^x*x^2) + 2*Log[x] + E^(3 + E^E^x*x)*(4 + E^E^x*(-80*x - 80*E^x*x^2 + (4*x + 4*E^x*x^2)*Log[x] ))))/x^41,x]
Output:
E^(4*(-10 + E^(3 + E^E^x*x))^2 + Log[x]^2)*x^(-40 + 4*E^(3 + E^E^x*x))
Leaf count is larger than twice the leaf count of optimal. \(209\) vs. \(2(22)=44\).
Time = 2.57 (sec) , antiderivative size = 209, normalized size of antiderivative = 9.50, 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.
\(\displaystyle \int \frac {\left (e^{2 e^{e^x} x+e^x+6} \left (8 e^x x^2+8 x\right )+e^{e^{e^x} x+3} \left (e^{e^x} \left (-80 e^x x^2+\left (4 e^x x^2+4 x\right ) \log (x)-80 x\right )+4\right )+2 \log (x)-40\right ) \exp \left (4 e^{2 e^{e^x} x+6}+\log ^2(x)+e^{e^{e^x} x+3} (4 \log (x)-80)+400\right )}{x^{41}} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {\left (4 e^{2 e^{e^x} x+e^x+6} \left (e^x x^2+x\right )+2 e^{e^{e^x} x+3} \left (1-e^{e^x} \left (20 e^x x^2-\left (e^x x^2+x\right ) \log (x)+20 x\right )\right )+\log (x)\right ) \exp \left (4 e^{2 e^{e^x} x+6}+\log ^2(x)-4 e^{e^{e^x} x+3} (20-\log (x))+400\right )}{x^{41} \left (4 e^{2 e^{e^x} x+6} \left (e^{x+e^x} x+e^{e^x}\right )+\frac {2 e^{e^{e^x} x+3}}{x}-2 e^{e^{e^x} x+3} \left (e^{x+e^x} x+e^{e^x}\right ) (20-\log (x))+\frac {\log (x)}{x}\right )}\) |
Input:
Int[(E^(400 + 4*E^(6 + 2*E^E^x*x) + Log[x]^2 + E^(3 + E^E^x*x)*(-80 + 4*Lo g[x]))*(-40 + E^(6 + E^x + 2*E^E^x*x)*(8*x + 8*E^x*x^2) + 2*Log[x] + E^(3 + E^E^x*x)*(4 + E^E^x*(-80*x - 80*E^x*x^2 + (4*x + 4*E^x*x^2)*Log[x]))))/x ^41,x]
Output:
(E^(400 + 4*E^(6 + 2*E^E^x*x) - 4*E^(3 + E^E^x*x)*(20 - Log[x]) + Log[x]^2 )*(4*E^(6 + E^x + 2*E^E^x*x)*(x + E^x*x^2) + Log[x] + 2*E^(3 + E^E^x*x)*(1 - E^E^x*(20*x + 20*E^x*x^2 - (x + E^x*x^2)*Log[x]))))/(x^41*((2*E^(3 + E^ E^x*x))/x + 4*E^(6 + 2*E^E^x*x)*(E^E^x + E^(E^x + x)*x) - 2*E^(3 + E^E^x*x )*(E^E^x + E^(E^x + x)*x)*(20 - Log[x]) + Log[x]/x))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(17)=34\).
Time = 7.90 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77
method | result | size |
parallelrisch | \({\mathrm e}^{4 \,{\mathrm e}^{2 x \,{\mathrm e}^{{\mathrm e}^{x}}+6}+\left (4 \ln \left (x \right )-80\right ) {\mathrm e}^{x \,{\mathrm e}^{{\mathrm e}^{x}}+3}+\ln \left (x \right )^{2}-40 \ln \left (x \right )+400}\) | \(39\) |
risch | \(\frac {x^{4 \,{\mathrm e}^{x \,{\mathrm e}^{{\mathrm e}^{x}}+3}} {\mathrm e}^{400+\ln \left (x \right )^{2}-80 \,{\mathrm e}^{x \,{\mathrm e}^{{\mathrm e}^{x}}+3}+4 \,{\mathrm e}^{2 x \,{\mathrm e}^{{\mathrm e}^{x}}+6}}}{x^{40}}\) | \(45\) |
Input:
int(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp(x)*x^2 +4*x)*ln(x)-80*exp(x)*x^2-80*x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3)+2*ln(x )-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*ln(x)-80)*exp(x*exp(exp(x))+3)+ln(x) ^2-40*ln(x)+400)/x,x,method=_RETURNVERBOSE)
Output:
exp(4*exp(x*exp(exp(x))+3)^2+(4*ln(x)-80)*exp(x*exp(exp(x))+3)+ln(x)^2-40* ln(x)+400)
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=e^{\left (4 \, {\left (\log \left (x\right ) - 20\right )} e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + \log \left (x\right )^{2} + 4 \, e^{\left (2 \, x e^{\left (e^{x}\right )} + 6\right )} - 40 \, \log \left (x\right ) + 400\right )} \] Input:
integrate(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp( x)*x^2+4*x)*log(x)-80*exp(x)*x^2-80*x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3) +2*log(x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*log(x)-80)*exp(x*exp(exp(x)) +3)+log(x)^2-40*log(x)+400)/x,x, algorithm="fricas")
Output:
e^(4*(log(x) - 20)*e^(x*e^(e^x) + 3) + log(x)^2 + 4*e^(2*x*e^(e^x) + 6) - 40*log(x) + 400)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 10.58 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=\frac {e^{\left (4 \log {\left (x \right )} - 80\right ) e^{x e^{e^{x}} + 3} + 4 e^{2 x e^{e^{x}} + 6} + \log {\left (x \right )}^{2} + 400}}{x^{40}} \] Input:
integrate(((8*exp(x)*x**2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)**2+(((4*ex p(x)*x**2+4*x)*ln(x)-80*exp(x)*x**2-80*x)*exp(exp(x))+4)*exp(x*exp(exp(x)) +3)+2*ln(x)-40)*exp(4*exp(x*exp(exp(x))+3)**2+(4*ln(x)-80)*exp(x*exp(exp(x ))+3)+ln(x)**2-40*ln(x)+400)/x,x)
Output:
exp((4*log(x) - 80)*exp(x*exp(exp(x)) + 3) + 4*exp(2*x*exp(exp(x)) + 6) + log(x)**2 + 400)/x**40
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=\frac {e^{\left (4 \, e^{\left (x e^{\left (e^{x}\right )} + 3\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 4 \, e^{\left (2 \, x e^{\left (e^{x}\right )} + 6\right )} - 80 \, e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + 400\right )}}{x^{40}} \] Input:
integrate(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp( x)*x^2+4*x)*log(x)-80*exp(x)*x^2-80*x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3) +2*log(x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*log(x)-80)*exp(x*exp(exp(x)) +3)+log(x)^2-40*log(x)+400)/x,x, algorithm="maxima")
Output:
e^(4*e^(x*e^(e^x) + 3)*log(x) + log(x)^2 + 4*e^(2*x*e^(e^x) + 6) - 80*e^(x *e^(e^x) + 3) + 400)/x^40
\[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=\int { \frac {2 \, {\left (4 \, {\left (x^{2} e^{x} + x\right )} e^{\left (2 \, x e^{\left (e^{x}\right )} + e^{x} + 6\right )} - 2 \, {\left ({\left (20 \, x^{2} e^{x} - {\left (x^{2} e^{x} + x\right )} \log \left (x\right ) + 20 \, x\right )} e^{\left (e^{x}\right )} - 1\right )} e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + \log \left (x\right ) - 20\right )} e^{\left (4 \, {\left (\log \left (x\right ) - 20\right )} e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + \log \left (x\right )^{2} + 4 \, e^{\left (2 \, x e^{\left (e^{x}\right )} + 6\right )} - 40 \, \log \left (x\right ) + 400\right )}}{x} \,d x } \] Input:
integrate(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp( x)*x^2+4*x)*log(x)-80*exp(x)*x^2-80*x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3) +2*log(x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*log(x)-80)*exp(x*exp(exp(x)) +3)+log(x)^2-40*log(x)+400)/x,x, algorithm="giac")
Output:
integrate(2*(4*(x^2*e^x + x)*e^(2*x*e^(e^x) + e^x + 6) - 2*((20*x^2*e^x - (x^2*e^x + x)*log(x) + 20*x)*e^(e^x) - 1)*e^(x*e^(e^x) + 3) + log(x) - 20) *e^(4*(log(x) - 20)*e^(x*e^(e^x) + 3) + log(x)^2 + 4*e^(2*x*e^(e^x) + 6) - 40*log(x) + 400)/x, x)
Time = 7.51 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=\frac {x^{4\,{\mathrm {e}}^3\,{\mathrm {e}}^{x\,{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^{400}\,{\mathrm {e}}^{{\ln \left (x\right )}^2}\,{\mathrm {e}}^{4\,{\mathrm {e}}^6\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^{-80\,{\mathrm {e}}^3\,{\mathrm {e}}^{x\,{\mathrm {e}}^{{\mathrm {e}}^x}}}}{x^{40}} \] Input:
int((exp(4*exp(2*x*exp(exp(x)) + 6) - 40*log(x) + log(x)^2 + exp(x*exp(exp (x)) + 3)*(4*log(x) - 80) + 400)*(2*log(x) - exp(x*exp(exp(x)) + 3)*(exp(e xp(x))*(80*x + 80*x^2*exp(x) - log(x)*(4*x + 4*x^2*exp(x))) - 4) + exp(exp (x))*exp(2*x*exp(exp(x)) + 6)*(8*x + 8*x^2*exp(x)) - 40))/x,x)
Output:
(x^(4*exp(3)*exp(x*exp(exp(x))))*exp(400)*exp(log(x)^2)*exp(4*exp(6)*exp(2 *x*exp(exp(x))))*exp(-80*exp(3)*exp(x*exp(exp(x)))))/x^40
Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} \left (-40+e^{6+e^x+2 e^{e^x} x} \left (8 x+8 e^x x^2\right )+2 \log (x)+e^{3+e^{e^x} x} \left (4+e^{e^x} \left (-80 x-80 e^x x^2+\left (4 x+4 e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41}} \, dx=\frac {e^{4 e^{2 e^{e^{x}} x} e^{6}+4 e^{e^{e^{x}} x} \mathrm {log}\left (x \right ) e^{3}+\mathrm {log}\left (x \right )^{2}} e^{400}}{e^{80 e^{e^{e^{x}} x} e^{3}} x^{40}} \] Input:
int(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp(x)*x^2 +4*x)*log(x)-80*exp(x)*x^2-80*x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3)+2*log (x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*log(x)-80)*exp(x*exp(exp(x))+3)+lo g(x)^2-40*log(x)+400)/x,x)
Output:
(e**(4*e**(2*e**(e**x)*x)*e**6 + 4*e**(e**(e**x)*x)*log(x)*e**3 + log(x)** 2)*e**400)/(e**(80*e**(e**(e**x)*x)*e**3)*x**40)