Integrand size = 97, antiderivative size = 30 \[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\frac {4 e^{8-\frac {2 \left (e^{(3+x)^2}-x\right )}{x \log (\log (x))}}}{x^2} \] Output:
4*exp(4)^2/x^2/exp((exp((3+x)^2)-x)/x/ln(ln(x)))^2
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\frac {4 e^{8+\frac {2-\frac {2 e^{(3+x)^2}}{x}}{\log (\log (x))}}}{x^2} \] Input:
Integrate[(8*E^(17 + 6*x + x^2) - 8*E^8*x + E^(17 + 6*x + x^2)*(8 - 48*x - 16*x^2)*Log[x]*Log[Log[x]] - 8*E^8*x*Log[x]*Log[Log[x]]^2)/(E^((2*(E^(9 + 6*x + x^2) - x))/(x*Log[Log[x]]))*x^4*Log[x]*Log[Log[x]]^2),x]
Output:
(4*E^(8 + (2 - (2*E^(3 + x)^2)/x)/Log[Log[x]]))/x^2
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(30)=60\).
Time = 1.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 5.40, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, 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.
\(\displaystyle \int \frac {e^{-\frac {2 \left (e^{x^2+6 x+9}-x\right )}{x \log (\log (x))}} \left (8 e^{x^2+6 x+17}+e^{x^2+6 x+17} \left (-16 x^2-48 x+8\right ) \log (x) \log (\log (x))-8 e^8 x-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {4 e^{-\frac {2 \left (e^{x^2+6 x+9}-x\right )}{x \log (\log (x))}} \left (e^{x^2+6 x+17}+e^{x^2+6 x+17} \left (-2 x^2-6 x+1\right ) \log (x) \log (\log (x))-e^8 x\right )}{x^4 \log (x) \left (\frac {e^{x^2+6 x+9}-x}{x^2 \log (x) \log ^2(\log (x))}+\frac {e^{x^2+6 x+9}-x}{x^2 \log (\log (x))}+\frac {1-2 e^{x^2+6 x+9} (x+3)}{x \log (\log (x))}\right ) \log ^2(\log (x))}\) |
Input:
Int[(8*E^(17 + 6*x + x^2) - 8*E^8*x + E^(17 + 6*x + x^2)*(8 - 48*x - 16*x^ 2)*Log[x]*Log[Log[x]] - 8*E^8*x*Log[x]*Log[Log[x]]^2)/(E^((2*(E^(9 + 6*x + x^2) - x))/(x*Log[Log[x]]))*x^4*Log[x]*Log[Log[x]]^2),x]
Output:
(4*(E^(17 + 6*x + x^2) - E^8*x + E^(17 + 6*x + x^2)*(1 - 6*x - 2*x^2)*Log[ x]*Log[Log[x]]))/(E^((2*(E^(9 + 6*x + x^2) - x))/(x*Log[Log[x]]))*x^4*Log[ x]*((E^(9 + 6*x + x^2) - x)/(x^2*Log[x]*Log[Log[x]]^2) + (E^(9 + 6*x + x^2 ) - x)/(x^2*Log[Log[x]]) + (1 - 2*E^(9 + 6*x + x^2)*(3 + x))/(x*Log[Log[x] ]))*Log[Log[x]]^2)
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 = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
\[\frac {4 \,{\mathrm e}^{\frac {8 x \ln \left (\ln \left (x \right )\right )-2 \,{\mathrm e}^{\left (3+x \right )^{2}}+2 x}{x \ln \left (\ln \left (x \right )\right )}}}{x^{2}}\]
Input:
int((-8*x*exp(4)^2*ln(x)*ln(ln(x))^2+(-16*x^2-48*x+8)*exp(4)^2*exp(x^2+6*x +9)*ln(x)*ln(ln(x))+8*exp(4)^2*exp(x^2+6*x+9)-8*x*exp(4)^2)/x^4/ln(x)/ln(l n(x))^2/exp((exp(x^2+6*x+9)-x)/x/ln(ln(x)))^2,x)
Output:
4/x^2*exp(2*(4*x*ln(ln(x))-exp((3+x)^2)+x)/x/ln(ln(x)))
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\frac {4 \, e^{\left (\frac {2 \, {\left (x e^{8} - e^{\left (x^{2} + 6 \, x + 17\right )}\right )} e^{\left (-8\right )}}{x \log \left (\log \left (x\right )\right )} + 8\right )}}{x^{2}} \] Input:
integrate((-8*x*exp(4)^2*log(x)*log(log(x))^2+(-16*x^2-48*x+8)*exp(4)^2*ex p(x^2+6*x+9)*log(x)*log(log(x))+8*exp(4)^2*exp(x^2+6*x+9)-8*x*exp(4)^2)/x^ 4/log(x)/log(log(x))^2/exp((exp(x^2+6*x+9)-x)/x/log(log(x)))^2,x, algorith m="fricas")
Output:
4*e^(2*(x*e^8 - e^(x^2 + 6*x + 17))*e^(-8)/(x*log(log(x))) + 8)/x^2
Time = 1.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\frac {4 e^{8} e^{- \frac {2 \left (- x + e^{x^{2} + 6 x + 9}\right )}{x \log {\left (\log {\left (x \right )} \right )}}}}{x^{2}} \] Input:
integrate((-8*x*exp(4)**2*ln(x)*ln(ln(x))**2+(-16*x**2-48*x+8)*exp(4)**2*e xp(x**2+6*x+9)*ln(x)*ln(ln(x))+8*exp(4)**2*exp(x**2+6*x+9)-8*x*exp(4)**2)/ x**4/ln(x)/ln(ln(x))**2/exp((exp(x**2+6*x+9)-x)/x/ln(ln(x)))**2,x)
Output:
4*exp(8)*exp(-2*(-x + exp(x**2 + 6*x + 9))/(x*log(log(x))))/x**2
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\frac {4 \, e^{\left (-\frac {2 \, e^{\left (x^{2} + 6 \, x + 9\right )}}{x \log \left (\log \left (x\right )\right )} + \frac {2}{\log \left (\log \left (x\right )\right )} + 8\right )}}{x^{2}} \] Input:
integrate((-8*x*exp(4)^2*log(x)*log(log(x))^2+(-16*x^2-48*x+8)*exp(4)^2*ex p(x^2+6*x+9)*log(x)*log(log(x))+8*exp(4)^2*exp(x^2+6*x+9)-8*x*exp(4)^2)/x^ 4/log(x)/log(log(x))^2/exp((exp(x^2+6*x+9)-x)/x/log(log(x)))^2,x, algorith m="maxima")
Output:
4*e^(-2*e^(x^2 + 6*x + 9)/(x*log(log(x))) + 2/log(log(x)) + 8)/x^2
\[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\int { -\frac {8 \, {\left (x e^{8} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + {\left (2 \, x^{2} + 6 \, x - 1\right )} e^{\left (x^{2} + 6 \, x + 17\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + x e^{8} - e^{\left (x^{2} + 6 \, x + 17\right )}\right )} e^{\left (\frac {2 \, {\left (x - e^{\left (x^{2} + 6 \, x + 9\right )}\right )}}{x \log \left (\log \left (x\right )\right )}\right )}}{x^{4} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2}} \,d x } \] Input:
integrate((-8*x*exp(4)^2*log(x)*log(log(x))^2+(-16*x^2-48*x+8)*exp(4)^2*ex p(x^2+6*x+9)*log(x)*log(log(x))+8*exp(4)^2*exp(x^2+6*x+9)-8*x*exp(4)^2)/x^ 4/log(x)/log(log(x))^2/exp((exp(x^2+6*x+9)-x)/x/log(log(x)))^2,x, algorith m="giac")
Output:
undef
Time = 3.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\frac {4\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^9}{x\,\ln \left (\ln \left (x\right )\right )}}\,{\mathrm {e}}^8\,{\mathrm {e}}^{\frac {2}{\ln \left (\ln \left (x\right )\right )}}}{x^2} \] Input:
int(-(exp((2*(x - exp(6*x + x^2 + 9)))/(x*log(log(x))))*(8*x*exp(8) - 8*ex p(8)*exp(6*x + x^2 + 9) + 8*x*log(log(x))^2*exp(8)*log(x) + log(log(x))*ex p(8)*exp(6*x + x^2 + 9)*log(x)*(48*x + 16*x^2 - 8)))/(x^4*log(log(x))^2*lo g(x)),x)
Output:
(4*exp(-(2*exp(6*x)*exp(x^2)*exp(9))/(x*log(log(x))))*exp(8)*exp(2/log(log (x))))/x^2
Time = 2.47 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-\frac {2 \left (e^{9+6 x+x^2}-x\right )}{x \log (\log (x))}} \left (8 e^{17+6 x+x^2}-8 e^8 x+e^{17+6 x+x^2} \left (8-48 x-16 x^2\right ) \log (x) \log (\log (x))-8 e^8 x \log (x) \log ^2(\log (x))\right )}{x^4 \log (x) \log ^2(\log (x))} \, dx=\frac {4 e^{\frac {2}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right )}} e^{8}}{e^{\frac {2 e^{x^{2}+6 x} e^{9}}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x}} x^{2}} \] Input:
int((-8*x*exp(4)^2*log(x)*log(log(x))^2+(-16*x^2-48*x+8)*exp(4)^2*exp(x^2+ 6*x+9)*log(x)*log(log(x))+8*exp(4)^2*exp(x^2+6*x+9)-8*x*exp(4)^2)/x^4/log( x)/log(log(x))^2/exp((exp(x^2+6*x+9)-x)/x/log(log(x)))^2,x)
Output:
(4*e**(2/log(log(x)))*e**8)/(e**((2*e**(x**2 + 6*x)*e**9)/(log(log(x))*x)) *x**2)