Integrand size = 176, antiderivative size = 34 \[ \int \frac {-2 e^{e^{x-x^2}}+e^{x-x^2} \left (-5 x+10 x^2\right )+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx=\frac {x}{x+\log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \]
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e^{e^{x-x^2}}+e^{x-x^2} \left (-5 x+10 x^2\right )+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx=\frac {x}{x+\log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \]
Integrate[(-2*E^E^(x - x^2) + E^(x - x^2)*(-5*x + 10*x^2) + E^E^(x - x^2)* Log[(2*x^2)/(E^(5/E^E^(x - x^2))*Log[Log[2]])])/(E^E^(x - x^2)*x^2 + 2*E^E ^(x - x^2)*x*Log[(2*x^2)/(E^(5/E^E^(x - x^2))*Log[Log[2]])] + E^E^(x - x^2 )*Log[(2*x^2)/(E^(5/E^E^(x - x^2))*Log[Log[2]])]^2),x]
Time = 1.52 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7292, 7262, 17}
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.
| \(\displaystyle \int \frac {e^{x-x^2} \left (10 x^2-5 x\right )-2 e^{e^{x-x^2}}+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{-e^{(1-x) x}} \left (e^{x-x^2} \left (10 x^2-5 x\right )-2 e^{e^{x-x^2}}+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )\right )}{\left (\log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+x\right )^2}dx\) |
| \(\Big \downarrow \) 7262 |
| \(\displaystyle \int \frac {1}{\left (\frac {x}{\log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}+1\right )^2}d\frac {x}{\log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {1}{\frac {x}{\log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}+1}\) |
Int[(-2*E^E^(x - x^2) + E^(x - x^2)*(-5*x + 10*x^2) + E^E^(x - x^2)*Log[(2 *x^2)/(E^(5/E^E^(x - x^2))*Log[Log[2]])])/(E^E^(x - x^2)*x^2 + 2*E^E^(x - x^2)*x*Log[(2*x^2)/(E^(5/E^E^(x - x^2))*Log[Log[2]])] + E^E^(x - x^2)*Log[ (2*x^2)/(E^(5/E^E^(x - x^2))*Log[Log[2]])]^2),x]
3.1.49.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[c*p Subst[Int[(b + a*x^p )^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]
Time = 1.68 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(\frac {x}{\ln \left (\frac {2 x^{2} {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x^{2}+x}}}}{\ln \left (\ln \left (2\right )\right )}\right )+x}\) | \(34\) |
| risch | \(\frac {2 x}{-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi -2 i \pi \operatorname {csgn}\left (i x^{2} {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right )+i \pi \operatorname {csgn}\left (i x^{2} {\mathrm e}^{-5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right )^{3}+2 \ln \left (2\right )-2 \ln \left (-\ln \left (\ln \left (2\right )\right )\right )+2 x +4 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{5 \,{\mathrm e}^{-{\mathrm e}^{-x \left (-1+x \right )}}}\right )}\) | \(271\) |
int((exp(exp(-x^2+x))*ln(2*x^2/ln(ln(2))/exp(5/exp(exp(-x^2+x))))-2*exp(ex p(-x^2+x))+(10*x^2-5*x)*exp(-x^2+x))/(exp(exp(-x^2+x))*ln(2*x^2/ln(ln(2))/ exp(5/exp(exp(-x^2+x))))^2+2*x*exp(exp(-x^2+x))*ln(2*x^2/ln(ln(2))/exp(5/e xp(exp(-x^2+x))))+x^2*exp(exp(-x^2+x))),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {-2 e^{e^{x-x^2}}+e^{x-x^2} \left (-5 x+10 x^2\right )+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx=\frac {x}{x + \log \left (\frac {2 \, x^{2} e^{\left (-5 \, e^{\left (-e^{\left (-x^{2} + x\right )}\right )}\right )}}{\log \left (\log \left (2\right )\right )}\right )} \]
integrate((exp(exp(-x^2+x))*log(2*x^2/log(log(2))/exp(5/exp(exp(-x^2+x)))) -2*exp(exp(-x^2+x))+(10*x^2-5*x)*exp(-x^2+x))/(exp(exp(-x^2+x))*log(2*x^2/ log(log(2))/exp(5/exp(exp(-x^2+x))))^2+2*x*exp(exp(-x^2+x))*log(2*x^2/log( log(2))/exp(5/exp(exp(-x^2+x))))+x^2*exp(exp(-x^2+x))),x, algorithm=\
Time = 0.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {-2 e^{e^{x-x^2}}+e^{x-x^2} \left (-5 x+10 x^2\right )+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx=\frac {x}{x + \log {\left (\frac {2 x^{2} e^{- 5 e^{- e^{- x^{2} + x}}}}{\log {\left (\log {\left (2 \right )} \right )}} \right )}} \]
integrate((exp(exp(-x**2+x))*ln(2*x**2/ln(ln(2))/exp(5/exp(exp(-x**2+x)))) -2*exp(exp(-x**2+x))+(10*x**2-5*x)*exp(-x**2+x))/(exp(exp(-x**2+x))*ln(2*x **2/ln(ln(2))/exp(5/exp(exp(-x**2+x))))**2+2*x*exp(exp(-x**2+x))*ln(2*x**2 /ln(ln(2))/exp(5/exp(exp(-x**2+x))))+x**2*exp(exp(-x**2+x))),x)
Time = 0.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {-2 e^{e^{x-x^2}}+e^{x-x^2} \left (-5 x+10 x^2\right )+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx=\frac {x e^{\left (e^{\left (-x^{2} + x\right )}\right )}}{{\left (x + \log \left (2\right ) + 2 \, \log \left (x\right ) - \log \left (\log \left (\log \left (2\right )\right )\right )\right )} e^{\left (e^{\left (-x^{2} + x\right )}\right )} - 5} \]
integrate((exp(exp(-x^2+x))*log(2*x^2/log(log(2))/exp(5/exp(exp(-x^2+x)))) -2*exp(exp(-x^2+x))+(10*x^2-5*x)*exp(-x^2+x))/(exp(exp(-x^2+x))*log(2*x^2/ log(log(2))/exp(5/exp(exp(-x^2+x))))^2+2*x*exp(exp(-x^2+x))*log(2*x^2/log( log(2))/exp(5/exp(exp(-x^2+x))))+x^2*exp(exp(-x^2+x))),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 2943 vs. \(2 (31) = 62\).
Time = 0.82 (sec) , antiderivative size = 2943, normalized size of antiderivative = 86.56 \[ \int \frac {-2 e^{e^{x-x^2}}+e^{x-x^2} \left (-5 x+10 x^2\right )+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx=\text {Too large to display} \]
integrate((exp(exp(-x^2+x))*log(2*x^2/log(log(2))/exp(5/exp(exp(-x^2+x)))) -2*exp(exp(-x^2+x))+(10*x^2-5*x)*exp(-x^2+x))/(exp(exp(-x^2+x))*log(2*x^2/ log(log(2))/exp(5/exp(exp(-x^2+x))))^2+2*x*exp(exp(-x^2+x))*log(2*x^2/log( log(2))/exp(5/exp(exp(-x^2+x))))+x^2*exp(exp(-x^2+x))),x, algorithm=\
(2*x^5*e^(-2*x^2 + 2*x) + 4*x^4*e^(-2*x^2 + 2*x)*log(2) + 2*x^3*e^(-2*x^2 + 2*x)*log(2)^2 + 4*x^4*e^(-2*x^2 + 2*x)*log(x^2) + 4*x^3*e^(-2*x^2 + 2*x) *log(2)*log(x^2) + 2*x^3*e^(-2*x^2 + 2*x)*log(x^2)^2 - 4*x^4*e^(-2*x^2 + 2 *x)*log(log(log(2))) - 4*x^3*e^(-2*x^2 + 2*x)*log(2)*log(log(log(2))) - 4* x^3*e^(-2*x^2 + 2*x)*log(x^2)*log(log(log(2))) + 2*x^3*e^(-2*x^2 + 2*x)*lo g(log(log(2)))^2 - 10*x^4*e^(-2*x^2 + 2*x - e^(-x^2 + x)) - x^4*e^(-2*x^2 + 2*x) - 10*x^3*e^(-2*x^2 + 2*x - e^(-x^2 + x))*log(2) - 2*x^3*e^(-2*x^2 + 2*x)*log(2) - x^2*e^(-2*x^2 + 2*x)*log(2)^2 - 10*x^3*e^(-2*x^2 + 2*x - e^ (-x^2 + x))*log(x^2) - 2*x^3*e^(-2*x^2 + 2*x)*log(x^2) - 2*x^2*e^(-2*x^2 + 2*x)*log(2)*log(x^2) - x^2*e^(-2*x^2 + 2*x)*log(x^2)^2 + 10*x^3*e^(-2*x^2 + 2*x - e^(-x^2 + x))*log(log(log(2))) + 2*x^3*e^(-2*x^2 + 2*x)*log(log(l og(2))) + 2*x^2*e^(-2*x^2 + 2*x)*log(2)*log(log(log(2))) + 2*x^2*e^(-2*x^2 + 2*x)*log(x^2)*log(log(log(2))) - x^2*e^(-2*x^2 + 2*x)*log(log(log(2)))^ 2 - x^3*e^(-x^2 + x) + 5*x^3*e^(-2*x^2 + 2*x - e^(-x^2 + x)) - x^2*e^(-x^2 + x)*log(2) + 5*x^2*e^(-2*x^2 + 2*x - e^(-x^2 + x))*log(2) - x^2*e^(-x^2 + x)*log(x^2) + 5*x^2*e^(-2*x^2 + 2*x - e^(-x^2 + x))*log(x^2) + x^2*e^(-x ^2 + x)*log(log(log(2))) - 5*x^2*e^(-2*x^2 + 2*x - e^(-x^2 + x))*log(log(l og(2))) + 5*x^2*e^(-x^2 + x - e^(-x^2 + x)) - 2*x^2*e^(-x^2 + x) - 2*x*e^( -x^2 + x)*log(2) - 2*x*e^(-x^2 + x)*log(x^2) + 2*x*e^(-x^2 + x)*log(log(lo g(2))) + 10*x*e^(-x^2 + x - e^(-x^2 + x)))/(2*x^5*e^(-2*x^2 + 2*x) + 6*...
Timed out. \[ \int \frac {-2 e^{e^{x-x^2}}+e^{x-x^2} \left (-5 x+10 x^2\right )+e^{e^{x-x^2}} \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )}{e^{e^{x-x^2}} x^2+2 e^{e^{x-x^2}} x \log \left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )+e^{e^{x-x^2}} \log ^2\left (\frac {2 e^{-5 e^{-e^{x-x^2}}} x^2}{\log (\log (2))}\right )} \, dx=-\int \frac {2\,{\mathrm {e}}^{{\mathrm {e}}^{x-x^2}}-{\mathrm {e}}^{{\mathrm {e}}^{x-x^2}}\,\ln \left (\frac {2\,x^2\,{\mathrm {e}}^{-5\,{\mathrm {e}}^{-{\mathrm {e}}^{x-x^2}}}}{\ln \left (\ln \left (2\right )\right )}\right )+{\mathrm {e}}^{x-x^2}\,\left (5\,x-10\,x^2\right )}{x^2\,{\mathrm {e}}^{{\mathrm {e}}^{x-x^2}}+{\mathrm {e}}^{{\mathrm {e}}^{x-x^2}}\,{\ln \left (\frac {2\,x^2\,{\mathrm {e}}^{-5\,{\mathrm {e}}^{-{\mathrm {e}}^{x-x^2}}}}{\ln \left (\ln \left (2\right )\right )}\right )}^2+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{x-x^2}}\,\ln \left (\frac {2\,x^2\,{\mathrm {e}}^{-5\,{\mathrm {e}}^{-{\mathrm {e}}^{x-x^2}}}}{\ln \left (\ln \left (2\right )\right )}\right )} \,d x \]
int(-(2*exp(exp(x - x^2)) - exp(exp(x - x^2))*log((2*x^2*exp(-5*exp(-exp(x - x^2))))/log(log(2))) + exp(x - x^2)*(5*x - 10*x^2))/(x^2*exp(exp(x - x^ 2)) + exp(exp(x - x^2))*log((2*x^2*exp(-5*exp(-exp(x - x^2))))/log(log(2)) )^2 + 2*x*exp(exp(x - x^2))*log((2*x^2*exp(-5*exp(-exp(x - x^2))))/log(log (2)))),x)
-int((2*exp(exp(x - x^2)) - exp(exp(x - x^2))*log((2*x^2*exp(-5*exp(-exp(x - x^2))))/log(log(2))) + exp(x - x^2)*(5*x - 10*x^2))/(x^2*exp(exp(x - x^ 2)) + exp(exp(x - x^2))*log((2*x^2*exp(-5*exp(-exp(x - x^2))))/log(log(2)) )^2 + 2*x*exp(exp(x - x^2))*log((2*x^2*exp(-5*exp(-exp(x - x^2))))/log(log (2)))), x)