Integrand size = 140, antiderivative size = 25 \[ \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx=\left (e^{-x}+\frac {19 x}{3}+\frac {2+x}{x}-\log (\log (x))\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(25)=50\).
Time = 1.82 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx=\frac {2}{9} \left (\frac {9 e^{-2 x}}{2}+\frac {18}{x^2}+\frac {18}{x}+57 x+\frac {361 x^2}{2}+e^{-x} \left (9+\frac {18}{x}+57 x\right )-9 \log (\log (x))-\frac {3 \left (6+3 e^{-x} x+19 x^2\right ) \log (\log (x))}{x}+\frac {9}{2} \log ^2(\log (x))\right ) \]
Integrate[(-18*E^x*x^2 + E^(2*x)*(-36*x - 18*x^2 - 114*x^3) + (-18*x^3 + E ^x*(-36*x - 36*x^2 + 96*x^3 - 114*x^4) + E^(2*x)*(-72 - 36*x + 114*x^3 + 7 22*x^4))*Log[x] + (18*E^(2*x)*x^2 + (18*E^x*x^3 + E^(2*x)*(36*x - 114*x^3) )*Log[x])*Log[Log[x]])/(9*E^(2*x)*x^3*Log[x]),x]
(2*(9/(2*E^(2*x)) + 18/x^2 + 18/x + 57*x + (361*x^2)/2 + (9 + 18/x + 57*x) /E^x - 9*Log[Log[x]] - (3*(6 + (3*x)/E^x + 19*x^2)*Log[Log[x]])/x + (9*Log [Log[x]]^2)/2))/9
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.
| \(\displaystyle \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-114 x^3-18 x^2-36 x\right )+\left (\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)+18 e^{2 x} x^2\right ) \log (\log (x))+\left (-18 x^3+e^{2 x} \left (722 x^4+114 x^3-36 x-72\right )+e^x \left (-114 x^4+96 x^3-36 x^2-36 x\right )\right ) \log (x)\right )}{9 x^3 \log (x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int -\frac {2 e^{-2 x} \left (9 e^x x^2+3 e^{2 x} \left (19 x^3+3 x^2+6 x\right )+\left (9 x^3+e^{2 x} \left (-361 x^4-57 x^3+18 x+36\right )+3 e^x \left (19 x^4-16 x^3+6 x^2+6 x\right )\right ) \log (x)-3 \left (3 e^{2 x} x^2+\left (3 e^x x^3+e^{2 x} \left (6 x-19 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{x^3 \log (x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{9} \int \frac {e^{-2 x} \left (9 e^x x^2+3 e^{2 x} \left (19 x^3+3 x^2+6 x\right )+\left (9 x^3+e^{2 x} \left (-361 x^4-57 x^3+18 x+36\right )+3 e^x \left (19 x^4-16 x^3+6 x^2+6 x\right )\right ) \log (x)-3 \left (3 e^{2 x} x^2+\left (3 e^x x^3+e^{2 x} \left (6 x-19 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{x^3 \log (x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2}{9} \int \frac {e^{-2 x} \left (3 e^x x-\left (e^x \left (19 x^2-6\right )-3 x^2\right ) \log (x)\right ) \left (-3 e^x \log (\log (x)) x+3 x+e^x \left (19 x^2+3 x+6\right )\right )}{x^3 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2}{9} \int \left (-\frac {\left (19 \log (x) x^2-3 x-6 \log (x)\right ) \left (19 x^2-3 \log (\log (x)) x+3 x+6\right )}{x^3 \log (x)}+9 e^{-2 x}+\frac {3 e^{-x} \left (19 \log (x) x^3-16 \log (x) x^2-3 \log (x) \log (\log (x)) x^2+6 \log (x) x+3 x+6 \log (x)\right )}{x^2 \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{9} \left (3 \int \frac {19 x^2+3 x+6}{x^2 \log (x)}dx+9 \int \frac {e^{-x}}{x \log (x)}dx-9 \int e^{-x} \log (\log (x))dx-18 \operatorname {ExpIntegralEi}(-\log (x))-57 \operatorname {LogIntegral}(x)-\frac {361 x^2}{2}-\frac {18}{x^2}-57 e^{-x} x-57 x-\frac {9 e^{-2 x}}{2}-9 e^{-x}-\frac {18 e^{-x}}{x}-\frac {18}{x}-\frac {9}{2} \log ^2(\log (x))+57 x \log (\log (x))+\frac {18 \log (\log (x))}{x}\right )\) |
Int[(-18*E^x*x^2 + E^(2*x)*(-36*x - 18*x^2 - 114*x^3) + (-18*x^3 + E^x*(-3 6*x - 36*x^2 + 96*x^3 - 114*x^4) + E^(2*x)*(-72 - 36*x + 114*x^3 + 722*x^4 ))*Log[x] + (18*E^(2*x)*x^2 + (18*E^x*x^3 + E^(2*x)*(36*x - 114*x^3))*Log[ x])*Log[Log[x]])/(9*E^(2*x)*x^3*Log[x]),x]
3.17.88.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(22)=44\).
Time = 0.64 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.96
| method | result | size |
| risch | \(\frac {361 x^{4}-114 \ln \left (\ln \left (x \right )\right ) x^{3}+114 x^{3} {\mathrm e}^{-x}-18 x^{2} \ln \left (\ln \left (x \right )\right )+114 x^{3}+18 x^{2} {\mathrm e}^{-x}-36 x \ln \left (\ln \left (x \right )\right )+9 x^{2} \ln \left (\ln \left (x \right )\right )^{2}-18 \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{-x} x^{2}+36 x \,{\mathrm e}^{-x}+36 x +9 \,{\mathrm e}^{-2 x} x^{2}+36}{9 x^{2}}\) | \(99\) |
| parallelrisch | \(\frac {\left (361 \,{\mathrm e}^{2 x} x^{4}-114 \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{2 x} x^{3}+9 \ln \left (\ln \left (x \right )\right )^{2} {\mathrm e}^{2 x} x^{2}+114 \,{\mathrm e}^{2 x} x^{3}-18 \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{2 x} x^{2}+114 \,{\mathrm e}^{x} x^{3}-18 \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{x} x^{2}-36 \ln \left (\ln \left (x \right )\right ) x \,{\mathrm e}^{2 x}+18 \,{\mathrm e}^{x} x^{2}+36 x \,{\mathrm e}^{2 x}+9 x^{2}+36 \,{\mathrm e}^{x} x +36 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}}{9 x^{2}}\) | \(124\) |
int(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*ln(x)+18*exp(x)^2*x^2)* ln(ln(x))+((722*x^4+114*x^3-36*x-72)*exp(x)^2+(-114*x^4+96*x^3-36*x^2-36*x )*exp(x)-18*x^3)*ln(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(x)*x^2)/x^3/ exp(x)^2/ln(x),x,method=_RETURNVERBOSE)
1/9*(361*x^4-114*ln(ln(x))*x^3+114*x^3*exp(-x)-18*x^2*ln(ln(x))+114*x^3+18 *x^2*exp(-x)-36*x*ln(ln(x))+9*x^2*ln(ln(x))^2-18*ln(ln(x))*exp(-x)*x^2+36* x*exp(-x)+36*x+9*exp(-2*x)*x^2+36)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (27) = 54\).
Time = 0.43 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.96 \[ \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx=\frac {{\left (9 \, x^{2} e^{\left (2 \, x\right )} \log \left (\log \left (x\right )\right )^{2} + 9 \, x^{2} + {\left (361 \, x^{4} + 114 \, x^{3} + 36 \, x + 36\right )} e^{\left (2 \, x\right )} + 6 \, {\left (19 \, x^{3} + 3 \, x^{2} + 6 \, x\right )} e^{x} - 6 \, {\left (3 \, x^{2} e^{x} + {\left (19 \, x^{3} + 3 \, x^{2} + 6 \, x\right )} e^{\left (2 \, x\right )}\right )} \log \left (\log \left (x\right )\right )\right )} e^{\left (-2 \, x\right )}}{9 \, x^{2}} \]
integrate(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*log(x)+18*exp(x)^ 2*x^2)*log(log(x))+((722*x^4+114*x^3-36*x-72)*exp(x)^2+(-114*x^4+96*x^3-36 *x^2-36*x)*exp(x)-18*x^3)*log(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(x) *x^2)/x^3/exp(x)^2/log(x),x, algorithm=\
1/9*(9*x^2*e^(2*x)*log(log(x))^2 + 9*x^2 + (361*x^4 + 114*x^3 + 36*x + 36) *e^(2*x) + 6*(19*x^3 + 3*x^2 + 6*x)*e^x - 6*(3*x^2*e^x + (19*x^3 + 3*x^2 + 6*x)*e^(2*x))*log(log(x)))*e^(-2*x)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (20) = 40\).
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.40 \[ \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx=\frac {361 x^{2}}{9} + \frac {38 x}{3} + \log {\left (\log {\left (x \right )} \right )}^{2} - 2 \log {\left (\log {\left (x \right )} \right )} + \frac {\left (- 38 x^{2} - 12\right ) \log {\left (\log {\left (x \right )} \right )}}{3 x} + \frac {3 x e^{- 2 x} + \left (38 x^{2} - 6 x \log {\left (\log {\left (x \right )} \right )} + 6 x + 12\right ) e^{- x}}{3 x} + \frac {36 x + 36}{9 x^{2}} \]
integrate(1/9*((((-114*x**3+36*x)*exp(x)**2+18*exp(x)*x**3)*ln(x)+18*exp(x )**2*x**2)*ln(ln(x))+((722*x**4+114*x**3-36*x-72)*exp(x)**2+(-114*x**4+96* x**3-36*x**2-36*x)*exp(x)-18*x**3)*ln(x)+(-114*x**3-18*x**2-36*x)*exp(x)** 2-18*exp(x)*x**2)/x**3/exp(x)**2/ln(x),x)
361*x**2/9 + 38*x/3 + log(log(x))**2 - 2*log(log(x)) + (-38*x**2 - 12)*log (log(x))/(3*x) + (3*x*exp(-2*x) + (38*x**2 - 6*x*log(log(x)) + 6*x + 12)*e xp(-x))/(3*x) + (36*x + 36)/(9*x**2)
\[ \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx=\int { -\frac {2 \, {\left (9 \, x^{2} e^{x} + 3 \, {\left (19 \, x^{3} + 3 \, x^{2} + 6 \, x\right )} e^{\left (2 \, x\right )} + {\left (9 \, x^{3} - {\left (361 \, x^{4} + 57 \, x^{3} - 18 \, x - 36\right )} e^{\left (2 \, x\right )} + 3 \, {\left (19 \, x^{4} - 16 \, x^{3} + 6 \, x^{2} + 6 \, x\right )} e^{x}\right )} \log \left (x\right ) - 3 \, {\left (3 \, x^{2} e^{\left (2 \, x\right )} + {\left (3 \, x^{3} e^{x} - {\left (19 \, x^{3} - 6 \, x\right )} e^{\left (2 \, x\right )}\right )} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )\right )} e^{\left (-2 \, x\right )}}{9 \, x^{3} \log \left (x\right )} \,d x } \]
integrate(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*log(x)+18*exp(x)^ 2*x^2)*log(log(x))+((722*x^4+114*x^3-36*x-72)*exp(x)^2+(-114*x^4+96*x^3-36 *x^2-36*x)*exp(x)-18*x^3)*log(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(x) *x^2)/x^3/exp(x)^2/log(x),x, algorithm=\
361/9*x^2 + 38/3*(x + 1)*e^(-x) + 38/3*x + 1/3*(3*x*e^x*log(log(x))^2 - 2* ((19*x^2 + 6)*e^x + 3*x)*log(log(x)))*e^(-x)/x + 4/x + 4/x^2 - 4*Ei(-x) - 4*Ei(-log(x)) - 38/3*Ei(log(x)) - 32/3*e^(-x) + e^(-2*x) + 4*gamma(-1, x) + 2/9*integrate(3*(19*x^2 + 6)/(x^2*log(x)), x) - 2*log(log(x))
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.92 \[ \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx=\frac {361 \, x^{4} + 114 \, x^{3} e^{\left (-x\right )} - 114 \, x^{3} \log \left (\log \left (x\right )\right ) - 18 \, x^{2} e^{\left (-x\right )} \log \left (\log \left (x\right )\right ) + 9 \, x^{2} \log \left (\log \left (x\right )\right )^{2} + 114 \, x^{3} + 18 \, x^{2} e^{\left (-x\right )} + 9 \, x^{2} e^{\left (-2 \, x\right )} - 18 \, x^{2} \log \left (\log \left (x\right )\right ) + 36 \, x e^{\left (-x\right )} - 36 \, x \log \left (\log \left (x\right )\right ) + 36 \, x + 36}{9 \, x^{2}} \]
integrate(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*log(x)+18*exp(x)^ 2*x^2)*log(log(x))+((722*x^4+114*x^3-36*x-72)*exp(x)^2+(-114*x^4+96*x^3-36 *x^2-36*x)*exp(x)-18*x^3)*log(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(x) *x^2)/x^3/exp(x)^2/log(x),x, algorithm=\
1/9*(361*x^4 + 114*x^3*e^(-x) - 114*x^3*log(log(x)) - 18*x^2*e^(-x)*log(lo g(x)) + 9*x^2*log(log(x))^2 + 114*x^3 + 18*x^2*e^(-x) + 9*x^2*e^(-2*x) - 1 8*x^2*log(log(x)) + 36*x*e^(-x) - 36*x*log(log(x)) + 36*x + 36)/x^2
Timed out. \[ \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx=\int -\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,x^2\,{\mathrm {e}}^x+\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (114\,x^4-96\,x^3+36\,x^2+36\,x\right )+{\mathrm {e}}^{2\,x}\,\left (-722\,x^4-114\,x^3+36\,x+72\right )+18\,x^3\right )}{9}+\frac {{\mathrm {e}}^{2\,x}\,\left (114\,x^3+18\,x^2+36\,x\right )}{9}-\frac {\ln \left (\ln \left (x\right )\right )\,\left (18\,x^2\,{\mathrm {e}}^{2\,x}+\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (36\,x-114\,x^3\right )+18\,x^3\,{\mathrm {e}}^x\right )\right )}{9}\right )}{x^3\,\ln \left (x\right )} \,d x \]
int(-(exp(-2*x)*(2*x^2*exp(x) + (log(x)*(exp(x)*(36*x + 36*x^2 - 96*x^3 + 114*x^4) + exp(2*x)*(36*x - 114*x^3 - 722*x^4 + 72) + 18*x^3))/9 + (exp(2* x)*(36*x + 18*x^2 + 114*x^3))/9 - (log(log(x))*(18*x^2*exp(2*x) + log(x)*( exp(2*x)*(36*x - 114*x^3) + 18*x^3*exp(x))))/9))/(x^3*log(x)),x)