Integrand size = 237, antiderivative size = 33 \[ \int \frac {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx=\frac {1}{9} \left (3+e^{e^x}-x\right )^2 x^2 \left (\frac {3}{4}+\log \left (\frac {4}{x}\right )\right ) \log ^2(x) \]
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx=\frac {1}{36} \left (3+e^{e^x}-x\right )^2 x^2 \left (3+4 \log \left (\frac {4}{x}\right )\right ) \log ^2(x) \]
Integrate[((27*x - 18*x^2 + 3*x^3 + (36*x - 24*x^2 + 4*x^3)*Log[4/x])*Log[ x] + (9*x - 15*x^2 + 4*x^3 + (36*x - 36*x^2 + 8*x^3)*Log[4/x])*Log[x]^2 + E^(2*E^x)*((3*x + 4*x*Log[4/x])*Log[x] + (x + 3*E^x*x^2 + (4*x + 4*E^x*x^2 )*Log[4/x])*Log[x]^2) + E^E^x*((18*x - 6*x^2 + (24*x - 8*x^2)*Log[4/x])*Lo g[x] + (6*x - 5*x^2 + E^x*(9*x^2 - 3*x^3) + (24*x - 12*x^2 + E^x*(12*x^2 - 4*x^3))*Log[4/x])*Log[x]^2))/18,x]
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.
| \(\displaystyle \int \frac {1}{18} \left (e^{2 e^x} \left (\left (3 e^x x^2+\left (4 e^x x^2+4 x\right ) \log \left (\frac {4}{x}\right )+x\right ) \log ^2(x)+\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)\right )+\left (4 x^3-15 x^2+\left (8 x^3-36 x^2+36 x\right ) \log \left (\frac {4}{x}\right )+9 x\right ) \log ^2(x)+e^{e^x} \left (\left (-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )+18 x\right ) \log (x)+\left (-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (-12 x^2+e^x \left (12 x^2-4 x^3\right )+24 x\right ) \log \left (\frac {4}{x}\right )+6 x\right ) \log ^2(x)\right )+\left (3 x^3-18 x^2+\left (4 x^3-24 x^2+36 x\right ) \log \left (\frac {4}{x}\right )+27 x\right ) \log (x)\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \int \left (\left (4 x^3-15 x^2+9 x+4 \left (2 x^3-9 x^2+9 x\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (3 x^3-18 x^2+27 x+4 \left (x^3-6 x^2+9 x\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+e^{2 e^x} \left (\left (3 e^x x^2+x+4 \left (e^x x^2+x\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (4 \log \left (\frac {4}{x}\right ) x+3 x\right ) \log (x)\right )+e^{e^x} \left (\left (-5 x^2+6 x+3 e^x \left (3 x^2-x^3\right )+4 \left (-3 x^2+6 x+e^x \left (3 x^2-x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+2 \left (-3 x^2+9 x+4 \left (3 x-x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{18} \left (-3 \int e^{x+e^x} x^3 \log ^2(x)dx-4 \int e^{x+e^x} x^3 \log \left (\frac {4}{x}\right ) \log ^2(x)dx+6 \int \frac {\int e^{e^x} x^2dx}{x}dx+16 \int \frac {\int \frac {\int e^{e^x} x^2dx}{x}dx}{x}dx-5 \int e^{e^x} x^2 \log ^2(x)dx+9 \int e^{x+e^x} x^2 \log ^2(x)dx-12 \int e^{e^x} x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)dx+12 \int e^{x+e^x} x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)dx-8 \log \left (\frac {4}{x}\right ) \log (x) \int e^{e^x} x^2dx-6 \log (x) \int e^{e^x} x^2dx+8 \log \left (\frac {4}{x}\right ) \int \frac {\int e^{e^x} x^2dx}{x}dx-8 \log (x) \int \frac {\int e^{e^x} x^2dx}{x}dx-18 \int \frac {\int e^{e^x} xdx}{x}dx-48 \int \frac {\int \frac {\int e^{e^x} xdx}{x}dx}{x}dx+6 \int e^{e^x} x \log ^2(x)dx+24 \int e^{e^x} x \log \left (\frac {4}{x}\right ) \log ^2(x)dx+24 \log \left (\frac {4}{x}\right ) \log (x) \int e^{e^x} xdx+18 \log (x) \int e^{e^x} xdx-24 \log \left (\frac {4}{x}\right ) \int \frac {\int e^{e^x} xdx}{x}dx+24 \log (x) \int \frac {\int e^{e^x} xdx}{x}dx+\frac {x^4}{4}+2 x^4 \log \left (\frac {4}{x}\right ) \log ^2(x)+\frac {3}{2} x^4 \log ^2(x)+\frac {1}{4} x^4 \log \left (\frac {4}{x}\right )-\frac {1}{4} x^4 \left (\log \left (\frac {4}{x}\right )+1\right )-x^4 \log \left (\frac {4}{x}\right ) \log (x)+\frac {1}{4} x^4 \left (4 \log \left (\frac {4}{x}\right )+3\right ) \log (x)-\frac {3}{4} x^4 \log (x)-\frac {26 x^3}{9}-12 x^3 \log \left (\frac {4}{x}\right ) \log ^2(x)-9 x^3 \log ^2(x)-\frac {8}{3} x^3 \log \left (\frac {4}{x}\right )+\frac {2}{9} x^3 \left (12 \log \left (\frac {4}{x}\right )+13\right )+8 x^3 \log \left (\frac {4}{x}\right ) \log (x)-2 x^3 \left (4 \log \left (\frac {4}{x}\right )+3\right ) \log (x)+6 x^3 \log (x)+\frac {45 x^2}{4}+18 x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)+\frac {27}{2} x^2 \log ^2(x)+9 x^2 \log \left (\frac {4}{x}\right )-\frac {9}{4} x^2 \left (4 \log \left (\frac {4}{x}\right )+5\right )-18 x^2 \log \left (\frac {4}{x}\right ) \log (x)+\frac {9}{2} x^2 \left (4 \log \left (\frac {4}{x}\right )+3\right ) \log (x)-\frac {27}{2} x^2 \log (x)+\frac {1}{2} e^{2 e^x-x} x \log (x) \left (3 e^x x \log (x)+4 e^x x \log \left (\frac {4}{x}\right ) \log (x)\right )\right )\) |
Int[((27*x - 18*x^2 + 3*x^3 + (36*x - 24*x^2 + 4*x^3)*Log[4/x])*Log[x] + ( 9*x - 15*x^2 + 4*x^3 + (36*x - 36*x^2 + 8*x^3)*Log[4/x])*Log[x]^2 + E^(2*E ^x)*((3*x + 4*x*Log[4/x])*Log[x] + (x + 3*E^x*x^2 + (4*x + 4*E^x*x^2)*Log[ 4/x])*Log[x]^2) + E^E^x*((18*x - 6*x^2 + (24*x - 8*x^2)*Log[4/x])*Log[x] + (6*x - 5*x^2 + E^x*(9*x^2 - 3*x^3) + (24*x - 12*x^2 + E^x*(12*x^2 - 4*x^3 ))*Log[4/x])*Log[x]^2))/18,x]
3.4.36.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(27)=54\).
Time = 6.82 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.06
| method | result | size |
| parallelrisch | \(\frac {x^{4} \ln \left (x \right )^{2}}{12}-\frac {x^{3} \ln \left (x \right )^{2}}{2}+\frac {3 x^{2} \ln \left (x \right )^{2}}{4}+x^{2} \ln \left (\frac {4}{x}\right ) \ln \left (x \right )^{2}+\frac {{\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{2}}{2}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{2}}{12}-\frac {{\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{3}}{6}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \left (\frac {4}{x}\right ) x^{2} \ln \left (x \right )^{2}}{9}+\frac {2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (\frac {4}{x}\right ) x^{2} \ln \left (x \right )^{2}}{3}-\frac {2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (\frac {4}{x}\right ) \ln \left (x \right )^{2} x^{3}}{9}-\frac {2 \ln \left (\frac {4}{x}\right ) \ln \left (x \right )^{2} x^{3}}{3}+\frac {\ln \left (\frac {4}{x}\right ) \ln \left (x \right )^{2} x^{4}}{9}\) | \(167\) |
| risch | \(-\frac {x^{3} \ln \left (x \right )^{2}}{2}+\frac {3 x^{2} \ln \left (x \right )^{2}}{4}+\frac {x^{4} \ln \left (x \right )^{2}}{12}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{2}}{12}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}} x^{2} \ln \left (x \right )^{3}}{9}+\frac {2 \ln \left (x \right )^{2} \ln \left (2\right ) x^{4}}{9}+\frac {2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{3} x^{3}}{9}-\frac {2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{3} x^{2}}{3}-\frac {4 \ln \left (2\right ) \ln \left (x \right )^{2} x^{3}}{3}+2 x^{2} \ln \left (2\right ) \ln \left (x \right )^{2}+\frac {{\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{2}}{2}-\frac {{\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{3}}{6}+\frac {4 \ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{2}}{3}+\frac {2 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \left (2\right ) \ln \left (x \right )^{2} x^{2}}{9}-\frac {4 \ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{x}} \ln \left (x \right )^{2} x^{3}}{9}-x^{2} \ln \left (x \right )^{3}+\frac {2 x^{3} \ln \left (x \right )^{3}}{3}-\frac {x^{4} \ln \left (x \right )^{3}}{9}\) | \(209\) |
int(1/18*(((4*exp(x)*x^2+4*x)*ln(4/x)+3*exp(x)*x^2+x)*ln(x)^2+(4*x*ln(4/x) +3*x)*ln(x))*exp(exp(x))^2+1/18*((((-4*x^3+12*x^2)*exp(x)-12*x^2+24*x)*ln( 4/x)+(-3*x^3+9*x^2)*exp(x)-5*x^2+6*x)*ln(x)^2+((-8*x^2+24*x)*ln(4/x)-6*x^2 +18*x)*ln(x))*exp(exp(x))+1/18*((8*x^3-36*x^2+36*x)*ln(4/x)+4*x^3-15*x^2+9 *x)*ln(x)^2+1/18*((4*x^3-24*x^2+36*x)*ln(4/x)+3*x^3-18*x^2+27*x)*ln(x),x,m ethod=_RETURNVERBOSE)
1/12*x^4*ln(x)^2-1/2*x^3*ln(x)^2+3/4*x^2*ln(x)^2+x^2*ln(4/x)*ln(x)^2+1/2*e xp(exp(x))*ln(x)^2*x^2+1/12*exp(exp(x))^2*ln(x)^2*x^2-1/6*exp(exp(x))*ln(x )^2*x^3+1/9*exp(exp(x))^2*ln(4/x)*x^2*ln(x)^2+2/3*exp(exp(x))*ln(4/x)*x^2* ln(x)^2-2/9*exp(exp(x))*ln(4/x)*ln(x)^2*x^3-2/3*ln(4/x)*ln(x)^2*x^3+1/9*ln (4/x)*ln(x)^2*x^4
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 324, normalized size of antiderivative = 9.82 \[ \int \frac {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx=\frac {1}{9} \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (\frac {4}{x}\right )^{3} + \frac {1}{3} \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (2\right )^{2} + \frac {1}{36} \, {\left (3 \, x^{4} - 18 \, x^{3} + 27 \, x^{2} - 16 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (\frac {4}{x}\right )^{2} + \frac {1}{36} \, {\left (4 \, x^{2} \log \left (\frac {4}{x}\right )^{3} + 12 \, x^{2} \log \left (2\right )^{2} - {\left (16 \, x^{2} \log \left (2\right ) - 3 \, x^{2}\right )} \log \left (\frac {4}{x}\right )^{2} + 4 \, {\left (4 \, x^{2} \log \left (2\right )^{2} - 3 \, x^{2} \log \left (2\right )\right )} \log \left (\frac {4}{x}\right )\right )} e^{\left (2 \, e^{x}\right )} - \frac {1}{18} \, {\left (4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (\frac {4}{x}\right )^{3} + 12 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (2\right )^{2} + {\left (3 \, x^{3} - 9 \, x^{2} - 16 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (\frac {4}{x}\right )^{2} + 4 \, {\left (4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (2\right )^{2} - 3 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (\frac {4}{x}\right )\right )} e^{\left (e^{x}\right )} + \frac {1}{9} \, {\left (4 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (2\right )^{2} - 3 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (\frac {4}{x}\right ) \]
integrate(1/18*(((4*exp(x)*x^2+4*x)*log(4/x)+3*exp(x)*x^2+x)*log(x)^2+(4*x *log(4/x)+3*x)*log(x))*exp(exp(x))^2+1/18*((((-4*x^3+12*x^2)*exp(x)-12*x^2 +24*x)*log(4/x)+(-3*x^3+9*x^2)*exp(x)-5*x^2+6*x)*log(x)^2+((-8*x^2+24*x)*l og(4/x)-6*x^2+18*x)*log(x))*exp(exp(x))+1/18*((8*x^3-36*x^2+36*x)*log(4/x) +4*x^3-15*x^2+9*x)*log(x)^2+1/18*((4*x^3-24*x^2+36*x)*log(4/x)+3*x^3-18*x^ 2+27*x)*log(x),x, algorithm=\
1/9*(x^4 - 6*x^3 + 9*x^2)*log(4/x)^3 + 1/3*(x^4 - 6*x^3 + 9*x^2)*log(2)^2 + 1/36*(3*x^4 - 18*x^3 + 27*x^2 - 16*(x^4 - 6*x^3 + 9*x^2)*log(2))*log(4/x )^2 + 1/36*(4*x^2*log(4/x)^3 + 12*x^2*log(2)^2 - (16*x^2*log(2) - 3*x^2)*l og(4/x)^2 + 4*(4*x^2*log(2)^2 - 3*x^2*log(2))*log(4/x))*e^(2*e^x) - 1/18*( 4*(x^3 - 3*x^2)*log(4/x)^3 + 12*(x^3 - 3*x^2)*log(2)^2 + (3*x^3 - 9*x^2 - 16*(x^3 - 3*x^2)*log(2))*log(4/x)^2 + 4*(4*(x^3 - 3*x^2)*log(2)^2 - 3*(x^3 - 3*x^2)*log(2))*log(4/x))*e^(e^x) + 1/9*(4*(x^4 - 6*x^3 + 9*x^2)*log(2)^ 2 - 3*(x^4 - 6*x^3 + 9*x^2)*log(2))*log(4/x)
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (27) = 54\).
Time = 5.61 (sec) , antiderivative size = 187, normalized size of antiderivative = 5.67 \[ \int \frac {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx=\left (- \frac {x^{4}}{9} + \frac {2 x^{3}}{3} - x^{2}\right ) \log {\left (x \right )}^{3} + \frac {\left (- 72 x^{2} \log {\left (x \right )}^{3} + 54 x^{2} \log {\left (x \right )}^{2} + 144 x^{2} \log {\left (2 \right )} \log {\left (x \right )}^{2}\right ) e^{2 e^{x}}}{648} + \left (\frac {x^{4}}{12} + \frac {2 x^{4} \log {\left (2 \right )}}{9} - \frac {4 x^{3} \log {\left (2 \right )}}{3} - \frac {x^{3}}{2} + \frac {3 x^{2}}{4} + 2 x^{2} \log {\left (2 \right )}\right ) \log {\left (x \right )}^{2} + \frac {\left (144 x^{3} \log {\left (x \right )}^{3} - 288 x^{3} \log {\left (2 \right )} \log {\left (x \right )}^{2} - 108 x^{3} \log {\left (x \right )}^{2} - 432 x^{2} \log {\left (x \right )}^{3} + 324 x^{2} \log {\left (x \right )}^{2} + 864 x^{2} \log {\left (2 \right )} \log {\left (x \right )}^{2}\right ) e^{e^{x}}}{648} \]
integrate(1/18*(((4*exp(x)*x**2+4*x)*ln(4/x)+3*exp(x)*x**2+x)*ln(x)**2+(4* x*ln(4/x)+3*x)*ln(x))*exp(exp(x))**2+1/18*((((-4*x**3+12*x**2)*exp(x)-12*x **2+24*x)*ln(4/x)+(-3*x**3+9*x**2)*exp(x)-5*x**2+6*x)*ln(x)**2+((-8*x**2+2 4*x)*ln(4/x)-6*x**2+18*x)*ln(x))*exp(exp(x))+1/18*((8*x**3-36*x**2+36*x)*l n(4/x)+4*x**3-15*x**2+9*x)*ln(x)**2+1/18*((4*x**3-24*x**2+36*x)*ln(4/x)+3* x**3-18*x**2+27*x)*ln(x),x)
(-x**4/9 + 2*x**3/3 - x**2)*log(x)**3 + (-72*x**2*log(x)**3 + 54*x**2*log( x)**2 + 144*x**2*log(2)*log(x)**2)*exp(2*exp(x))/648 + (x**4/12 + 2*x**4*l og(2)/9 - 4*x**3*log(2)/3 - x**3/2 + 3*x**2/4 + 2*x**2*log(2))*log(x)**2 + (144*x**3*log(x)**3 - 288*x**3*log(2)*log(x)**2 - 108*x**3*log(x)**2 - 43 2*x**2*log(x)**3 + 324*x**2*log(x)**2 + 864*x**2*log(2)*log(x)**2)*exp(exp (x))/648
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (29) = 58\).
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 6.76 \[ \int \frac {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx=\frac {1}{36} \, {\left (3 \, x^{4} - 18 \, x^{3} + 27 \, x^{2} + 4 \, {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (\frac {4}{x}\right )\right )} \log \left (x\right )^{2} + \frac {1}{36} \, {\left (x^{2} {\left (8 \, \log \left (2\right ) + 3\right )} \log \left (x\right )^{2} - 4 \, x^{2} \log \left (x\right )^{3}\right )} e^{\left (2 \, e^{x}\right )} + \frac {1}{18} \, {\left (4 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x\right )^{3} - {\left (x^{3} {\left (8 \, \log \left (2\right ) + 3\right )} - 3 \, x^{2} {\left (8 \, \log \left (2\right ) + 3\right )}\right )} \log \left (x\right )^{2}\right )} e^{\left (e^{x}\right )} - \frac {1}{108} \, {\left (6 \, x^{4} {\left (2 \, \log \left (2\right ) + 1\right )} - 4 \, x^{3} {\left (24 \, \log \left (2\right ) + 13\right )} + 27 \, x^{2} {\left (8 \, \log \left (2\right ) + 5\right )} - 6 \, {\left (x^{4} - 8 \, x^{3} + 18 \, x^{2}\right )} \log \left (x\right )\right )} \log \left (x\right ) + \frac {1}{108} \, {\left (6 \, x^{4} - 52 \, x^{3} + 135 \, x^{2} + 6 \, {\left (x^{4} - 8 \, x^{3} + 18 \, x^{2}\right )} \log \left (\frac {4}{x}\right )\right )} \log \left (x\right ) \]
integrate(1/18*(((4*exp(x)*x^2+4*x)*log(4/x)+3*exp(x)*x^2+x)*log(x)^2+(4*x *log(4/x)+3*x)*log(x))*exp(exp(x))^2+1/18*((((-4*x^3+12*x^2)*exp(x)-12*x^2 +24*x)*log(4/x)+(-3*x^3+9*x^2)*exp(x)-5*x^2+6*x)*log(x)^2+((-8*x^2+24*x)*l og(4/x)-6*x^2+18*x)*log(x))*exp(exp(x))+1/18*((8*x^3-36*x^2+36*x)*log(4/x) +4*x^3-15*x^2+9*x)*log(x)^2+1/18*((4*x^3-24*x^2+36*x)*log(4/x)+3*x^3-18*x^ 2+27*x)*log(x),x, algorithm=\
1/36*(3*x^4 - 18*x^3 + 27*x^2 + 4*(x^4 - 6*x^3 + 9*x^2)*log(4/x))*log(x)^2 + 1/36*(x^2*(8*log(2) + 3)*log(x)^2 - 4*x^2*log(x)^3)*e^(2*e^x) + 1/18*(4 *(x^3 - 3*x^2)*log(x)^3 - (x^3*(8*log(2) + 3) - 3*x^2*(8*log(2) + 3))*log( x)^2)*e^(e^x) - 1/108*(6*x^4*(2*log(2) + 1) - 4*x^3*(24*log(2) + 13) + 27* x^2*(8*log(2) + 5) - 6*(x^4 - 8*x^3 + 18*x^2)*log(x))*log(x) + 1/108*(6*x^ 4 - 52*x^3 + 135*x^2 + 6*(x^4 - 8*x^3 + 18*x^2)*log(4/x))*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (29) = 58\).
Time = 0.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 6.88 \[ \int \frac {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx=\frac {2}{9} \, x^{4} \log \left (2\right ) \log \left (x\right )^{2} - \frac {1}{9} \, x^{4} \log \left (x\right )^{3} + \frac {1}{12} \, x^{4} \log \left (x\right )^{2} - \frac {4}{3} \, x^{3} \log \left (2\right ) \log \left (x\right )^{2} + \frac {2}{9} \, x^{2} e^{\left (2 \, e^{x}\right )} \log \left (2\right ) \log \left (x\right )^{2} + \frac {2}{3} \, x^{3} \log \left (x\right )^{3} - \frac {1}{9} \, x^{2} e^{\left (2 \, e^{x}\right )} \log \left (x\right )^{3} - \frac {1}{2} \, x^{3} \log \left (x\right )^{2} + \frac {1}{12} \, x^{2} e^{\left (2 \, e^{x}\right )} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (2\right ) \log \left (x\right )^{2} - x^{2} \log \left (x\right )^{3} + \frac {3}{4} \, x^{2} \log \left (x\right )^{2} - \frac {1}{18} \, {\left (8 \, x^{3} e^{\left (x + e^{x}\right )} \log \left (2\right ) \log \left (x\right )^{2} - 4 \, x^{3} e^{\left (x + e^{x}\right )} \log \left (x\right )^{3} + 3 \, x^{3} e^{\left (x + e^{x}\right )} \log \left (x\right )^{2} - 24 \, x^{2} e^{\left (x + e^{x}\right )} \log \left (2\right ) \log \left (x\right )^{2} + 12 \, x^{2} e^{\left (x + e^{x}\right )} \log \left (x\right )^{3} - 9 \, x^{2} e^{\left (x + e^{x}\right )} \log \left (x\right )^{2}\right )} e^{\left (-x\right )} \]
integrate(1/18*(((4*exp(x)*x^2+4*x)*log(4/x)+3*exp(x)*x^2+x)*log(x)^2+(4*x *log(4/x)+3*x)*log(x))*exp(exp(x))^2+1/18*((((-4*x^3+12*x^2)*exp(x)-12*x^2 +24*x)*log(4/x)+(-3*x^3+9*x^2)*exp(x)-5*x^2+6*x)*log(x)^2+((-8*x^2+24*x)*l og(4/x)-6*x^2+18*x)*log(x))*exp(exp(x))+1/18*((8*x^3-36*x^2+36*x)*log(4/x) +4*x^3-15*x^2+9*x)*log(x)^2+1/18*((4*x^3-24*x^2+36*x)*log(4/x)+3*x^3-18*x^ 2+27*x)*log(x),x, algorithm=\
2/9*x^4*log(2)*log(x)^2 - 1/9*x^4*log(x)^3 + 1/12*x^4*log(x)^2 - 4/3*x^3*l og(2)*log(x)^2 + 2/9*x^2*e^(2*e^x)*log(2)*log(x)^2 + 2/3*x^3*log(x)^3 - 1/ 9*x^2*e^(2*e^x)*log(x)^3 - 1/2*x^3*log(x)^2 + 1/12*x^2*e^(2*e^x)*log(x)^2 + 2*x^2*log(2)*log(x)^2 - x^2*log(x)^3 + 3/4*x^2*log(x)^2 - 1/18*(8*x^3*e^ (x + e^x)*log(2)*log(x)^2 - 4*x^3*e^(x + e^x)*log(x)^3 + 3*x^3*e^(x + e^x) *log(x)^2 - 24*x^2*e^(x + e^x)*log(2)*log(x)^2 + 12*x^2*e^(x + e^x)*log(x) ^3 - 9*x^2*e^(x + e^x)*log(x)^2)*e^(-x)
Timed out. \[ \int \frac {1}{18} \left (\left (27 x-18 x^2+3 x^3+\left (36 x-24 x^2+4 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (9 x-15 x^2+4 x^3+\left (36 x-36 x^2+8 x^3\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)+e^{2 e^x} \left (\left (3 x+4 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (x+3 e^x x^2+\left (4 x+4 e^x x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )+e^{e^x} \left (\left (18 x-6 x^2+\left (24 x-8 x^2\right ) \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (6 x-5 x^2+e^x \left (9 x^2-3 x^3\right )+\left (24 x-12 x^2+e^x \left (12 x^2-4 x^3\right )\right ) \log \left (\frac {4}{x}\right )\right ) \log ^2(x)\right )\right ) \, dx=\int \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (\left (x+3\,x^2\,{\mathrm {e}}^x+\ln \left (\frac {4}{x}\right )\,\left (4\,x+4\,x^2\,{\mathrm {e}}^x\right )\right )\,{\ln \left (x\right )}^2+\left (3\,x+4\,x\,\ln \left (\frac {4}{x}\right )\right )\,\ln \left (x\right )\right )}{18}+\frac {\ln \left (x\right )\,\left (27\,x+\ln \left (\frac {4}{x}\right )\,\left (4\,x^3-24\,x^2+36\,x\right )-18\,x^2+3\,x^3\right )}{18}+\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (\left (6\,x+{\mathrm {e}}^x\,\left (9\,x^2-3\,x^3\right )+\ln \left (\frac {4}{x}\right )\,\left (24\,x+{\mathrm {e}}^x\,\left (12\,x^2-4\,x^3\right )-12\,x^2\right )-5\,x^2\right )\,{\ln \left (x\right )}^2+\left (18\,x+\ln \left (\frac {4}{x}\right )\,\left (24\,x-8\,x^2\right )-6\,x^2\right )\,\ln \left (x\right )\right )}{18}+\frac {{\ln \left (x\right )}^2\,\left (9\,x+\ln \left (\frac {4}{x}\right )\,\left (8\,x^3-36\,x^2+36\,x\right )-15\,x^2+4\,x^3\right )}{18} \,d x \]
int((exp(2*exp(x))*(log(x)^2*(x + 3*x^2*exp(x) + log(4/x)*(4*x + 4*x^2*exp (x))) + log(x)*(3*x + 4*x*log(4/x))))/18 + (log(x)*(27*x + log(4/x)*(36*x - 24*x^2 + 4*x^3) - 18*x^2 + 3*x^3))/18 + (exp(exp(x))*(log(x)*(18*x + log (4/x)*(24*x - 8*x^2) - 6*x^2) + log(x)^2*(6*x + exp(x)*(9*x^2 - 3*x^3) + l og(4/x)*(24*x + exp(x)*(12*x^2 - 4*x^3) - 12*x^2) - 5*x^2)))/18 + (log(x)^ 2*(9*x + log(4/x)*(36*x - 36*x^2 + 8*x^3) - 15*x^2 + 4*x^3))/18,x)
int((exp(2*exp(x))*(log(x)^2*(x + 3*x^2*exp(x) + log(4/x)*(4*x + 4*x^2*exp (x))) + log(x)*(3*x + 4*x*log(4/x))))/18 + (log(x)*(27*x + log(4/x)*(36*x - 24*x^2 + 4*x^3) - 18*x^2 + 3*x^3))/18 + (exp(exp(x))*(log(x)*(18*x + log (4/x)*(24*x - 8*x^2) - 6*x^2) + log(x)^2*(6*x + exp(x)*(9*x^2 - 3*x^3) + l og(4/x)*(24*x + exp(x)*(12*x^2 - 4*x^3) - 12*x^2) - 5*x^2)))/18 + (log(x)^ 2*(9*x + log(4/x)*(36*x - 36*x^2 + 8*x^3) - 15*x^2 + 4*x^3))/18, x)