Integrand size = 207, antiderivative size = 32 \begin {dmath*} \int \frac {e^{-\frac {x^2+\left (16 x+4 x^2\right ) \log ^2(x)+\left (64+36 x+4 x^2\right ) \log ^4(x)}{4 \log ^4(x)}} \left (e^5 \left (-2 x^2+2 x^3\right ) \log \left (x-x^2\right )+e^5 \left (x^2-x^3\right ) \log (x) \log \left (x-x^2\right )+e^5 \left (-16 x+12 x^2+4 x^3\right ) \log ^2(x) \log \left (x-x^2\right )+e^5 \left (8 x-4 x^2-4 x^3\right ) \log ^3(x) \log \left (x-x^2\right )+\log ^5(x) \left (e^5 (-2+4 x)+e^5 \left (18 x-14 x^2-4 x^3\right ) \log \left (x-x^2\right )\right )\right )}{\left (-2 x+2 x^2\right ) \log ^5(x)} \, dx=e^{5-x-\left (4+x+\frac {x}{2 \log ^2(x)}\right )^2} \log \left (x-x^2\right ) \end {dmath*}
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \begin {dmath*} \int \frac {e^{-\frac {x^2+\left (16 x+4 x^2\right ) \log ^2(x)+\left (64+36 x+4 x^2\right ) \log ^4(x)}{4 \log ^4(x)}} \left (e^5 \left (-2 x^2+2 x^3\right ) \log \left (x-x^2\right )+e^5 \left (x^2-x^3\right ) \log (x) \log \left (x-x^2\right )+e^5 \left (-16 x+12 x^2+4 x^3\right ) \log ^2(x) \log \left (x-x^2\right )+e^5 \left (8 x-4 x^2-4 x^3\right ) \log ^3(x) \log \left (x-x^2\right )+\log ^5(x) \left (e^5 (-2+4 x)+e^5 \left (18 x-14 x^2-4 x^3\right ) \log \left (x-x^2\right )\right )\right )}{\left (-2 x+2 x^2\right ) \log ^5(x)} \, dx=e^{-11-9 x-x^2-\frac {x^2}{4 \log ^4(x)}-\frac {x (4+x)}{\log ^2(x)}} \log (-((-1+x) x)) \end {dmath*}
Integrate[(E^5*(-2*x^2 + 2*x^3)*Log[x - x^2] + E^5*(x^2 - x^3)*Log[x]*Log[ x - x^2] + E^5*(-16*x + 12*x^2 + 4*x^3)*Log[x]^2*Log[x - x^2] + E^5*(8*x - 4*x^2 - 4*x^3)*Log[x]^3*Log[x - x^2] + Log[x]^5*(E^5*(-2 + 4*x) + E^5*(18 *x - 14*x^2 - 4*x^3)*Log[x - x^2]))/(E^((x^2 + (16*x + 4*x^2)*Log[x]^2 + ( 64 + 36*x + 4*x^2)*Log[x]^4)/(4*Log[x]^4))*(-2*x + 2*x^2)*Log[x]^5),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.
| \(\displaystyle \int \frac {\left (\left (e^5 \left (-4 x^3-14 x^2+18 x\right ) \log \left (x-x^2\right )+e^5 (4 x-2)\right ) \log ^5(x)+e^5 \left (-4 x^3-4 x^2+8 x\right ) \log \left (x-x^2\right ) \log ^3(x)+e^5 \left (4 x^3+12 x^2-16 x\right ) \log \left (x-x^2\right ) \log ^2(x)+e^5 \left (x^2-x^3\right ) \log \left (x-x^2\right ) \log (x)+e^5 \left (2 x^3-2 x^2\right ) \log \left (x-x^2\right )\right ) \exp \left (-\frac {x^2+\left (4 x^2+36 x+64\right ) \log ^4(x)+\left (4 x^2+16 x\right ) \log ^2(x)}{4 \log ^4(x)}\right )}{\left (2 x^2-2 x\right ) \log ^5(x)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (\left (e^5 \left (-4 x^3-14 x^2+18 x\right ) \log \left (x-x^2\right )+e^5 (4 x-2)\right ) \log ^5(x)+e^5 \left (-4 x^3-4 x^2+8 x\right ) \log \left (x-x^2\right ) \log ^3(x)+e^5 \left (4 x^3+12 x^2-16 x\right ) \log \left (x-x^2\right ) \log ^2(x)+e^5 \left (x^2-x^3\right ) \log \left (x-x^2\right ) \log (x)+e^5 \left (2 x^3-2 x^2\right ) \log \left (x-x^2\right )\right ) \exp \left (-\frac {x^2+\left (4 x^2+36 x+64\right ) \log ^4(x)+\left (4 x^2+16 x\right ) \log ^2(x)}{4 \log ^4(x)}\right )}{x (2 x-2) \log ^5(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(2 x-1) \exp \left (5-\frac {x^2+\left (4 x^2+36 x+64\right ) \log ^4(x)+\left (4 x^2+16 x\right ) \log ^2(x)}{4 \log ^4(x)}\right )}{(x-1) x}+\frac {\left (2 x-4 x \log ^5(x)-18 \log ^5(x)-4 x \log ^3(x)-8 \log ^3(x)+4 x \log ^2(x)+16 \log ^2(x)-x \log (x)\right ) \log ((1-x) x) \exp \left (5-\frac {x^2+\left (4 x^2+36 x+64\right ) \log ^4(x)+\left (4 x^2+16 x\right ) \log ^2(x)}{4 \log ^4(x)}\right )}{2 \log ^5(x)}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(2 x-1) \exp \left (-x^2-\frac {x^2}{4 \log ^4(x)}-9 x-\frac {(x+4) x}{\log ^2(x)}-11\right )}{(x-1) x}+\frac {\left (2 x-4 x \log ^5(x)-18 \log ^5(x)-4 x \log ^3(x)-8 \log ^3(x)+4 x \log ^2(x)+16 \log ^2(x)-x \log (x)\right ) \log ((1-x) x) \exp \left (-x^2-\frac {x^2}{4 \log ^4(x)}-9 x-\frac {(x+4) x}{\log ^2(x)}-11\right )}{2 \log ^5(x)}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {(2 x-1) \exp \left (-x^2-\frac {x^2}{4 \log ^4(x)}-9 x-\frac {(x+4) x}{\log ^2(x)}-11\right )}{(x-1) x}+\frac {\left (2 x-4 x \log ^5(x)-18 \log ^5(x)-4 x \log ^3(x)-8 \log ^3(x)+4 x \log ^2(x)+16 \log ^2(x)-x \log (x)\right ) \log ((1-x) x) \exp \left (-x^2-\frac {x^2}{4 \log ^4(x)}-9 x-\frac {(x+4) x}{\log ^2(x)}-11\right )}{2 \log ^5(x)}\right )dx\) |
Int[(E^5*(-2*x^2 + 2*x^3)*Log[x - x^2] + E^5*(x^2 - x^3)*Log[x]*Log[x - x^ 2] + E^5*(-16*x + 12*x^2 + 4*x^3)*Log[x]^2*Log[x - x^2] + E^5*(8*x - 4*x^2 - 4*x^3)*Log[x]^3*Log[x - x^2] + Log[x]^5*(E^5*(-2 + 4*x) + E^5*(18*x - 1 4*x^2 - 4*x^3)*Log[x - x^2]))/(E^((x^2 + (16*x + 4*x^2)*Log[x]^2 + (64 + 3 6*x + 4*x^2)*Log[x]^4)/(4*Log[x]^4))*(-2*x + 2*x^2)*Log[x]^5),x]
3.10.53.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 5.38
\[\left (-i {\mathrm e}^{5} \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2}+\frac {i {\mathrm e}^{5} \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{3}}{2}+\frac {i {\mathrm e}^{5} \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )}{2}+\frac {i {\mathrm e}^{5} \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-1+x \right )\right )}{2}-\frac {i {\mathrm e}^{5} \pi \,\operatorname {csgn}\left (i x \left (-1+x \right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (-1+x \right )\right )}{2}+i {\mathrm e}^{5} \pi +{\mathrm e}^{5} \ln \left (x \right )+{\mathrm e}^{5} \ln \left (-1+x \right )\right ) {\mathrm e}^{-\frac {4 x^{2} \ln \left (x \right )^{4}+36 x \ln \left (x \right )^{4}+64 \ln \left (x \right )^{4}+4 x^{2} \ln \left (x \right )^{2}+16 x \ln \left (x \right )^{2}+x^{2}}{4 \ln \left (x \right )^{4}}}\]
int((((-4*x^3-14*x^2+18*x)*exp(5)*ln(-x^2+x)+(4*x-2)*exp(5))*ln(x)^5+(-4*x ^3-4*x^2+8*x)*exp(5)*ln(-x^2+x)*ln(x)^3+(4*x^3+12*x^2-16*x)*exp(5)*ln(-x^2 +x)*ln(x)^2+(-x^3+x^2)*exp(5)*ln(-x^2+x)*ln(x)+(2*x^3-2*x^2)*exp(5)*ln(-x^ 2+x))/(2*x^2-2*x)/ln(x)^5/exp(1/4*((4*x^2+36*x+64)*ln(x)^4+(4*x^2+16*x)*ln (x)^2+x^2)/ln(x)^4),x)
(-I*exp(5)*Pi*csgn(I*x*(-1+x))^2+1/2*I*exp(5)*Pi*csgn(I*x*(-1+x))^3+1/2*I* exp(5)*Pi*csgn(I*x*(-1+x))^2*csgn(I*x)+1/2*I*exp(5)*Pi*csgn(I*x*(-1+x))^2* csgn(I*(-1+x))-1/2*I*exp(5)*Pi*csgn(I*x*(-1+x))*csgn(I*x)*csgn(I*(-1+x))+I *exp(5)*Pi+exp(5)*ln(x)+exp(5)*ln(-1+x))*exp(-1/4*(4*x^2*ln(x)^4+36*x*ln(x )^4+64*ln(x)^4+4*x^2*ln(x)^2+16*x*ln(x)^2+x^2)/ln(x)^4)
Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \begin {dmath*} \int \frac {e^{-\frac {x^2+\left (16 x+4 x^2\right ) \log ^2(x)+\left (64+36 x+4 x^2\right ) \log ^4(x)}{4 \log ^4(x)}} \left (e^5 \left (-2 x^2+2 x^3\right ) \log \left (x-x^2\right )+e^5 \left (x^2-x^3\right ) \log (x) \log \left (x-x^2\right )+e^5 \left (-16 x+12 x^2+4 x^3\right ) \log ^2(x) \log \left (x-x^2\right )+e^5 \left (8 x-4 x^2-4 x^3\right ) \log ^3(x) \log \left (x-x^2\right )+\log ^5(x) \left (e^5 (-2+4 x)+e^5 \left (18 x-14 x^2-4 x^3\right ) \log \left (x-x^2\right )\right )\right )}{\left (-2 x+2 x^2\right ) \log ^5(x)} \, dx=e^{\left (-\frac {4 \, {\left (x^{2} + 9 \, x + 16\right )} \log \left (x\right )^{4} + 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} + x^{2}}{4 \, \log \left (x\right )^{4}} + 5\right )} \log \left (-x^{2} + x\right ) \end {dmath*}
integrate((((-4*x^3-14*x^2+18*x)*exp(5)*log(-x^2+x)+(4*x-2)*exp(5))*log(x) ^5+(-4*x^3-4*x^2+8*x)*exp(5)*log(-x^2+x)*log(x)^3+(4*x^3+12*x^2-16*x)*exp( 5)*log(-x^2+x)*log(x)^2+(-x^3+x^2)*exp(5)*log(-x^2+x)*log(x)+(2*x^3-2*x^2) *exp(5)*log(-x^2+x))/(2*x^2-2*x)/log(x)^5/exp(1/4*((4*x^2+36*x+64)*log(x)^ 4+(4*x^2+16*x)*log(x)^2+x^2)/log(x)^4),x, algorithm=\
e^(-1/4*(4*(x^2 + 9*x + 16)*log(x)^4 + 4*(x^2 + 4*x)*log(x)^2 + x^2)/log(x )^4 + 5)*log(-x^2 + x)
Timed out. \begin {dmath*} \int \frac {e^{-\frac {x^2+\left (16 x+4 x^2\right ) \log ^2(x)+\left (64+36 x+4 x^2\right ) \log ^4(x)}{4 \log ^4(x)}} \left (e^5 \left (-2 x^2+2 x^3\right ) \log \left (x-x^2\right )+e^5 \left (x^2-x^3\right ) \log (x) \log \left (x-x^2\right )+e^5 \left (-16 x+12 x^2+4 x^3\right ) \log ^2(x) \log \left (x-x^2\right )+e^5 \left (8 x-4 x^2-4 x^3\right ) \log ^3(x) \log \left (x-x^2\right )+\log ^5(x) \left (e^5 (-2+4 x)+e^5 \left (18 x-14 x^2-4 x^3\right ) \log \left (x-x^2\right )\right )\right )}{\left (-2 x+2 x^2\right ) \log ^5(x)} \, dx=\text {Timed out} \end {dmath*}
integrate((((-4*x**3-14*x**2+18*x)*exp(5)*ln(-x**2+x)+(4*x-2)*exp(5))*ln(x )**5+(-4*x**3-4*x**2+8*x)*exp(5)*ln(-x**2+x)*ln(x)**3+(4*x**3+12*x**2-16*x )*exp(5)*ln(-x**2+x)*ln(x)**2+(-x**3+x**2)*exp(5)*ln(-x**2+x)*ln(x)+(2*x** 3-2*x**2)*exp(5)*ln(-x**2+x))/(2*x**2-2*x)/ln(x)**5/exp(1/4*((4*x**2+36*x+ 64)*ln(x)**4+(4*x**2+16*x)*ln(x)**2+x**2)/ln(x)**4),x)
Time = 0.47 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \begin {dmath*} \int \frac {e^{-\frac {x^2+\left (16 x+4 x^2\right ) \log ^2(x)+\left (64+36 x+4 x^2\right ) \log ^4(x)}{4 \log ^4(x)}} \left (e^5 \left (-2 x^2+2 x^3\right ) \log \left (x-x^2\right )+e^5 \left (x^2-x^3\right ) \log (x) \log \left (x-x^2\right )+e^5 \left (-16 x+12 x^2+4 x^3\right ) \log ^2(x) \log \left (x-x^2\right )+e^5 \left (8 x-4 x^2-4 x^3\right ) \log ^3(x) \log \left (x-x^2\right )+\log ^5(x) \left (e^5 (-2+4 x)+e^5 \left (18 x-14 x^2-4 x^3\right ) \log \left (x-x^2\right )\right )\right )}{\left (-2 x+2 x^2\right ) \log ^5(x)} \, dx={\left (\log \left (x\right ) + \log \left (-x + 1\right )\right )} e^{\left (-x^{2} - 9 \, x - \frac {x^{2}}{\log \left (x\right )^{2}} - \frac {4 \, x}{\log \left (x\right )^{2}} - \frac {x^{2}}{4 \, \log \left (x\right )^{4}} - 11\right )} \end {dmath*}
integrate((((-4*x^3-14*x^2+18*x)*exp(5)*log(-x^2+x)+(4*x-2)*exp(5))*log(x) ^5+(-4*x^3-4*x^2+8*x)*exp(5)*log(-x^2+x)*log(x)^3+(4*x^3+12*x^2-16*x)*exp( 5)*log(-x^2+x)*log(x)^2+(-x^3+x^2)*exp(5)*log(-x^2+x)*log(x)+(2*x^3-2*x^2) *exp(5)*log(-x^2+x))/(2*x^2-2*x)/log(x)^5/exp(1/4*((4*x^2+36*x+64)*log(x)^ 4+(4*x^2+16*x)*log(x)^2+x^2)/log(x)^4),x, algorithm=\
\begin {dmath*} \int \frac {e^{-\frac {x^2+\left (16 x+4 x^2\right ) \log ^2(x)+\left (64+36 x+4 x^2\right ) \log ^4(x)}{4 \log ^4(x)}} \left (e^5 \left (-2 x^2+2 x^3\right ) \log \left (x-x^2\right )+e^5 \left (x^2-x^3\right ) \log (x) \log \left (x-x^2\right )+e^5 \left (-16 x+12 x^2+4 x^3\right ) \log ^2(x) \log \left (x-x^2\right )+e^5 \left (8 x-4 x^2-4 x^3\right ) \log ^3(x) \log \left (x-x^2\right )+\log ^5(x) \left (e^5 (-2+4 x)+e^5 \left (18 x-14 x^2-4 x^3\right ) \log \left (x-x^2\right )\right )\right )}{\left (-2 x+2 x^2\right ) \log ^5(x)} \, dx=\int { -\frac {{\left (4 \, {\left (x^{3} + x^{2} - 2 \, x\right )} e^{5} \log \left (-x^{2} + x\right ) \log \left (x\right )^{3} + 2 \, {\left ({\left (2 \, x^{3} + 7 \, x^{2} - 9 \, x\right )} e^{5} \log \left (-x^{2} + x\right ) - {\left (2 \, x - 1\right )} e^{5}\right )} \log \left (x\right )^{5} - 4 \, {\left (x^{3} + 3 \, x^{2} - 4 \, x\right )} e^{5} \log \left (-x^{2} + x\right ) \log \left (x\right )^{2} + {\left (x^{3} - x^{2}\right )} e^{5} \log \left (-x^{2} + x\right ) \log \left (x\right ) - 2 \, {\left (x^{3} - x^{2}\right )} e^{5} \log \left (-x^{2} + x\right )\right )} e^{\left (-\frac {4 \, {\left (x^{2} + 9 \, x + 16\right )} \log \left (x\right )^{4} + 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} + x^{2}}{4 \, \log \left (x\right )^{4}}\right )}}{2 \, {\left (x^{2} - x\right )} \log \left (x\right )^{5}} \,d x } \end {dmath*}
integrate((((-4*x^3-14*x^2+18*x)*exp(5)*log(-x^2+x)+(4*x-2)*exp(5))*log(x) ^5+(-4*x^3-4*x^2+8*x)*exp(5)*log(-x^2+x)*log(x)^3+(4*x^3+12*x^2-16*x)*exp( 5)*log(-x^2+x)*log(x)^2+(-x^3+x^2)*exp(5)*log(-x^2+x)*log(x)+(2*x^3-2*x^2) *exp(5)*log(-x^2+x))/(2*x^2-2*x)/log(x)^5/exp(1/4*((4*x^2+36*x+64)*log(x)^ 4+(4*x^2+16*x)*log(x)^2+x^2)/log(x)^4),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {e^{-\frac {x^2+\left (16 x+4 x^2\right ) \log ^2(x)+\left (64+36 x+4 x^2\right ) \log ^4(x)}{4 \log ^4(x)}} \left (e^5 \left (-2 x^2+2 x^3\right ) \log \left (x-x^2\right )+e^5 \left (x^2-x^3\right ) \log (x) \log \left (x-x^2\right )+e^5 \left (-16 x+12 x^2+4 x^3\right ) \log ^2(x) \log \left (x-x^2\right )+e^5 \left (8 x-4 x^2-4 x^3\right ) \log ^3(x) \log \left (x-x^2\right )+\log ^5(x) \left (e^5 (-2+4 x)+e^5 \left (18 x-14 x^2-4 x^3\right ) \log \left (x-x^2\right )\right )\right )}{\left (-2 x+2 x^2\right ) \log ^5(x)} \, dx=\int -\frac {{\mathrm {e}}^{-\frac {\frac {{\ln \left (x\right )}^2\,\left (4\,x^2+16\,x\right )}{4}+\frac {{\ln \left (x\right )}^4\,\left (4\,x^2+36\,x+64\right )}{4}+\frac {x^2}{4}}{{\ln \left (x\right )}^4}}\,\left (\left ({\mathrm {e}}^5\,\left (4\,x-2\right )-{\mathrm {e}}^5\,\ln \left (x-x^2\right )\,\left (4\,x^3+14\,x^2-18\,x\right )\right )\,{\ln \left (x\right )}^5-{\mathrm {e}}^5\,\ln \left (x-x^2\right )\,\left (4\,x^3+4\,x^2-8\,x\right )\,{\ln \left (x\right )}^3+{\mathrm {e}}^5\,\ln \left (x-x^2\right )\,\left (4\,x^3+12\,x^2-16\,x\right )\,{\ln \left (x\right )}^2+{\mathrm {e}}^5\,\ln \left (x-x^2\right )\,\left (x^2-x^3\right )\,\ln \left (x\right )-{\mathrm {e}}^5\,\ln \left (x-x^2\right )\,\left (2\,x^2-2\,x^3\right )\right )}{{\ln \left (x\right )}^5\,\left (2\,x-2\,x^2\right )} \,d x \end {dmath*}
int(-(exp(-((log(x)^2*(16*x + 4*x^2))/4 + (log(x)^4*(36*x + 4*x^2 + 64))/4 + x^2/4)/log(x)^4)*(log(x)^5*(exp(5)*(4*x - 2) - exp(5)*log(x - x^2)*(14* x^2 - 18*x + 4*x^3)) - exp(5)*log(x - x^2)*(2*x^2 - 2*x^3) - exp(5)*log(x - x^2)*log(x)^3*(4*x^2 - 8*x + 4*x^3) + exp(5)*log(x - x^2)*log(x)^2*(12*x ^2 - 16*x + 4*x^3) + exp(5)*log(x - x^2)*log(x)*(x^2 - x^3)))/(log(x)^5*(2 *x - 2*x^2)),x)
int(-(exp(-((log(x)^2*(16*x + 4*x^2))/4 + (log(x)^4*(36*x + 4*x^2 + 64))/4 + x^2/4)/log(x)^4)*(log(x)^5*(exp(5)*(4*x - 2) - exp(5)*log(x - x^2)*(14* x^2 - 18*x + 4*x^3)) - exp(5)*log(x - x^2)*(2*x^2 - 2*x^3) - exp(5)*log(x - x^2)*log(x)^3*(4*x^2 - 8*x + 4*x^3) + exp(5)*log(x - x^2)*log(x)^2*(12*x ^2 - 16*x + 4*x^3) + exp(5)*log(x - x^2)*log(x)*(x^2 - x^3)))/(log(x)^5*(2 *x - 2*x^2)), x)