Integrand size = 183, antiderivative size = 31 \[ \int \frac {e^{\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}} \left (4 e^{2/x} x^3+e^{2/x} \left (2 x^2-x^3+(-2+x) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right )}{\left (-x^3+x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx=-5+e^{x \left (-1+\frac {e^{2/x}}{\log \left (3 \left (x^2-\log (6)\right )^2\right )}\right )} \]
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}} \left (4 e^{2/x} x^3+e^{2/x} \left (2 x^2-x^3+(-2+x) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right )}{\left (-x^3+x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx=e^{x \left (-1+\frac {e^{2/x}}{\log \left (3 \left (x^2-\log (6)\right )^2\right )}\right )} \]
Integrate[(E^((E^(2/x)*x - x*Log[3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2])/Log[3 *x^4 - 6*x^2*Log[6] + 3*Log[6]^2])*(4*E^(2/x)*x^3 + E^(2/x)*(2*x^2 - x^3 + (-2 + x)*Log[6])*Log[3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2] + (x^3 - x*Log[6] )*Log[3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2]^2))/((-x^3 + x*Log[6])*Log[3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2]^2),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.
| \(\displaystyle \int \frac {\left (4 e^{2/x} x^3+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+e^{2/x} \left (-x^3+2 x^2+(x-2) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right ) \exp \left (\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}\right )}{\left (x \log (6)-x^3\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (4 e^{2/x} x^3+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+e^{2/x} \left (-x^3+2 x^2+(x-2) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right ) \exp \left (\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}\right )}{x \left (\log (6)-x^2\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\left (-4 x^3-2 x^2 \log \left (3 \left (x^2-\log (6)\right )^2\right )-x \log (6) \log \left (3 \left (x^2-\log (6)\right )^2\right )+2 \log (6) \log \left (3 \left (x^2-\log (6)\right )^2\right )+x^3 \log \left (3 \left (x^2-\log (6)\right )^2\right )\right ) \exp \left (\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}+\frac {2}{x}\right )}{x \left (x^2-\log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}-\exp \left (\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{\frac {e^{2/x} x}{\log \left (3 \left (x^2-\log (6)\right )^2\right )}-x} \left (-4 e^{2/x} x^3+e^{2/x} (x-2) \left (x^2-\log (6)\right ) \log \left (3 \left (x^2-\log (6)\right )^2\right )+\left (x \log (6)-x^3\right ) \log ^2\left (3 \left (x^2-\log (6)\right )^2\right )\right )}{x \left (x^2-\log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {e^{\frac {e^{2/x} x}{\log \left (3 \left (x^2-\log (6)\right )^2\right )}-x+\frac {2}{x}} \left (-4 x^3-2 x^2 \log \left (3 \left (x^2-\log (6)\right )^2\right )-x \log (6) \log \left (3 \left (x^2-\log (6)\right )^2\right )+2 \log (6) \log \left (3 \left (x^2-\log (6)\right )^2\right )+x^3 \log \left (3 \left (x^2-\log (6)\right )^2\right )\right )}{x \left (x^2-\log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}-e^{\frac {e^{2/x} x}{\log \left (3 \left (x^2-\log (6)\right )^2\right )}-x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {e^{\frac {e^{2/x} x}{\log \left (3 \left (x^2-\log (6)\right )^2\right )}-x+\frac {2}{x}} \left (-4 x^3-2 x^2 \log \left (3 \left (x^2-\log (6)\right )^2\right )-x \log (6) \log \left (3 \left (x^2-\log (6)\right )^2\right )+2 \log (6) \log \left (3 \left (x^2-\log (6)\right )^2\right )+x^3 \log \left (3 \left (x^2-\log (6)\right )^2\right )\right )}{x \left (x^2-\log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}-e^{\frac {e^{2/x} x}{\log \left (3 \left (x^2-\log (6)\right )^2\right )}-x}\right )dx\) |
Int[(E^((E^(2/x)*x - x*Log[3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2])/Log[3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2])*(4*E^(2/x)*x^3 + E^(2/x)*(2*x^2 - x^3 + (-2 + x)*Log[6])*Log[3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2] + (x^3 - x*Log[6])*Log[ 3*x^4 - 6*x^2*Log[6] + 3*Log[6]^2]^2))/((-x^3 + x*Log[6])*Log[3*x^4 - 6*x^ 2*Log[6] + 3*Log[6]^2]^2),x]
3.12.16.3.1 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])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 137.58 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81
| method | result | size |
| parallelrisch | \({\mathrm e}^{-\frac {x \left (\ln \left (3 \ln \left (6\right )^{2}-6 x^{2} \ln \left (6\right )+3 x^{4}\right )-{\mathrm e}^{\frac {2}{x}}\right )}{\ln \left (3 \ln \left (6\right )^{2}-6 x^{2} \ln \left (6\right )+3 x^{4}\right )}}\) | \(56\) |
| risch | \({\mathrm e}^{-\frac {x \left (-i \pi {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )^{2}\right )}^{3}+2 i \pi {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )\right )-i \pi \,\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )^{2}\right ) {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )\right )}^{2}-2 \,{\mathrm e}^{\frac {2}{x}}+2 \ln \left (3\right )+4 \ln \left (-x^{2}+\ln \left (2\right )+\ln \left (3\right )\right )\right )}{-i \pi {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )^{2}\right )}^{3}+2 i \pi {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )\right )-i \pi \,\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )^{2}\right ) {\operatorname {csgn}\left (i \left (-x^{2}+\ln \left (6\right )\right )\right )}^{2}+2 \ln \left (3\right )+4 \ln \left (-x^{2}+\ln \left (2\right )+\ln \left (3\right )\right )}}\) | \(219\) |
int(((-x*ln(6)+x^3)*ln(3*ln(6)^2-6*x^2*ln(6)+3*x^4)^2+((-2+x)*ln(6)-x^3+2* x^2)*exp(2/x)*ln(3*ln(6)^2-6*x^2*ln(6)+3*x^4)+4*x^3*exp(2/x))*exp((-x*ln(3 *ln(6)^2-6*x^2*ln(6)+3*x^4)+x*exp(2/x))/ln(3*ln(6)^2-6*x^2*ln(6)+3*x^4))/( x*ln(6)-x^3)/ln(3*ln(6)^2-6*x^2*ln(6)+3*x^4)^2,x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {e^{\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}} \left (4 e^{2/x} x^3+e^{2/x} \left (2 x^2-x^3+(-2+x) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right )}{\left (-x^3+x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx=e^{\left (\frac {x e^{\frac {2}{x}} - x \log \left (3 \, x^{4} - 6 \, x^{2} \log \left (6\right ) + 3 \, \log \left (6\right )^{2}\right )}{\log \left (3 \, x^{4} - 6 \, x^{2} \log \left (6\right ) + 3 \, \log \left (6\right )^{2}\right )}\right )} \]
integrate(((-x*log(6)+x^3)*log(3*log(6)^2-6*x^2*log(6)+3*x^4)^2+((-2+x)*lo g(6)-x^3+2*x^2)*exp(2/x)*log(3*log(6)^2-6*x^2*log(6)+3*x^4)+4*x^3*exp(2/x) )*exp((-x*log(3*log(6)^2-6*x^2*log(6)+3*x^4)+x*exp(2/x))/log(3*log(6)^2-6* x^2*log(6)+3*x^4))/(x*log(6)-x^3)/log(3*log(6)^2-6*x^2*log(6)+3*x^4)^2,x, algorithm=\
e^((x*e^(2/x) - x*log(3*x^4 - 6*x^2*log(6) + 3*log(6)^2))/log(3*x^4 - 6*x^ 2*log(6) + 3*log(6)^2))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.54 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {e^{\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}} \left (4 e^{2/x} x^3+e^{2/x} \left (2 x^2-x^3+(-2+x) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right )}{\left (-x^3+x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx=e^{\frac {x e^{\frac {2}{x}} - x \log {\left (3 x^{4} - 6 x^{2} \log {\left (6 \right )} + 3 \log {\left (6 \right )}^{2} \right )}}{\log {\left (3 x^{4} - 6 x^{2} \log {\left (6 \right )} + 3 \log {\left (6 \right )}^{2} \right )}}} \]
integrate(((-x*ln(6)+x**3)*ln(3*ln(6)**2-6*x**2*ln(6)+3*x**4)**2+((-2+x)*l n(6)-x**3+2*x**2)*exp(2/x)*ln(3*ln(6)**2-6*x**2*ln(6)+3*x**4)+4*x**3*exp(2 /x))*exp((-x*ln(3*ln(6)**2-6*x**2*ln(6)+3*x**4)+x*exp(2/x))/ln(3*ln(6)**2- 6*x**2*ln(6)+3*x**4))/(x*ln(6)-x**3)/ln(3*ln(6)**2-6*x**2*ln(6)+3*x**4)**2 ,x)
exp((x*exp(2/x) - x*log(3*x**4 - 6*x**2*log(6) + 3*log(6)**2))/log(3*x**4 - 6*x**2*log(6) + 3*log(6)**2))
Exception generated. \[ \int \frac {e^{\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}} \left (4 e^{2/x} x^3+e^{2/x} \left (2 x^2-x^3+(-2+x) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right )}{\left (-x^3+x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx=\text {Exception raised: RuntimeError} \]
integrate(((-x*log(6)+x^3)*log(3*log(6)^2-6*x^2*log(6)+3*x^4)^2+((-2+x)*lo g(6)-x^3+2*x^2)*exp(2/x)*log(3*log(6)^2-6*x^2*log(6)+3*x^4)+4*x^3*exp(2/x) )*exp((-x*log(3*log(6)^2-6*x^2*log(6)+3*x^4)+x*exp(2/x))/log(3*log(6)^2-6* x^2*log(6)+3*x^4))/(x*log(6)-x^3)/log(3*log(6)^2-6*x^2*log(6)+3*x^4)^2,x, algorithm=\
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
Time = 6.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {e^{\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}} \left (4 e^{2/x} x^3+e^{2/x} \left (2 x^2-x^3+(-2+x) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right )}{\left (-x^3+x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx=e^{\left (-x + \frac {x e^{\frac {2}{x}}}{\log \left (3 \, x^{4} - 6 \, x^{2} \log \left (6\right ) + 3 \, \log \left (6\right )^{2}\right )}\right )} \]
integrate(((-x*log(6)+x^3)*log(3*log(6)^2-6*x^2*log(6)+3*x^4)^2+((-2+x)*lo g(6)-x^3+2*x^2)*exp(2/x)*log(3*log(6)^2-6*x^2*log(6)+3*x^4)+4*x^3*exp(2/x) )*exp((-x*log(3*log(6)^2-6*x^2*log(6)+3*x^4)+x*exp(2/x))/log(3*log(6)^2-6* x^2*log(6)+3*x^4))/(x*log(6)-x^3)/log(3*log(6)^2-6*x^2*log(6)+3*x^4)^2,x, algorithm=\
Time = 13.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {e^{2/x} x-x \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}{\log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )}} \left (4 e^{2/x} x^3+e^{2/x} \left (2 x^2-x^3+(-2+x) \log (6)\right ) \log \left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )+\left (x^3-x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )\right )}{\left (-x^3+x \log (6)\right ) \log ^2\left (3 x^4-6 x^2 \log (6)+3 \log ^2(6)\right )} \, dx={\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{2/x}}{\ln \left (3\,x^4-6\,\ln \left (6\right )\,x^2+3\,{\ln \left (6\right )}^2\right )}}\,{\mathrm {e}}^{-x} \]
int((exp((x*exp(2/x) - x*log(3*log(6)^2 - 6*x^2*log(6) + 3*x^4))/log(3*log (6)^2 - 6*x^2*log(6) + 3*x^4))*(4*x^3*exp(2/x) - log(3*log(6)^2 - 6*x^2*lo g(6) + 3*x^4)^2*(x*log(6) - x^3) + exp(2/x)*log(3*log(6)^2 - 6*x^2*log(6) + 3*x^4)*(log(6)*(x - 2) + 2*x^2 - x^3)))/(log(3*log(6)^2 - 6*x^2*log(6) + 3*x^4)^2*(x*log(6) - x^3)),x)