Integrand size = 205, antiderivative size = 34 \[ \int \frac {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx=e^{\frac {3}{3+\log \left (\frac {(3-x)^2 x^3}{4 \left (-x+x^2\right )^2}\right )}} x \]
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx=e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} x \]
Integrate[(E^(3/(3 + Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]))*(18 - 36 *x + 6*x^2 + (18 - 24*x + 6*x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2) ] + (3 - 4*x + x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]^2))/(27 - 3 6*x + 9*x^2 + (18 - 24*x + 6*x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2 )] + (3 - 4*x + x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^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.
\(\displaystyle \int \frac {e^{\frac {3}{\log \left (\frac {x^3-6 x^2+9 x}{4 x^2-8 x+4}\right )+3}} \left (6 x^2+\left (x^2-4 x+3\right ) \log ^2\left (\frac {x^3-6 x^2+9 x}{4 x^2-8 x+4}\right )+\left (6 x^2-24 x+18\right ) \log \left (\frac {x^3-6 x^2+9 x}{4 x^2-8 x+4}\right )-36 x+18\right )}{9 x^2+\left (x^2-4 x+3\right ) \log ^2\left (\frac {x^3-6 x^2+9 x}{4 x^2-8 x+4}\right )+\left (6 x^2-24 x+18\right ) \log \left (\frac {x^3-6 x^2+9 x}{4 x^2-8 x+4}\right )-36 x+27} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{\frac {3}{\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3}} \left (6 \left (x^2-6 x+3\right )+\left (x^2-4 x+3\right ) \log ^2\left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+6 \left (x^2-4 x+3\right ) \log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )\right )}{\left (x^2-4 x+3\right ) \left (\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3\right )^2}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (e^{\frac {3}{\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3}}-\frac {3 \left (x^2+3\right ) e^{\frac {3}{\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3}}}{(x-3) (x-1) \left (\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{\frac {3}{\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3}}dx-3 \int \frac {e^{\frac {3}{\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3}}}{\left (\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3\right )^2}dx-18 \int \frac {e^{\frac {3}{\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3}}}{(x-3) \left (\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3\right )^2}dx+6 \int \frac {e^{\frac {3}{\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3}}}{(x-1) \left (\log \left (\frac {(x-3)^2 x}{4 (x-1)^2}\right )+3\right )^2}dx\) |
Int[(E^(3/(3 + Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]))*(18 - 36*x + 6 *x^2 + (18 - 24*x + 6*x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)] + (3 - 4*x + x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]^2))/(27 - 36*x + 9*x^2 + (18 - 24*x + 6*x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)] + ( 3 - 4*x + x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]^2),x]
3.18.26.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 13.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06
method | result | size |
risch | \(x \,{\mathrm e}^{\frac {3}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}}\) | \(36\) |
norman | \(\frac {x \ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right ) {\mathrm e}^{\frac {3}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}}+3 x \,{\mathrm e}^{\frac {3}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}}}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}\) | \(130\) |
int(((x^2-4*x+3)*ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18)*ln((x ^3-6*x^2+9*x)/(4*x^2-8*x+4))+6*x^2-36*x+18)*exp(3/(ln((x^3-6*x^2+9*x)/(4*x ^2-8*x+4))+3))/((x^2-4*x+3)*ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24* x+18)*ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x,method=_RETURNVER BOSE)
Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx=x e^{\left (\frac {3}{\log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) + 3}\right )} \]
integrate(((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18 )*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+6*x^2-36*x+18)*exp(3/(log((x^3-6*x^2+ 9*x)/(4*x^2-8*x+4))+3))/((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+ (6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x, algor ithm=\
Time = 8.72 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx=x e^{\frac {3}{\log {\left (\frac {x^{3} - 6 x^{2} + 9 x}{4 x^{2} - 8 x + 4} \right )} + 3}} \]
integrate(((x**2-4*x+3)*ln((x**3-6*x**2+9*x)/(4*x**2-8*x+4))**2+(6*x**2-24 *x+18)*ln((x**3-6*x**2+9*x)/(4*x**2-8*x+4))+6*x**2-36*x+18)*exp(3/(ln((x** 3-6*x**2+9*x)/(4*x**2-8*x+4))+3))/((x**2-4*x+3)*ln((x**3-6*x**2+9*x)/(4*x* *2-8*x+4))**2+(6*x**2-24*x+18)*ln((x**3-6*x**2+9*x)/(4*x**2-8*x+4))+9*x**2 -36*x+27),x)
\[ \int \frac {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx=\int { \frac {{\left ({\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )^{2} + 6 \, x^{2} + 6 \, {\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - 36 \, x + 18\right )} e^{\left (\frac {3}{\log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) + 3}\right )}}{{\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )^{2} + 9 \, x^{2} + 6 \, {\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - 36 \, x + 27} \,d x } \]
integrate(((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18 )*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+6*x^2-36*x+18)*exp(3/(log((x^3-6*x^2+ 9*x)/(4*x^2-8*x+4))+3))/((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+ (6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x, algor ithm=\
-2*x^3*e^(-3/(2*log(2) + 2*log(x - 1) - 2*log(x - 3) - log(x) - 3))/(x^2 + 3) + 12*x^2*e^(-3/(2*log(2) + 2*log(x - 1) - 2*log(x - 3) - log(x) - 3))/ (x^2 + 3) - 6*x*e^(-3/(2*log(2) + 2*log(x - 1) - 2*log(x - 3) - log(x) - 3 ))/(x^2 + 3) + integrate((4*log(2)^2 + 4*(2*log(2) - 2*log(x - 3) - log(x) - 3)*log(x - 1) + 4*log(x - 1)^2 - 4*(2*log(2) - log(x) - 3)*log(x - 3) + 4*log(x - 3)^2 - 2*(2*log(2) - 3)*log(x) + log(x)^2 - 12*log(2))*e^(-3/(2 *log(2) + 2*log(x - 1) - 2*log(x - 3) - log(x) - 3))/(4*log(2)^2 + 4*(2*lo g(2) - 2*log(x - 3) - log(x) - 3)*log(x - 1) + 4*log(x - 1)^2 - 4*(2*log(2 ) - log(x) - 3)*log(x - 3) + 4*log(x - 3)^2 - 2*(2*log(2) - 3)*log(x) + lo g(x)^2 - 12*log(2) + 9), x)
\[ \int \frac {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx=\int { \frac {{\left ({\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )^{2} + 6 \, x^{2} + 6 \, {\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - 36 \, x + 18\right )} e^{\left (\frac {3}{\log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) + 3}\right )}}{{\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )^{2} + 9 \, x^{2} + 6 \, {\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - 36 \, x + 27} \,d x } \]
integrate(((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18 )*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+6*x^2-36*x+18)*exp(3/(log((x^3-6*x^2+ 9*x)/(4*x^2-8*x+4))+3))/((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+ (6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x, algor ithm=\
integrate(((x^2 - 4*x + 3)*log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1))^2 + 6*x^2 + 6*(x^2 - 4*x + 3)*log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1)) - 36*x + 18)*e^(3/(log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1)) + 3))/((x^2 - 4*x + 3)*log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1))^2 + 9*x^2 + 6*(x^ 2 - 4*x + 3)*log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1)) - 36*x + 27), x)
Timed out. \[ \int \frac {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx=\int \frac {{\mathrm {e}}^{\frac {3}{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )+3}}\,\left (\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )\,\left (6\,x^2-24\,x+18\right )-36\,x+{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )}^2\,\left (x^2-4\,x+3\right )+6\,x^2+18\right )}{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )\,\left (6\,x^2-24\,x+18\right )-36\,x+{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )}^2\,\left (x^2-4\,x+3\right )+9\,x^2+27} \,d x \]
int((exp(3/(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4)) + 3))*(log((9*x - 6 *x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^2 - 24*x + 18) - 36*x + log((9*x - 6*x ^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x^2 - 4*x + 3) + 6*x^2 + 18))/(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^2 - 24*x + 18) - 36*x + log((9*x - 6* x^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x^2 - 4*x + 3) + 9*x^2 + 27),x)
int((exp(3/(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4)) + 3))*(log((9*x - 6 *x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^2 - 24*x + 18) - 36*x + log((9*x - 6*x ^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x^2 - 4*x + 3) + 6*x^2 + 18))/(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^2 - 24*x + 18) - 36*x + log((9*x - 6* x^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x^2 - 4*x + 3) + 9*x^2 + 27), x)