Integrand size = 97, antiderivative size = 28 \[ \int \frac {e^{-4-x+\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}-\log ^2(x)} \left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-4 e-2 e \log \left (\frac {3}{x}\right )\right ) \log (x)\right )}{4 x+4 x \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=-4+e^{\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}} \]
Time = 5.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-4-x+\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}-\log ^2(x)} \left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-4 e-2 e \log \left (\frac {3}{x}\right )\right ) \log (x)\right )}{4 x+4 x \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}} \]
Integrate[(E^(-4 - x + E^(-3 - x - Log[x]^2)/(2 + Log[3/x]) - Log[x]^2)*(E *(1 - 2*x) - E*x*Log[3/x] + (-4*E - 2*E*Log[3/x])*Log[x]))/(4*x + 4*x*Log[ 3/x] + x*Log[3/x]^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.
\(\displaystyle \int \frac {\left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-2 e \log \left (\frac {3}{x}\right )-4 e\right ) \log (x)\right ) \exp \left (-x-\log ^2(x)+\frac {e^{-x-\log ^2(x)-3}}{\log \left (\frac {3}{x}\right )+2}-4\right )}{4 x+x \log ^2\left (\frac {3}{x}\right )+4 x \log \left (\frac {3}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-2 x+x \left (-\log \left (\frac {3}{x}\right )\right )-2 \log \left (\frac {3}{x}\right ) \log (x)-4 \log (x)+1\right ) \exp \left (-x-\log ^2(x)+\frac {e^{-x-\log ^2(x)-3}}{\log \left (\frac {3}{x}\right )+2}-3\right )}{x \left (\log \left (\frac {3}{x}\right )+2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (-2 x+x \left (-\log \left (\frac {3}{x}\right )\right )+1\right ) \exp \left (-x-\log ^2(x)+\frac {e^{-x-\log ^2(x)-3}}{\log \left (\frac {3}{x}\right )+2}-3\right )}{x \left (\log \left (\frac {3}{x}\right )+2\right )^2}-\frac {2 \log (x) \exp \left (-x-\log ^2(x)+\frac {e^{-x-\log ^2(x)-3}}{\log \left (\frac {3}{x}\right )+2}-3\right )}{x \left (\log \left (\frac {3}{x}\right )+2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {e^{-\log ^2(x)-x+\frac {e^{-\log ^2(x)-x-3}}{\log \left (\frac {3}{x}\right )+2}-3}}{-\log \left (\frac {3}{x}\right )-2}dx+\int \frac {e^{-\log ^2(x)-x+\frac {e^{-\log ^2(x)-x-3}}{\log \left (\frac {3}{x}\right )+2}-3}}{x \left (\log \left (\frac {3}{x}\right )+2\right )^2}dx-2 \int \frac {e^{-\log ^2(x)-x+\frac {e^{-\log ^2(x)-x-3}}{\log \left (\frac {3}{x}\right )+2}-3} \log (x)}{x \left (\log \left (\frac {3}{x}\right )+2\right )}dx\) |
Int[(E^(-4 - x + E^(-3 - x - Log[x]^2)/(2 + Log[3/x]) - Log[x]^2)*(E*(1 - 2*x) - E*x*Log[3/x] + (-4*E - 2*E*Log[3/x])*Log[x]))/(4*x + 4*x*Log[3/x] + x*Log[3/x]^2),x]
3.15.44.3.1 Defintions of rubi rules used
Time = 104.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
risch | \({\mathrm e}^{\frac {{\mathrm e}^{-3-\ln \left (x \right )^{2}-x}}{\ln \left (3\right )-\ln \left (x \right )+2}}\) | \(25\) |
parallelrisch | \({\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{-\ln \left (x \right )^{2}-4-x}}{\ln \left (\frac {3}{x}\right )+2}}\) | \(25\) |
int(((-2*exp(1)*ln(3/x)-4*exp(1))*ln(x)-x*exp(1)*ln(3/x)+(1-2*x)*exp(1))*e xp(exp(1)/(ln(3/x)+2)/exp(ln(x)^2+4+x))/(x*ln(3/x)^2+4*x*ln(3/x)+4*x)/exp( ln(x)^2+4+x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.39 \[ \int \frac {e^{-4-x+\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}-\log ^2(x)} \left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-4 e-2 e \log \left (\frac {3}{x}\right )\right ) \log (x)\right )}{4 x+4 x \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\frac {3}{x}\right ) + \log \left (\frac {3}{x}\right )^{2} + x + \frac {2 \, {\left (\log \left (3\right ) - 1\right )} \log \left (\frac {3}{x}\right )^{2} - \log \left (\frac {3}{x}\right )^{3} - 2 \, \log \left (3\right )^{2} - {\left (\log \left (3\right )^{2} + x - 4 \, \log \left (3\right ) + 4\right )} \log \left (\frac {3}{x}\right ) - 2 \, x + e^{\left (-\log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (\frac {3}{x}\right ) - \log \left (\frac {3}{x}\right )^{2} - x - 3\right )} - 8}{\log \left (\frac {3}{x}\right ) + 2} + 4\right )} \]
integrate(((-2*exp(1)*log(3/x)-4*exp(1))*log(x)-x*exp(1)*log(3/x)+(1-2*x)* exp(1))*exp(exp(1)/(log(3/x)+2)/exp(log(x)^2+4+x))/(x*log(3/x)^2+4*x*log(3 /x)+4*x)/exp(log(x)^2+4+x),x, algorithm=\
e^(log(3)^2 - 2*log(3)*log(3/x) + log(3/x)^2 + x + (2*(log(3) - 1)*log(3/x )^2 - log(3/x)^3 - 2*log(3)^2 - (log(3)^2 + x - 4*log(3) + 4)*log(3/x) - 2 *x + e^(-log(3)^2 + 2*log(3)*log(3/x) - log(3/x)^2 - x - 3) - 8)/(log(3/x) + 2) + 4)
Time = 1.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-4-x+\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}-\log ^2(x)} \left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-4 e-2 e \log \left (\frac {3}{x}\right )\right ) \log (x)\right )}{4 x+4 x \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\frac {e e^{- x - \log {\left (x \right )}^{2} - 4}}{- \log {\left (x \right )} + \log {\left (3 \right )} + 2}} \]
integrate(((-2*exp(1)*ln(3/x)-4*exp(1))*ln(x)-x*exp(1)*ln(3/x)+(1-2*x)*exp (1))*exp(exp(1)/(ln(3/x)+2)/exp(ln(x)**2+4+x))/(x*ln(3/x)**2+4*x*ln(3/x)+4 *x)/exp(ln(x)**2+4+x),x)
Time = 1.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-4-x+\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}-\log ^2(x)} \left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-4 e-2 e \log \left (\frac {3}{x}\right )\right ) \log (x)\right )}{4 x+4 x \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=e^{\left (\frac {1}{{\left (e^{3} \log \left (3\right ) + 2 \, e^{3}\right )} e^{\left (\log \left (x\right )^{2} + x\right )} - e^{\left (\log \left (x\right )^{2} + x + 3\right )} \log \left (x\right )}\right )} \]
integrate(((-2*exp(1)*log(3/x)-4*exp(1))*log(x)-x*exp(1)*log(3/x)+(1-2*x)* exp(1))*exp(exp(1)/(log(3/x)+2)/exp(log(x)^2+4+x))/(x*log(3/x)^2+4*x*log(3 /x)+4*x)/exp(log(x)^2+4+x),x, algorithm=\
\[ \int \frac {e^{-4-x+\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}-\log ^2(x)} \left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-4 e-2 e \log \left (\frac {3}{x}\right )\right ) \log (x)\right )}{4 x+4 x \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx=\int { -\frac {{\left (x e \log \left (\frac {3}{x}\right ) + {\left (2 \, x - 1\right )} e + 2 \, {\left (e \log \left (\frac {3}{x}\right ) + 2 \, e\right )} \log \left (x\right )\right )} e^{\left (-\log \left (x\right )^{2} - x + \frac {e^{\left (-\log \left (x\right )^{2} - x - 3\right )}}{\log \left (\frac {3}{x}\right ) + 2} - 4\right )}}{x \log \left (\frac {3}{x}\right )^{2} + 4 \, x \log \left (\frac {3}{x}\right ) + 4 \, x} \,d x } \]
integrate(((-2*exp(1)*log(3/x)-4*exp(1))*log(x)-x*exp(1)*log(3/x)+(1-2*x)* exp(1))*exp(exp(1)/(log(3/x)+2)/exp(log(x)^2+4+x))/(x*log(3/x)^2+4*x*log(3 /x)+4*x)/exp(log(x)^2+4+x),x, algorithm=\
integrate(-(x*e*log(3/x) + (2*x - 1)*e + 2*(e*log(3/x) + 2*e)*log(x))*e^(- log(x)^2 - x + e^(-log(x)^2 - x - 3)/(log(3/x) + 2) - 4)/(x*log(3/x)^2 + 4 *x*log(3/x) + 4*x), x)
Time = 7.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-4-x+\frac {e^{-3-x-\log ^2(x)}}{2+\log \left (\frac {3}{x}\right )}-\log ^2(x)} \left (e (1-2 x)-e x \log \left (\frac {3}{x}\right )+\left (-4 e-2 e \log \left (\frac {3}{x}\right )\right ) \log (x)\right )}{4 x+4 x \log \left (\frac {3}{x}\right )+x \log ^2\left (\frac {3}{x}\right )} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{-{\ln \left (x\right )}^2}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-3}}{\ln \left (\frac {1}{x}\right )+\ln \left (3\right )+2}} \]