Integrand size = 155, antiderivative size = 29 \[ \int \frac {\left (-12 x^2-2 e^x x^2\right ) \log (x)+2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )} \left (6+\left (6+6 e^x x\right ) \log (x)+\left (-2+\left (-2-2 e^x x\right ) \log (x)\right ) \log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6}}{\left (-6 x-e^x x\right ) \log (x)+x \log (x) \log \left (\frac {3}{x \log (x)}\right )} \, dx=5+e^{\left (-3+\log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )^2}+x^2 \]
Time = 0.47 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {\left (-12 x^2-2 e^x x^2\right ) \log (x)+2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )} \left (6+\left (6+6 e^x x\right ) \log (x)+\left (-2+\left (-2-2 e^x x\right ) \log (x)\right ) \log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6}}{\left (-6 x-e^x x\right ) \log (x)+x \log (x) \log \left (\frac {3}{x \log (x)}\right )} \, dx=x^2+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )}}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6} \]
Integrate[((-12*x^2 - 2*E^x*x^2)*Log[x] + 2*x^2*Log[x]*Log[3/(x*Log[x])] + (E^(9 + Log[6 + E^x - Log[3/(x*Log[x])]]^2)*(6 + (6 + 6*E^x*x)*Log[x] + ( -2 + (-2 - 2*E^x*x)*Log[x])*Log[6 + E^x - Log[3/(x*Log[x])]]))/(6 + E^x - Log[3/(x*Log[x])])^6)/((-6*x - E^x*x)*Log[x] + x*Log[x]*Log[3/(x*Log[x])]) ,x]
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(29)=58\).
Time = 10.63 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7292, 7293, 2009}
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.
| \(\displaystyle \int \frac {2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\left (-2 e^x x^2-12 x^2\right ) \log (x)+\frac {e^{\log ^2\left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )+9} \left (\left (6 e^x x+6\right ) \log (x)+\left (\left (-2 e^x x-2\right ) \log (x)-2\right ) \log \left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )+6\right )}{\left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )^6}}{\left (-e^x x-6 x\right ) \log (x)+x \log \left (\frac {3}{x \log (x)}\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )-\left (-2 e^x x^2-12 x^2\right ) \log (x)-\frac {e^{\log ^2\left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )+9} \left (\left (6 e^x x+6\right ) \log (x)+\left (\left (-2 e^x x-2\right ) \log (x)-2\right ) \log \left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )+6\right )}{\left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )^6}}{x \log (x) \left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x+\frac {2 e^{\log ^2\left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )+9} \left (e^x x \log (x)+\log (x)+1\right ) \left (\log \left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )-3\right )}{x \log (x) \left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^2+\frac {e^{\log ^2\left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )+9} \left (e^x x \log (x)+\log (x)+1\right )}{x \log (x) \left (x \left (\frac {1}{x^2 \log ^2(x)}+\frac {1}{x^2 \log (x)}\right ) \log (x)+e^x\right ) \left (e^x-\log \left (\frac {3}{x \log (x)}\right )+6\right )^6}\) |
Int[((-12*x^2 - 2*E^x*x^2)*Log[x] + 2*x^2*Log[x]*Log[3/(x*Log[x])] + (E^(9 + Log[6 + E^x - Log[3/(x*Log[x])]]^2)*(6 + (6 + 6*E^x*x)*Log[x] + (-2 + ( -2 - 2*E^x*x)*Log[x])*Log[6 + E^x - Log[3/(x*Log[x])]]))/(6 + E^x - Log[3/ (x*Log[x])])^6)/((-6*x - E^x*x)*Log[x] + x*Log[x]*Log[3/(x*Log[x])]),x]
x^2 + (E^(9 + Log[6 + E^x - Log[3/(x*Log[x])]]^2)*(1 + Log[x] + E^x*x*Log[ x]))/(x*Log[x]*(E^x + x*(1/(x^2*Log[x]^2) + 1/(x^2*Log[x]))*Log[x])*(6 + E ^x - Log[3/(x*Log[x])])^6)
3.23.97.3.1 Defintions of rubi rules used
Time = 58.47 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59
| method | result | size |
| parallelrisch | \(x^{2}+{\mathrm e}^{\ln \left (-\ln \left (\frac {3}{x \ln \left (x \right )}\right )+{\mathrm e}^{x}+6\right )^{2}-6 \ln \left (-\ln \left (\frac {3}{x \ln \left (x \right )}\right )+{\mathrm e}^{x}+6\right )+9}\) | \(46\) |
| risch | \(x^{2}+\frac {{\mathrm e}^{9} {\mathrm e}^{\ln \left (-\ln \left (3\right )+\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x \ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i}{x \ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i}{x \ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right )}{2}+6+{\mathrm e}^{x}\right )^{2}}}{\left (-\ln \left (3\right )+\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x \ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i}{x \ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i}{x \ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right )}{2}+6+{\mathrm e}^{x}\right )^{6}}\) | \(156\) |
int(((((-2*exp(x)*x-2)*ln(x)-2)*ln(-ln(3/x/ln(x))+exp(x)+6)+(6*exp(x)*x+6) *ln(x)+6)*exp(ln(-ln(3/x/ln(x))+exp(x)+6)^2-6*ln(-ln(3/x/ln(x))+exp(x)+6)+ 9)+2*x^2*ln(x)*ln(3/x/ln(x))+(-2*exp(x)*x^2-12*x^2)*ln(x))/(x*ln(x)*ln(3/x /ln(x))+(-exp(x)*x-6*x)*ln(x)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {\left (-12 x^2-2 e^x x^2\right ) \log (x)+2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )} \left (6+\left (6+6 e^x x\right ) \log (x)+\left (-2+\left (-2-2 e^x x\right ) \log (x)\right ) \log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6}}{\left (-6 x-e^x x\right ) \log (x)+x \log (x) \log \left (\frac {3}{x \log (x)}\right )} \, dx=x^{2} + e^{\left (\log \left (e^{x} - \log \left (\frac {3}{x \log \left (x\right )}\right ) + 6\right )^{2} - 6 \, \log \left (e^{x} - \log \left (\frac {3}{x \log \left (x\right )}\right ) + 6\right ) + 9\right )} \]
integrate(((((-2*exp(x)*x-2)*log(x)-2)*log(-log(3/x/log(x))+exp(x)+6)+(6*e xp(x)*x+6)*log(x)+6)*exp(log(-log(3/x/log(x))+exp(x)+6)^2-6*log(-log(3/x/l og(x))+exp(x)+6)+9)+2*x^2*log(x)*log(3/x/log(x))+(-2*exp(x)*x^2-12*x^2)*lo g(x))/(x*log(x)*log(3/x/log(x))+(-exp(x)*x-6*x)*log(x)),x, algorithm=\
Timed out. \[ \int \frac {\left (-12 x^2-2 e^x x^2\right ) \log (x)+2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )} \left (6+\left (6+6 e^x x\right ) \log (x)+\left (-2+\left (-2-2 e^x x\right ) \log (x)\right ) \log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6}}{\left (-6 x-e^x x\right ) \log (x)+x \log (x) \log \left (\frac {3}{x \log (x)}\right )} \, dx=\text {Timed out} \]
integrate(((((-2*exp(x)*x-2)*ln(x)-2)*ln(-ln(3/x/ln(x))+exp(x)+6)+(6*exp(x )*x+6)*ln(x)+6)*exp(ln(-ln(3/x/ln(x))+exp(x)+6)**2-6*ln(-ln(3/x/ln(x))+exp (x)+6)+9)+2*x**2*ln(x)*ln(3/x/ln(x))+(-2*exp(x)*x**2-12*x**2)*ln(x))/(x*ln (x)*ln(3/x/ln(x))+(-exp(x)*x-6*x)*ln(x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 2490 vs. \(2 (27) = 54\).
Time = 0.47 (sec) , antiderivative size = 2490, normalized size of antiderivative = 85.86 \[ \int \frac {\left (-12 x^2-2 e^x x^2\right ) \log (x)+2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )} \left (6+\left (6+6 e^x x\right ) \log (x)+\left (-2+\left (-2-2 e^x x\right ) \log (x)\right ) \log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6}}{\left (-6 x-e^x x\right ) \log (x)+x \log (x) \log \left (\frac {3}{x \log (x)}\right )} \, dx=\text {Too large to display} \]
integrate(((((-2*exp(x)*x-2)*log(x)-2)*log(-log(3/x/log(x))+exp(x)+6)+(6*e xp(x)*x+6)*log(x)+6)*exp(log(-log(3/x/log(x))+exp(x)+6)^2-6*log(-log(3/x/l og(x))+exp(x)+6)+9)+2*x^2*log(x)*log(3/x/log(x))+(-2*exp(x)*x^2-12*x^2)*lo g(x))/(x*log(x)*log(3/x/log(x))+(-exp(x)*x-6*x)*log(x)),x, algorithm=\
-(6*x^2*(log(3) - 6)*log(x)^5 - x^2*log(x)^6 - x^2*log(log(x))^6 - 15*(log (3)^2 - 12*log(3) + 36)*x^2*log(x)^4 + 20*(log(3)^3 - 18*log(3)^2 + 108*lo g(3) - 216)*x^2*log(x)^3 + 6*(x^2*(log(3) - 6) - x^2*e^x - x^2*log(x))*log (log(x))^5 - 15*(log(3)^4 - 24*log(3)^3 + 216*log(3)^2 - 864*log(3) + 1296 )*x^2*log(x)^2 + 15*(2*x^2*(log(3) - 6)*log(x) - x^2*log(x)^2 - (log(3)^2 - 12*log(3) + 36)*x^2 - x^2*e^(2*x) + 2*(x^2*(log(3) - 6) - x^2*log(x))*e^ x)*log(log(x))^4 + 6*(log(3)^5 - 30*log(3)^4 + 360*log(3)^3 - 2160*log(3)^ 2 + 6480*log(3) - 7776)*x^2*log(x) + 20*(3*x^2*(log(3) - 6)*log(x)^2 - x^2 *log(x)^3 - 3*(log(3)^2 - 12*log(3) + 36)*x^2*log(x) + (log(3)^3 - 18*log( 3)^2 + 108*log(3) - 216)*x^2 - x^2*e^(3*x) + 3*(x^2*(log(3) - 6) - x^2*log (x))*e^(2*x) + 3*(2*x^2*(log(3) - 6)*log(x) - x^2*log(x)^2 - (log(3)^2 - 1 2*log(3) + 36)*x^2)*e^x)*log(log(x))^3 - (log(3)^6 - 36*log(3)^5 + 540*log (3)^4 - 4320*log(3)^3 + 19440*log(3)^2 - 46656*log(3) + 46656)*x^2 - x^2*e ^(6*x) + 15*(4*x^2*(log(3) - 6)*log(x)^3 - x^2*log(x)^4 - 6*(log(3)^2 - 12 *log(3) + 36)*x^2*log(x)^2 + 4*(log(3)^3 - 18*log(3)^2 + 108*log(3) - 216) *x^2*log(x) - (log(3)^4 - 24*log(3)^3 + 216*log(3)^2 - 864*log(3) + 1296)* x^2 - x^2*e^(4*x) + 4*(x^2*(log(3) - 6) - x^2*log(x))*e^(3*x) + 6*(2*x^2*( log(3) - 6)*log(x) - x^2*log(x)^2 - (log(3)^2 - 12*log(3) + 36)*x^2)*e^(2* x) + 4*(3*x^2*(log(3) - 6)*log(x)^2 - x^2*log(x)^3 - 3*(log(3)^2 - 12*log( 3) + 36)*x^2*log(x) + (log(3)^3 - 18*log(3)^2 + 108*log(3) - 216)*x^2)*...
\[ \int \frac {\left (-12 x^2-2 e^x x^2\right ) \log (x)+2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )} \left (6+\left (6+6 e^x x\right ) \log (x)+\left (-2+\left (-2-2 e^x x\right ) \log (x)\right ) \log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6}}{\left (-6 x-e^x x\right ) \log (x)+x \log (x) \log \left (\frac {3}{x \log (x)}\right )} \, dx=\int { \frac {2 \, {\left (x^{2} \log \left (x\right ) \log \left (\frac {3}{x \log \left (x\right )}\right ) + {\left (3 \, {\left (x e^{x} + 1\right )} \log \left (x\right ) - {\left ({\left (x e^{x} + 1\right )} \log \left (x\right ) + 1\right )} \log \left (e^{x} - \log \left (\frac {3}{x \log \left (x\right )}\right ) + 6\right ) + 3\right )} e^{\left (\log \left (e^{x} - \log \left (\frac {3}{x \log \left (x\right )}\right ) + 6\right )^{2} - 6 \, \log \left (e^{x} - \log \left (\frac {3}{x \log \left (x\right )}\right ) + 6\right ) + 9\right )} - {\left (x^{2} e^{x} + 6 \, x^{2}\right )} \log \left (x\right )\right )}}{x \log \left (x\right ) \log \left (\frac {3}{x \log \left (x\right )}\right ) - {\left (x e^{x} + 6 \, x\right )} \log \left (x\right )} \,d x } \]
integrate(((((-2*exp(x)*x-2)*log(x)-2)*log(-log(3/x/log(x))+exp(x)+6)+(6*e xp(x)*x+6)*log(x)+6)*exp(log(-log(3/x/log(x))+exp(x)+6)^2-6*log(-log(3/x/l og(x))+exp(x)+6)+9)+2*x^2*log(x)*log(3/x/log(x))+(-2*exp(x)*x^2-12*x^2)*lo g(x))/(x*log(x)*log(3/x/log(x))+(-exp(x)*x-6*x)*log(x)),x, algorithm=\
Time = 14.23 (sec) , antiderivative size = 1132, normalized size of antiderivative = 39.03 \[ \int \frac {\left (-12 x^2-2 e^x x^2\right ) \log (x)+2 x^2 \log (x) \log \left (\frac {3}{x \log (x)}\right )+\frac {e^{9+\log ^2\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )} \left (6+\left (6+6 e^x x\right ) \log (x)+\left (-2+\left (-2-2 e^x x\right ) \log (x)\right ) \log \left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )\right )}{\left (6+e^x-\log \left (\frac {3}{x \log (x)}\right )\right )^6}}{\left (-6 x-e^x x\right ) \log (x)+x \log (x) \log \left (\frac {3}{x \log (x)}\right )} \, dx=\text {Too large to display} \]
int(-(exp(log(exp(x) - log(3/(x*log(x))) + 6)^2 - 6*log(exp(x) - log(3/(x* log(x))) + 6) + 9)*(log(x)*(6*x*exp(x) + 6) - log(exp(x) - log(3/(x*log(x) )) + 6)*(log(x)*(2*x*exp(x) + 2) + 2) + 6) - log(x)*(2*x^2*exp(x) + 12*x^2 ) + 2*x^2*log(x)*log(3/(x*log(x))))/(log(x)*(6*x + x*exp(x)) - x*log(x)*lo g(3/(x*log(x)))),x)
x^2 + (exp(9)*exp(log(exp(x) - log(3) - log(1/(x*log(x))) + 6)^2))/(19440* exp(2*x) + 4320*exp(3*x) + 540*exp(4*x) + 36*exp(5*x) + exp(6*x) - 46656*l og(3) + 46656*exp(x) - 46656*log(1/(x*log(x))) - 12960*exp(2*x)*log(3) - 2 160*exp(3*x)*log(3) - 180*exp(4*x)*log(3) - 6*exp(5*x)*log(3) + 12960*exp( x)*log(3)^2 - 2160*exp(x)*log(3)^3 + 180*exp(x)*log(3)^4 - 6*exp(x)*log(3) ^5 - 12960*log(3)*log(1/(x*log(x)))^2 - 12960*log(3)^2*log(1/(x*log(x))) + 2160*log(3)*log(1/(x*log(x)))^3 + 2160*log(3)^3*log(1/(x*log(x))) - 180*l og(3)*log(1/(x*log(x)))^4 - 180*log(3)^4*log(1/(x*log(x))) + 6*log(3)*log( 1/(x*log(x)))^5 + 6*log(3)^5*log(1/(x*log(x))) - 38880*exp(x)*log(1/(x*log (x))) + 19440*log(1/(x*log(x)))^2 - 4320*log(1/(x*log(x)))^3 + 540*log(1/( x*log(x)))^4 - 36*log(1/(x*log(x)))^5 + log(1/(x*log(x)))^6 + 3240*exp(2*x )*log(3)^2 - 360*exp(2*x)*log(3)^3 + 360*exp(3*x)*log(3)^2 + 15*exp(2*x)*l og(3)^4 - 20*exp(3*x)*log(3)^3 + 15*exp(4*x)*log(3)^2 + 3240*log(3)^2*log( 1/(x*log(x)))^2 - 360*log(3)^2*log(1/(x*log(x)))^3 - 360*log(3)^3*log(1/(x *log(x)))^2 + 15*log(3)^2*log(1/(x*log(x)))^4 + 20*log(3)^3*log(1/(x*log(x )))^3 + 15*log(3)^4*log(1/(x*log(x)))^2 - 12960*exp(2*x)*log(1/(x*log(x))) - 2160*exp(3*x)*log(1/(x*log(x))) - 180*exp(4*x)*log(1/(x*log(x))) - 6*ex p(5*x)*log(1/(x*log(x))) + 12960*exp(x)*log(1/(x*log(x)))^2 - 2160*exp(x)* log(1/(x*log(x)))^3 + 180*exp(x)*log(1/(x*log(x)))^4 - 6*exp(x)*log(1/(x*l og(x)))^5 - 38880*exp(x)*log(3) + 38880*log(3)*log(1/(x*log(x))) + 1944...