Integrand size = 187, antiderivative size = 22 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=\left (e^{1+x+\frac {x}{\log (x-\log (x))}}-x\right )^2 \] Output:
(-x+exp(1+x/ln(x-ln(x))+x))^2
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=\left (e^{1+x+\frac {x}{\log (x-\log (x))}}-x\right )^2 \] Input:
Integrate[((-2*x^2 + 2*x*Log[x])*Log[x - Log[x]]^2 + E^((2*(x + (1 + x)*Lo g[x - Log[x]]))/Log[x - Log[x]])*(-2 + 2*x + (-2*x + 2*Log[x])*Log[x - Log [x]] + (-2*x + 2*Log[x])*Log[x - Log[x]]^2) + E^((x + (1 + x)*Log[x - Log[ x]])/Log[x - Log[x]])*(2*x - 2*x^2 + (2*x^2 - 2*x*Log[x])*Log[x - Log[x]] + (2*x + 2*x^2 + (-2 - 2*x)*Log[x])*Log[x - Log[x]]^2))/((-x + Log[x])*Log [x - Log[x]]^2),x]
Output:
(E^(1 + x + x/Log[x - Log[x]]) - x)^2
Leaf count is larger than twice the leaf count of optimal. \(155\) vs. \(2(22)=44\).
Time = 2.76 (sec) , antiderivative size = 155, normalized size of antiderivative = 7.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {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.
\(\displaystyle \int \frac {\left (2 x+(2 \log (x)-2 x) \log ^2(x-\log (x))+(2 \log (x)-2 x) \log (x-\log (x))-2\right ) \exp \left (\frac {2 (x+(x+1) \log (x-\log (x)))}{\log (x-\log (x))}\right )+\left (2 x \log (x)-2 x^2\right ) \log ^2(x-\log (x))+e^{\frac {x+(x+1) \log (x-\log (x))}{\log (x-\log (x))}} \left (-2 x^2+\left (2 x^2+2 x+(-2 x-2) \log (x)\right ) \log ^2(x-\log (x))+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+2 x\right )}{(\log (x)-x) \log ^2(x-\log (x))} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 e^{x+\frac {x}{\log (x-\log (x))}+1} \left (-x^2+x^2 \log ^2(x-\log (x))+x^2 \log (x-\log (x))+x-x \log (x) \log ^2(x-\log (x))+x \log ^2(x-\log (x))-\log (x) \log ^2(x-\log (x))-x \log (x) \log (x-\log (x))\right )}{(x-\log (x)) \log ^2(x-\log (x))}+2 x+\frac {2 e^{2 x+\frac {2 x}{\log (x-\log (x))}+2} \left (-x+x \log ^2(x-\log (x))-\log (x) \log ^2(x-\log (x))+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{(x-\log (x)) \log ^2(x-\log (x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2-\frac {2 e^{x+\frac {x}{\log (x-\log (x))}+1} \left (-x^2+x^2 \log ^2(x-\log (x))+x^2 \log (x-\log (x))+x-x \log (x) \log ^2(x-\log (x))-x \log (x) \log (x-\log (x))\right )}{(x-\log (x)) \left (-\frac {\left (1-\frac {1}{x}\right ) x}{(x-\log (x)) \log ^2(x-\log (x))}+\frac {1}{\log (x-\log (x))}+1\right ) \log ^2(x-\log (x))}+e^{2 x+\frac {2 x}{\log (x-\log (x))}+2}\) |
Input:
Int[((-2*x^2 + 2*x*Log[x])*Log[x - Log[x]]^2 + E^((2*(x + (1 + x)*Log[x - Log[x]]))/Log[x - Log[x]])*(-2 + 2*x + (-2*x + 2*Log[x])*Log[x - Log[x]] + (-2*x + 2*Log[x])*Log[x - Log[x]]^2) + E^((x + (1 + x)*Log[x - Log[x]])/L og[x - Log[x]])*(2*x - 2*x^2 + (2*x^2 - 2*x*Log[x])*Log[x - Log[x]] + (2*x + 2*x^2 + (-2 - 2*x)*Log[x])*Log[x - Log[x]]^2))/((-x + Log[x])*Log[x - L og[x]]^2),x]
Output:
E^(2 + 2*x + (2*x)/Log[x - Log[x]]) + x^2 - (2*E^(1 + x + x/Log[x - Log[x] ])*(x - x^2 + x^2*Log[x - Log[x]] - x*Log[x]*Log[x - Log[x]] + x^2*Log[x - Log[x]]^2 - x*Log[x]*Log[x - Log[x]]^2))/((x - Log[x])*(1 - ((1 - x^(-1)) *x)/((x - Log[x])*Log[x - Log[x]]^2) + Log[x - Log[x]]^(-1))*Log[x - Log[x ]]^2)
Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(21)=42\).
Time = 2.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64
method | result | size |
parallelrisch | \(x^{2}-2 x \,{\mathrm e}^{\frac {\left (1+x \right ) \ln \left (x -\ln \left (x \right )\right )+x}{\ln \left (x -\ln \left (x \right )\right )}}+{\mathrm e}^{\frac {2 \left (1+x \right ) \ln \left (x -\ln \left (x \right )\right )+2 x}{\ln \left (x -\ln \left (x \right )\right )}}\) | \(58\) |
risch | \(x^{2}-2 x \,{\mathrm e}^{\frac {x \ln \left (x -\ln \left (x \right )\right )+\ln \left (x -\ln \left (x \right )\right )+x}{\ln \left (x -\ln \left (x \right )\right )}}+{\mathrm e}^{\frac {2 x \ln \left (x -\ln \left (x \right )\right )+2 \ln \left (x -\ln \left (x \right )\right )+2 x}{\ln \left (x -\ln \left (x \right )\right )}}\) | \(67\) |
Input:
int((((2*ln(x)-2*x)*ln(x-ln(x))^2+(2*ln(x)-2*x)*ln(x-ln(x))+2*x-2)*exp(((1 +x)*ln(x-ln(x))+x)/ln(x-ln(x)))^2+(((-2-2*x)*ln(x)+2*x^2+2*x)*ln(x-ln(x))^ 2+(-2*x*ln(x)+2*x^2)*ln(x-ln(x))-2*x^2+2*x)*exp(((1+x)*ln(x-ln(x))+x)/ln(x -ln(x)))+(2*x*ln(x)-2*x^2)*ln(x-ln(x))^2)/(ln(x)-x)/ln(x-ln(x))^2,x,method =_RETURNVERBOSE)
Output:
x^2-2*x*exp(((1+x)*ln(x-ln(x))+x)/ln(x-ln(x)))+exp(((1+x)*ln(x-ln(x))+x)/l n(x-ln(x)))^2
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^{2} - 2 \, x e^{\left (\frac {{\left (x + 1\right )} \log \left (x - \log \left (x\right )\right ) + x}{\log \left (x - \log \left (x\right )\right )}\right )} + e^{\left (\frac {2 \, {\left ({\left (x + 1\right )} \log \left (x - \log \left (x\right )\right ) + x\right )}}{\log \left (x - \log \left (x\right )\right )}\right )} \] Input:
integrate((((2*log(x)-2*x)*log(x-log(x))^2+(2*log(x)-2*x)*log(x-log(x))+2* x-2)*exp(((1+x)*log(x-log(x))+x)/log(x-log(x)))^2+(((-2-2*x)*log(x)+2*x^2+ 2*x)*log(x-log(x))^2+(-2*x*log(x)+2*x^2)*log(x-log(x))-2*x^2+2*x)*exp(((1+ x)*log(x-log(x))+x)/log(x-log(x)))+(2*x*log(x)-2*x^2)*log(x-log(x))^2)/(lo g(x)-x)/log(x-log(x))^2,x, algorithm="fricas")
Output:
x^2 - 2*x*e^(((x + 1)*log(x - log(x)) + x)/log(x - log(x))) + e^(2*((x + 1 )*log(x - log(x)) + x)/log(x - log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).
Time = 1.62 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^{2} - 2 x e^{\frac {x + \left (x + 1\right ) \log {\left (x - \log {\left (x \right )} \right )}}{\log {\left (x - \log {\left (x \right )} \right )}}} + e^{\frac {2 \left (x + \left (x + 1\right ) \log {\left (x - \log {\left (x \right )} \right )}\right )}{\log {\left (x - \log {\left (x \right )} \right )}}} \] Input:
integrate((((2*ln(x)-2*x)*ln(x-ln(x))**2+(2*ln(x)-2*x)*ln(x-ln(x))+2*x-2)* exp(((1+x)*ln(x-ln(x))+x)/ln(x-ln(x)))**2+(((-2-2*x)*ln(x)+2*x**2+2*x)*ln( x-ln(x))**2+(-2*x*ln(x)+2*x**2)*ln(x-ln(x))-2*x**2+2*x)*exp(((1+x)*ln(x-ln (x))+x)/ln(x-ln(x)))+(2*x*ln(x)-2*x**2)*ln(x-ln(x))**2)/(ln(x)-x)/ln(x-ln( x))**2,x)
Output:
x**2 - 2*x*exp((x + (x + 1)*log(x - log(x)))/log(x - log(x))) + exp(2*(x + (x + 1)*log(x - log(x)))/log(x - log(x)))
Exception generated. \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((((2*log(x)-2*x)*log(x-log(x))^2+(2*log(x)-2*x)*log(x-log(x))+2* x-2)*exp(((1+x)*log(x-log(x))+x)/log(x-log(x)))^2+(((-2-2*x)*log(x)+2*x^2+ 2*x)*log(x-log(x))^2+(-2*x*log(x)+2*x^2)*log(x-log(x))-2*x^2+2*x)*exp(((1+ x)*log(x-log(x))+x)/log(x-log(x)))+(2*x*log(x)-2*x^2)*log(x-log(x))^2)/(lo g(x)-x)/log(x-log(x))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).
Time = 0.40 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^{2} - 2 \, x e^{\left (\frac {x \log \left (x - \log \left (x\right )\right ) + x + \log \left (x - \log \left (x\right )\right )}{\log \left (x - \log \left (x\right )\right )}\right )} + e^{\left (2 \, x + \frac {2 \, x}{\log \left (x - \log \left (x\right )\right )} + 2\right )} \] Input:
integrate((((2*log(x)-2*x)*log(x-log(x))^2+(2*log(x)-2*x)*log(x-log(x))+2* x-2)*exp(((1+x)*log(x-log(x))+x)/log(x-log(x)))^2+(((-2-2*x)*log(x)+2*x^2+ 2*x)*log(x-log(x))^2+(-2*x*log(x)+2*x^2)*log(x-log(x))-2*x^2+2*x)*exp(((1+ x)*log(x-log(x))+x)/log(x-log(x)))+(2*x*log(x)-2*x^2)*log(x-log(x))^2)/(lo g(x)-x)/log(x-log(x))^2,x, algorithm="giac")
Output:
x^2 - 2*x*e^((x*log(x - log(x)) + x + log(x - log(x)))/log(x - log(x))) + e^(2*x + 2*x/log(x - log(x)) + 2)
Time = 3.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=x^2+{\mathrm {e}}^{\frac {2\,x}{\ln \left (x-\ln \left (x\right )\right )}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2-2\,x\,{\mathrm {e}}^{\frac {x}{\ln \left (x-\ln \left (x\right )\right )}}\,\mathrm {e}\,{\mathrm {e}}^x \] Input:
int(-(exp((x + log(x - log(x))*(x + 1))/log(x - log(x)))*(2*x + log(x - lo g(x))^2*(2*x - log(x)*(2*x + 2) + 2*x^2) - log(x - log(x))*(2*x*log(x) - 2 *x^2) - 2*x^2) + log(x - log(x))^2*(2*x*log(x) - 2*x^2) - exp((2*(x + log( x - log(x))*(x + 1)))/log(x - log(x)))*(log(x - log(x))*(2*x - 2*log(x)) - 2*x + log(x - log(x))^2*(2*x - 2*log(x)) + 2))/(log(x - log(x))^2*(x - lo g(x))),x)
Output:
x^2 + exp((2*x)/log(x - log(x)))*exp(2*x)*exp(2) - 2*x*exp(x/log(x - log(x )))*exp(1)*exp(x)
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {\left (-2 x^2+2 x \log (x)\right ) \log ^2(x-\log (x))+e^{\frac {2 (x+(1+x) \log (x-\log (x)))}{\log (x-\log (x))}} \left (-2+2 x+(-2 x+2 \log (x)) \log (x-\log (x))+(-2 x+2 \log (x)) \log ^2(x-\log (x))\right )+e^{\frac {x+(1+x) \log (x-\log (x))}{\log (x-\log (x))}} \left (2 x-2 x^2+\left (2 x^2-2 x \log (x)\right ) \log (x-\log (x))+\left (2 x+2 x^2+(-2-2 x) \log (x)\right ) \log ^2(x-\log (x))\right )}{(-x+\log (x)) \log ^2(x-\log (x))} \, dx=e^{\frac {2 \,\mathrm {log}\left (x -\mathrm {log}\left (x \right )\right ) x +2 x}{\mathrm {log}\left (x -\mathrm {log}\left (x \right )\right )}} e^{2}-2 e^{\frac {\mathrm {log}\left (x -\mathrm {log}\left (x \right )\right ) x +x}{\mathrm {log}\left (x -\mathrm {log}\left (x \right )\right )}} e x +x^{2} \] Input:
int((((2*log(x)-2*x)*log(x-log(x))^2+(2*log(x)-2*x)*log(x-log(x))+2*x-2)*e xp(((1+x)*log(x-log(x))+x)/log(x-log(x)))^2+(((-2-2*x)*log(x)+2*x^2+2*x)*l og(x-log(x))^2+(-2*x*log(x)+2*x^2)*log(x-log(x))-2*x^2+2*x)*exp(((1+x)*log (x-log(x))+x)/log(x-log(x)))+(2*x*log(x)-2*x^2)*log(x-log(x))^2)/(log(x)-x )/log(x-log(x))^2,x)
Output:
e**((2*log( - log(x) + x)*x + 2*x)/log( - log(x) + x))*e**2 - 2*e**((log( - log(x) + x)*x + x)/log( - log(x) + x))*e*x + x**2