Integrand size = 60, antiderivative size = 22 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\log \left (2 e^x+e^{-\frac {1}{\log (64)}}-x-\log (x)\right ) \] Output:
ln(2*exp(x)-x+exp(-1/6/ln(2))-ln(x))
Time = 0.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\log \left (1+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x)\right ) \] Input:
Integrate[(-2*E^(x + Log[64]^(-1))*x + E^Log[64]^(-1)*(1 + x))/(-x - 2*E^( x + Log[64]^(-1))*x + E^Log[64]^(-1)*x^2 + E^Log[64]^(-1)*x*Log[x]),x]
Output:
Log[1 + 2*E^(x + Log[64]^(-1)) - E^Log[64]^(-1)*x - E^Log[64]^(-1)*Log[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.
\(\displaystyle \int \frac {(x+1) e^{\frac {1}{\log (64)}}-2 x e^{x+\frac {1}{\log (64)}}}{x^2 e^{\frac {1}{\log (64)}}-x-2 x e^{x+\frac {1}{\log (64)}}+x e^{\frac {1}{\log (64)}} \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (2 e^x x-x-1\right ) e^{\frac {1}{\log (64)}}}{x^2 \left (-e^{\frac {1}{\log (64)}}\right )+x+2 x e^{x+\frac {1}{\log (64)}}-x e^{\frac {1}{\log (64)}} \log (x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^{\frac {1}{\log (64)}} \int -\frac {-2 e^x x+x+1}{-e^{\frac {1}{\log (64)}} x^2+2 e^{x+\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x) x+x}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -e^{\frac {1}{\log (64)}} \int \frac {-2 e^x x+x+1}{-e^{\frac {1}{\log (64)}} x^2+2 e^{x+\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x) x+x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -e^{\frac {1}{\log (64)}} \int \left (-\frac {1}{x \left (e^{\frac {1}{\log (64)}} x-2 e^{x+\frac {1}{\log (64)}}+e^{\frac {1}{\log (64)}} \log (x)-1\right )}-\frac {2 e^x}{-e^{\frac {1}{\log (64)}} x+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} \log (x)+1}+\frac {1}{-e^{\frac {1}{\log (64)}} x+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} \log (x)+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -e^{\frac {1}{\log (64)}} \left (\int \frac {1}{-e^{\frac {1}{\log (64)}} x+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} \log (x)+1}dx-2 \int \frac {e^x}{-e^{\frac {1}{\log (64)}} x+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} \log (x)+1}dx-\int \frac {1}{x \left (e^{\frac {1}{\log (64)}} x-2 e^{x+\frac {1}{\log (64)}}+e^{\frac {1}{\log (64)}} \log (x)-1\right )}dx\right )\) |
Input:
Int[(-2*E^(x + Log[64]^(-1))*x + E^Log[64]^(-1)*(1 + x))/(-x - 2*E^(x + Lo g[64]^(-1))*x + E^Log[64]^(-1)*x^2 + E^Log[64]^(-1)*x*Log[x]),x]
Output:
$Aborted
Time = 0.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\ln \left (x -{\mathrm e}^{-\frac {1}{6 \ln \left (2\right )}}-2 \,{\mathrm e}^{x}+\ln \left (x \right )\right )\) | \(19\) |
norman | \(\ln \left (x \,{\mathrm e}^{\frac {1}{6 \ln \left (2\right )}}-2 \,{\mathrm e}^{\frac {1}{6 \ln \left (2\right )}} {\mathrm e}^{x}+{\mathrm e}^{\frac {1}{6 \ln \left (2\right )}} \ln \left (x \right )-1\right )\) | \(34\) |
parallelrisch | \(\ln \left (\left (x \,{\mathrm e}^{\frac {1}{6 \ln \left (2\right )}}-2 \,{\mathrm e}^{\frac {1}{6 \ln \left (2\right )}} {\mathrm e}^{x}+{\mathrm e}^{\frac {1}{6 \ln \left (2\right )}} \ln \left (x \right )-1\right ) {\mathrm e}^{-\frac {1}{6 \ln \left (2\right )}}\right )\) | \(44\) |
Input:
int((-2*x*exp(1/6/ln(2))*exp(x)+(1+x)*exp(1/6/ln(2)))/(x*exp(1/6/ln(2))*ln (x)-2*x*exp(1/6/ln(2))*exp(x)+x^2*exp(1/6/ln(2))-x),x,method=_RETURNVERBOS E)
Output:
ln(x-exp(-1/6/ln(2))-2*exp(x)+ln(x))
Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\log \left (x e^{\left (\frac {1}{6 \, \log \left (2\right )}\right )} + e^{\left (\frac {1}{6 \, \log \left (2\right )}\right )} \log \left (x\right ) - 2 \, e^{\left (\frac {6 \, x \log \left (2\right ) + 1}{6 \, \log \left (2\right )}\right )} - 1\right ) \] Input:
integrate((-2*x*exp(1/6/log(2))*exp(x)+(1+x)*exp(1/6/log(2)))/(x*exp(1/6/l og(2))*log(x)-2*x*exp(1/6/log(2))*exp(x)+x^2*exp(1/6/log(2))-x),x, algorit hm="fricas")
Output:
log(x*e^(1/6/log(2)) + e^(1/6/log(2))*log(x) - 2*e^(1/6*(6*x*log(2) + 1)/l og(2)) - 1)
Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\log {\left (\frac {- x e^{\frac {1}{6 \log {\left (2 \right )}}} - e^{\frac {1}{6 \log {\left (2 \right )}}} \log {\left (x \right )} + 1}{2 e^{\frac {1}{6 \log {\left (2 \right )}}}} + e^{x} \right )} \] Input:
integrate((-2*x*exp(1/6/ln(2))*exp(x)+(1+x)*exp(1/6/ln(2)))/(x*exp(1/6/ln( 2))*ln(x)-2*x*exp(1/6/ln(2))*exp(x)+x**2*exp(1/6/ln(2))-x),x)
Output:
log((-x*exp(1/(6*log(2))) - exp(1/(6*log(2)))*log(x) + 1)*exp(-1/(6*log(2) ))/2 + exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\log \left (-\frac {1}{2} \, {\left (x e^{\left (\frac {1}{6 \, \log \left (2\right )}\right )} + e^{\left (\frac {1}{6 \, \log \left (2\right )}\right )} \log \left (x\right ) - 2 \, e^{\left (x + \frac {1}{6 \, \log \left (2\right )}\right )} - 1\right )} e^{\left (-\frac {1}{6 \, \log \left (2\right )}\right )}\right ) \] Input:
integrate((-2*x*exp(1/6/log(2))*exp(x)+(1+x)*exp(1/6/log(2)))/(x*exp(1/6/l og(2))*log(x)-2*x*exp(1/6/log(2))*exp(x)+x^2*exp(1/6/log(2))-x),x, algorit hm="maxima")
Output:
log(-1/2*(x*e^(1/6/log(2)) + e^(1/6/log(2))*log(x) - 2*e^(x + 1/6/log(2)) - 1)*e^(-1/6/log(2)))
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\log \left (-x e^{\left (\frac {1}{6 \, \log \left (2\right )}\right )} - e^{\left (\frac {1}{6 \, \log \left (2\right )}\right )} \log \left (x\right ) + 2 \, e^{\left (x + \frac {1}{6 \, \log \left (2\right )}\right )} + 1\right ) \] Input:
integrate((-2*x*exp(1/6/log(2))*exp(x)+(1+x)*exp(1/6/log(2)))/(x*exp(1/6/l og(2))*log(x)-2*x*exp(1/6/log(2))*exp(x)+x^2*exp(1/6/log(2))-x),x, algorit hm="giac")
Output:
log(-x*e^(1/6/log(2)) - e^(1/6/log(2))*log(x) + 2*e^(x + 1/6/log(2)) + 1)
Time = 2.96 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\ln \left (x-{\mathrm {e}}^{-\frac {1}{6\,\ln \left (2\right )}}-2\,{\mathrm {e}}^x+\ln \left (x\right )\right ) \] Input:
int(-(exp(1/(6*log(2)))*(x + 1) - 2*x*exp(1/(6*log(2)))*exp(x))/(x - x^2*e xp(1/(6*log(2))) + 2*x*exp(1/(6*log(2)))*exp(x) - x*exp(1/(6*log(2)))*log( x)),x)
Output:
log(x - exp(-1/(6*log(2))) - 2*exp(x) + log(x))
Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx=\mathrm {log}\left (2 e^{\frac {6 \,\mathrm {log}\left (2\right ) x +1}{6 \,\mathrm {log}\left (2\right )}}-e^{\frac {1}{6 \,\mathrm {log}\left (2\right )}} \mathrm {log}\left (x \right )-e^{\frac {1}{6 \,\mathrm {log}\left (2\right )}} x +1\right ) \] Input:
int((-2*x*exp(1/6/log(2))*exp(x)+(1+x)*exp(1/6/log(2)))/(x*exp(1/6/log(2)) *log(x)-2*x*exp(1/6/log(2))*exp(x)+x^2*exp(1/6/log(2))-x),x)
Output:
log(2*e**((6*log(2)*x + 1)/(6*log(2))) - e**(1/(6*log(2)))*log(x) - e**(1/ (6*log(2)))*x + 1)