Integrand size = 74, antiderivative size = 15 \[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=x \log \left (\log \left (e^x-x-\log (3)\right )\right ) \] Output:
ln(ln(exp(x)-ln(3)-x))*x
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=x \log \left (\log \left (e^x-x-\log (3)\right )\right ) \] Input:
Integrate[(-x + E^x*x + (E^x - x - Log[3])*Log[E^x - x - Log[3]]*Log[Log[E ^x - x - Log[3]]])/((E^x - x - Log[3])*Log[E^x - x - Log[3]]),x]
Output:
x*Log[Log[E^x - x - Log[3]]]
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.
\(\displaystyle \int \frac {e^x x-x+\left (-x+e^x-\log (3)\right ) \log \left (-x+e^x-\log (3)\right ) \log \left (\log \left (-x+e^x-\log (3)\right )\right )}{\left (-x+e^x-\log (3)\right ) \log \left (-x+e^x-\log (3)\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x+\log \left (-x+e^x-\log (3)\right ) \log \left (\log \left (-x+e^x-\log (3)\right )\right )}{\log \left (-x+e^x-\log (3)\right )}-\frac {x (x-1+\log (3))}{\left (x-e^x+\log (3)\right ) \log \left (-x+e^x-\log (3)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {x^2}{\left (x-e^x+\log (3)\right ) \log \left (-x+e^x-\log (3)\right )}dx+\int \frac {x}{\log \left (-x+e^x-\log (3)\right )}dx+(1-\log (3)) \int \frac {x}{\left (x-e^x+\log (3)\right ) \log \left (-x+e^x-\log (3)\right )}dx+\int \log \left (\log \left (-x+e^x-\log (3)\right )\right )dx\) |
Input:
Int[(-x + E^x*x + (E^x - x - Log[3])*Log[E^x - x - Log[3]]*Log[Log[E^x - x - Log[3]]])/((E^x - x - Log[3])*Log[E^x - x - Log[3]]),x]
Output:
$Aborted
Time = 0.63 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\ln \left (\ln \left ({\mathrm e}^{x}-\ln \left (3\right )-x \right )\right ) x\) | \(15\) |
parallelrisch | \(\ln \left (\ln \left ({\mathrm e}^{x}-\ln \left (3\right )-x \right )\right ) x\) | \(15\) |
Input:
int(((exp(x)-ln(3)-x)*ln(exp(x)-ln(3)-x)*ln(ln(exp(x)-ln(3)-x))+exp(x)*x-x )/(exp(x)-ln(3)-x)/ln(exp(x)-ln(3)-x),x,method=_RETURNVERBOSE)
Output:
ln(ln(exp(x)-ln(3)-x))*x
Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=x \log \left (\log \left (-x + e^{x} - \log \left (3\right )\right )\right ) \] Input:
integrate(((exp(x)-log(3)-x)*log(exp(x)-log(3)-x)*log(log(exp(x)-log(3)-x) )+exp(x)*x-x)/(exp(x)-log(3)-x)/log(exp(x)-log(3)-x),x, algorithm="fricas" )
Output:
x*log(log(-x + e^x - log(3)))
Time = 0.67 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=x \log {\left (\log {\left (- x + e^{x} - \log {\left (3 \right )} \right )} \right )} \] Input:
integrate(((exp(x)-ln(3)-x)*ln(exp(x)-ln(3)-x)*ln(ln(exp(x)-ln(3)-x))+exp( x)*x-x)/(exp(x)-ln(3)-x)/ln(exp(x)-ln(3)-x),x)
Output:
x*log(log(-x + exp(x) - log(3)))
Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=x \log \left (\log \left (-x + e^{x} - \log \left (3\right )\right )\right ) \] Input:
integrate(((exp(x)-log(3)-x)*log(exp(x)-log(3)-x)*log(log(exp(x)-log(3)-x) )+exp(x)*x-x)/(exp(x)-log(3)-x)/log(exp(x)-log(3)-x),x, algorithm="maxima" )
Output:
x*log(log(-x + e^x - log(3)))
\[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=\int { \frac {{\left (x - e^{x} + \log \left (3\right )\right )} \log \left (-x + e^{x} - \log \left (3\right )\right ) \log \left (\log \left (-x + e^{x} - \log \left (3\right )\right )\right ) - x e^{x} + x}{{\left (x - e^{x} + \log \left (3\right )\right )} \log \left (-x + e^{x} - \log \left (3\right )\right )} \,d x } \] Input:
integrate(((exp(x)-log(3)-x)*log(exp(x)-log(3)-x)*log(log(exp(x)-log(3)-x) )+exp(x)*x-x)/(exp(x)-log(3)-x)/log(exp(x)-log(3)-x),x, algorithm="giac")
Output:
integrate(((x - e^x + log(3))*log(-x + e^x - log(3))*log(log(-x + e^x - lo g(3))) - x*e^x + x)/((x - e^x + log(3))*log(-x + e^x - log(3))), x)
Time = 4.63 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=x\,\ln \left (\ln \left ({\mathrm {e}}^x-\ln \left (3\right )-x\right )\right ) \] Input:
int((x - x*exp(x) + log(exp(x) - log(3) - x)*log(log(exp(x) - log(3) - x)) *(x + log(3) - exp(x)))/(log(exp(x) - log(3) - x)*(x + log(3) - exp(x))),x )
Output:
x*log(log(exp(x) - log(3) - x))
Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^x x+\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right ) \log \left (\log \left (e^x-x-\log (3)\right )\right )}{\left (e^x-x-\log (3)\right ) \log \left (e^x-x-\log (3)\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (e^{x}-\mathrm {log}\left (3\right )-x \right )\right ) x \] Input:
int(((exp(x)-log(3)-x)*log(exp(x)-log(3)-x)*log(log(exp(x)-log(3)-x))+exp( x)*x-x)/(exp(x)-log(3)-x)/log(exp(x)-log(3)-x),x)
Output:
log(log(e**x - log(3) - x))*x