Integrand size = 233, antiderivative size = 29 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\log \left (x \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )\right ) \] Output:
ln(x*(256+ln((exp(2*x)-ln(x))/(1-exp(exp(2))+x))))
Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\log (x)+\log \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right ) \] Input:
Integrate[(-1 + E^E^2*(1 + E^(2*x)*(-256 - 2*x)) - x + E^(2*x)*(256 + 257* x + 2*x^2) + (-256 + 256*E^E^2 - 255*x)*Log[x] + (-E^(E^2 + 2*x) + E^(2*x) *(1 + x) + (-1 + E^E^2 - x)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2 - x)])/(-256*E^(E^2 + 2*x)*x + E^(2*x)*(256*x + 256*x^2) + (-256*x + 256*E^E ^2*x - 256*x^2)*Log[x] + (-(E^(E^2 + 2*x)*x) + E^(2*x)*(x + x^2) + (-x + E ^E^2*x - x^2)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2 - x)]),x]
Output:
Log[x] + Log[256 + Log[(E^(2*x) - Log[x])/(1 - E^E^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.
| \(\displaystyle \int \frac {e^{2 x} \left (2 x^2+257 x+256\right )+e^{e^2} \left (e^{2 x} (-2 x-256)+1\right )-x+\left (-255 x+256 e^{e^2}-256\right ) \log (x)+\left (e^{2 x} (x+1)-e^{2 x+e^2}+\left (-x+e^{e^2}-1\right ) \log (x)\right ) \log \left (\frac {\log (x)-e^{2 x}}{-x+e^{e^2}-1}\right )-1}{e^{2 x} \left (256 x^2+256 x\right )+\left (-256 x^2+256 e^{e^2} x-256 x\right ) \log (x)+\left (e^{2 x} \left (x^2+x\right )+\left (-x^2+e^{e^2} x-x\right ) \log (x)-e^{2 x+e^2} x\right ) \log \left (\frac {\log (x)-e^{2 x}}{-x+e^{e^2}-1}\right )-256 e^{2 x+e^2} x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 x} \left (2 x^2+257 x+256\right )+e^{e^2} \left (e^{2 x} (-2 x-256)+1\right )-x+\left (-255 x+256 e^{e^2}-256\right ) \log (x)+\left (e^{2 x} (x+1)-e^{2 x+e^2}+\left (-x+e^{e^2}-1\right ) \log (x)\right ) \log \left (\frac {\log (x)-e^{2 x}}{-x+e^{e^2}-1}\right )-1}{x \left (x-e^{e^2}+1\right ) \left (e^{2 x}-\log (x)\right ) \left (\log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x^2+257 \left (1-\frac {2 e^{e^2}}{257}\right ) x+x \log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+\left (1-e^{e^2}\right ) \log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256 \left (1-e^{e^2}\right )}{x \left (x-e^{e^2}+1\right ) \left (\log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256\right )}+\frac {2 x \log (x)-1}{x \left (e^{2 x}-\log (x)\right ) \left (\log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {1}{\log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256}dx+\int \frac {1}{\left (-x+e^{e^2}-1\right ) \left (\log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256\right )}dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (\log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256\right )}dx+2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (\log \left (\frac {e^{2 x}-\log (x)}{x-e^{e^2}+1}\right )+256\right )}dx+\log (x)\) |
Input:
Int[(-1 + E^E^2*(1 + E^(2*x)*(-256 - 2*x)) - x + E^(2*x)*(256 + 257*x + 2* x^2) + (-256 + 256*E^E^2 - 255*x)*Log[x] + (-E^(E^2 + 2*x) + E^(2*x)*(1 + x) + (-1 + E^E^2 - x)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2 - x)])/( -256*E^(E^2 + 2*x)*x + E^(2*x)*(256*x + 256*x^2) + (-256*x + 256*E^E^2*x - 256*x^2)*Log[x] + (-(E^(E^2 + 2*x)*x) + E^(2*x)*(x + x^2) + (-x + E^E^2*x - x^2)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2 - x)]),x]
Output:
$Aborted
Time = 24.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
| method | result | size |
| parallelrisch | \(\ln \left (\ln \left (\frac {\ln \left (x \right )-{\mathrm e}^{2 x}}{{\mathrm e}^{{\mathrm e}^{2}}-x -1}\right )+256\right )+\ln \left (x \right )\) | \(28\) |
| risch | \(\ln \left (x \right )+\ln \left (\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )-\frac {i \left (-2 \pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (\frac {i}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )}^{3}+2 \pi -2 i \ln \left ({\mathrm e}^{2 x}-\ln \left (x \right )\right )-512 i\right )}{2}\right )\) | \(228\) |
Input:
int((((exp(exp(2))-x-1)*ln(x)-exp(2*x)*exp(exp(2))+(1+x)*exp(2*x))*ln((ln( x)-exp(2*x))/(exp(exp(2))-x-1))+(256*exp(exp(2))-255*x-256)*ln(x)+((-2*x-2 56)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/(((x*exp(exp(2 ))-x^2-x)*ln(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*ln((ln(x)-exp(2*x ))/(exp(exp(2))-x-1))+(256*x*exp(exp(2))-256*x^2-256*x)*ln(x)-256*x*exp(2* x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x,method=_RETURNVERBOSE)
Output:
ln(ln((ln(x)-exp(2*x))/(exp(exp(2))-x-1))+256)+ln(x)
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\log \left (x\right ) + \log \left (\log \left (-\frac {e^{\left (e^{2}\right )} \log \left (x\right ) - e^{\left (2 \, x + e^{2}\right )}}{{\left (x + 1\right )} e^{\left (e^{2}\right )} - e^{\left (2 \, e^{2}\right )}}\right ) + 256\right ) \] Input:
integrate((((exp(exp(2))-x-1)*log(x)-exp(2*x)*exp(exp(2))+(1+x)*exp(2*x))* log((log(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*exp(exp(2))-255*x-256)*log(x )+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/(((x *exp(exp(2))-x^2-x)*log(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*log((l og(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*x*exp(exp(2))-256*x^2-256*x)*log(x )-256*x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x, algorithm="frica s")
Output:
log(x) + log(log(-(e^(e^2)*log(x) - e^(2*x + e^2))/((x + 1)*e^(e^2) - e^(2 *e^2))) + 256)
Time = 2.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\log {\left (x \right )} + \log {\left (\log {\left (\frac {- e^{2 x} + \log {\left (x \right )}}{- x - 1 + e^{e^{2}}} \right )} + 256 \right )} \] Input:
integrate((((exp(exp(2))-x-1)*ln(x)-exp(2*x)*exp(exp(2))+(1+x)*exp(2*x))*l n((ln(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*exp(exp(2))-255*x-256)*ln(x)+(( -2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x**2+257*x+256)*exp(2*x)-x-1)/(((x*ex p(exp(2))-x**2-x)*ln(x)-x*exp(2*x)*exp(exp(2))+(x**2+x)*exp(2*x))*ln((ln(x )-exp(2*x))/(exp(exp(2))-x-1))+(256*x*exp(exp(2))-256*x**2-256*x)*ln(x)-25 6*x*exp(2*x)*exp(exp(2))+(256*x**2+256*x)*exp(2*x)),x)
Output:
log(x) + log(log((-exp(2*x) + log(x))/(-x - 1 + exp(exp(2)))) + 256)
Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\log \left (x\right ) + \log \left (-\log \left (-x + e^{\left (e^{2}\right )} - 1\right ) + \log \left (-e^{\left (2 \, x\right )} + \log \left (x\right )\right ) + 256\right ) \] Input:
integrate((((exp(exp(2))-x-1)*log(x)-exp(2*x)*exp(exp(2))+(1+x)*exp(2*x))* log((log(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*exp(exp(2))-255*x-256)*log(x )+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/(((x *exp(exp(2))-x^2-x)*log(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*log((l og(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*x*exp(exp(2))-256*x^2-256*x)*log(x )-256*x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x, algorithm="maxim a")
Output:
log(x) + log(-log(-x + e^(e^2) - 1) + log(-e^(2*x) + log(x)) + 256)
Time = 0.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\log \left (x\right ) + \log \left (\log \left (x - e^{\left (e^{2}\right )} + 1\right ) - \log \left (e^{\left (2 \, x\right )} - \log \left (x\right )\right ) - 256\right ) \] Input:
integrate((((exp(exp(2))-x-1)*log(x)-exp(2*x)*exp(exp(2))+(1+x)*exp(2*x))* log((log(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*exp(exp(2))-255*x-256)*log(x )+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/(((x *exp(exp(2))-x^2-x)*log(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*log((l og(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*x*exp(exp(2))-256*x^2-256*x)*log(x )-256*x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x, algorithm="giac" )
Output:
log(x) + log(log(x - e^(e^2) + 1) - log(e^(2*x) - log(x)) - 256)
Time = 3.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\ln \left (\ln \left (\frac {{\mathrm {e}}^{2\,x}-\ln \left (x\right )}{x-{\mathrm {e}}^{{\mathrm {e}}^2}+1}\right )+256\right )+\ln \left (x\right ) \] Input:
int((x - exp(2*x)*(257*x + 2*x^2 + 256) + exp(exp(2))*(exp(2*x)*(2*x + 256 ) - 1) + log(x)*(255*x - 256*exp(exp(2)) + 256) + log((exp(2*x) - log(x))/ (x - exp(exp(2)) + 1))*(log(x)*(x - exp(exp(2)) + 1) + exp(2*x)*exp(exp(2) ) - exp(2*x)*(x + 1)) + 1)/(log(x)*(256*x - 256*x*exp(exp(2)) + 256*x^2) - exp(2*x)*(256*x + 256*x^2) + log((exp(2*x) - log(x))/(x - exp(exp(2)) + 1 ))*(log(x)*(x - x*exp(exp(2)) + x^2) - exp(2*x)*(x + x^2) + x*exp(2*x)*exp (exp(2))) + 256*x*exp(2*x)*exp(exp(2))),x)
Output:
log(log((exp(2*x) - log(x))/(x - exp(exp(2)) + 1)) + 256) + log(x)
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {-e^{2 x}+\mathrm {log}\left (x \right )}{e^{e^{2}}-x -1}\right )+256\right )+\mathrm {log}\left (x \right ) \] Input:
int((((exp(exp(2))-x-1)*log(x)-exp(2*x)*exp(exp(2))+(1+x)*exp(2*x))*log((l og(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*exp(exp(2))-255*x-256)*log(x)+((-2 *x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/(((x*exp(e xp(2))-x^2-x)*log(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*log((log(x)- exp(2*x))/(exp(exp(2))-x-1))+(256*x*exp(exp(2))-256*x^2-256*x)*log(x)-256* x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x)
Output:
log(log(( - e**(2*x) + log(x))/(e**(e**2) - x - 1)) + 256) + log(x)