Integrand size = 74, antiderivative size = 25 \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=\log \left (5-\frac {e^{\frac {e^3}{x}}+x}{x \log (6 x)}\right ) \] Output:
ln(5-(x+exp(exp(3)/x))/x/ln(6*x))
Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=-\log (x)-\log (\log (6 x))+\log \left (e^{\frac {e^3}{x}}+x-5 x \log (6 x)\right ) \] Input:
Integrate[(E^(E^3/x)*x + x^2 + E^(E^3/x)*(E^3 + x)*Log[6*x])/((-(E^(E^3/x) *x^2) - x^3)*Log[6*x] + 5*x^3*Log[6*x]^2),x]
Output:
-Log[x] - Log[Log[6*x]] + Log[E^(E^3/x) + x - 5*x*Log[6*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.
\(\displaystyle \int \frac {x^2+e^{\frac {e^3}{x}} x+e^{\frac {e^3}{x}} \left (x+e^3\right ) \log (6 x)}{5 x^3 \log ^2(6 x)+\left (-x^3-e^{\frac {e^3}{x}} x^2\right ) \log (6 x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-x^2-e^{\frac {e^3}{x}} x-e^{\frac {e^3}{x}} \left (x+e^3\right ) \log (6 x)}{x^2 \log (6 x) \left (x+e^{\frac {e^3}{x}}-5 x \log (6 x)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-x+x (-\log (6 x))-e^3 \log (6 x)}{x^2 \log (6 x)}+\frac {4 x+5 x \log (6 x)+5 e^3 \log (6 x)-e^3}{x \left (-x-e^{\frac {e^3}{x}}+5 x \log (6 x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {1}{-5 \log (6 x) x+x+e^{\frac {e^3}{x}}}dx-e^3 \int \frac {1}{x \left (5 \log (6 x) x-x-e^{\frac {e^3}{x}}\right )}dx+5 \int \frac {\log (6 x)}{5 \log (6 x) x-x-e^{\frac {e^3}{x}}}dx+5 e^3 \int \frac {\log (6 x)}{x \left (5 \log (6 x) x-x-e^{\frac {e^3}{x}}\right )}dx+\frac {e^3}{x}-\log (x)-\log (\log (6 x))\) |
Input:
Int[(E^(E^3/x)*x + x^2 + E^(E^3/x)*(E^3 + x)*Log[6*x])/((-(E^(E^3/x)*x^2) - x^3)*Log[6*x] + 5*x^3*Log[6*x]^2),x]
Output:
$Aborted
Time = 0.80 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\ln \left (\ln \left (6 x \right )\right )+\ln \left (\ln \left (6 x \right )-\frac {x +{\mathrm e}^{\frac {{\mathrm e}^{3}}{x}}}{5 x}\right )\) | \(29\) |
parallelrisch | \(-\ln \left (\ln \left (6 x \right )\right )+\ln \left (\ln \left (6 x \right ) x -\frac {x}{5}-\frac {{\mathrm e}^{\frac {{\mathrm e}^{3}}{x}}}{5}\right )-\ln \left (6 x \right )\) | \(35\) |
Input:
int(((exp(3)+x)*exp(exp(3)/x)*ln(6*x)+x*exp(exp(3)/x)+x^2)/(5*x^3*ln(6*x)^ 2+(-x^2*exp(exp(3)/x)-x^3)*ln(6*x)),x,method=_RETURNVERBOSE)
Output:
-ln(ln(6*x))+ln(ln(6*x)-1/5*(x+exp(exp(3)/x))/x)
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=\log \left (\frac {5 \, x \log \left (6 \, x\right ) - x - e^{\left (\frac {e^{3}}{x}\right )}}{x}\right ) - \log \left (\log \left (6 \, x\right )\right ) \] Input:
integrate(((exp(3)+x)*exp(exp(3)/x)*log(6*x)+x*exp(exp(3)/x)+x^2)/(5*x^3*l og(6*x)^2+(-x^2*exp(exp(3)/x)-x^3)*log(6*x)),x, algorithm="fricas")
Output:
log((5*x*log(6*x) - x - e^(e^3/x))/x) - log(log(6*x))
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=- \log {\left (x \right )} + \log {\left (- 5 x \log {\left (6 x \right )} + x + e^{\frac {e^{3}}{x}} \right )} - \log {\left (\log {\left (6 x \right )} \right )} \] Input:
integrate(((exp(3)+x)*exp(exp(3)/x)*ln(6*x)+x*exp(exp(3)/x)+x**2)/(5*x**3* ln(6*x)**2+(-x**2*exp(exp(3)/x)-x**3)*ln(6*x)),x)
Output:
-log(x) + log(-5*x*log(6*x) + x + exp(exp(3)/x)) - log(log(6*x))
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=\log \left (-x {\left (5 \, \log \left (3\right ) + 5 \, \log \left (2\right ) - 1\right )} - 5 \, x \log \left (x\right ) + e^{\left (\frac {e^{3}}{x}\right )}\right ) - \log \left (x\right ) - \log \left (\log \left (3\right ) + \log \left (2\right ) + \log \left (x\right )\right ) \] Input:
integrate(((exp(3)+x)*exp(exp(3)/x)*log(6*x)+x*exp(exp(3)/x)+x^2)/(5*x^3*l og(6*x)^2+(-x^2*exp(exp(3)/x)-x^3)*log(6*x)),x, algorithm="maxima")
Output:
log(-x*(5*log(3) + 5*log(2) - 1) - 5*x*log(x) + e^(e^3/x)) - log(x) - log( log(3) + log(2) + log(x))
Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=\log \left (5 \, x e^{3} \log \left (6 \, x\right ) - x e^{3} - e^{\left (\frac {3 \, x + e^{3}}{x}\right )}\right ) - \log \left (x\right ) - \log \left (\log \left (6 \, x\right )\right ) \] Input:
integrate(((exp(3)+x)*exp(exp(3)/x)*log(6*x)+x*exp(exp(3)/x)+x^2)/(5*x^3*l og(6*x)^2+(-x^2*exp(exp(3)/x)-x^3)*log(6*x)),x, algorithm="giac")
Output:
log(5*x*e^3*log(6*x) - x*e^3 - e^((3*x + e^3)/x)) - log(x) - log(log(6*x))
Timed out. \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=\int -\frac {x^2+x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{x}}+\ln \left (6\,x\right )\,{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{x}}\,\left (x+{\mathrm {e}}^3\right )}{\ln \left (6\,x\right )\,\left (x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{x}}+x^3\right )-5\,x^3\,{\ln \left (6\,x\right )}^2} \,d x \] Input:
int(-(x^2 + x*exp(exp(3)/x) + log(6*x)*exp(exp(3)/x)*(x + exp(3)))/(log(6* x)*(x^2*exp(exp(3)/x) + x^3) - 5*x^3*log(6*x)^2),x)
Output:
int(-(x^2 + x*exp(exp(3)/x) + log(6*x)*exp(exp(3)/x)*(x + exp(3)))/(log(6* x)*(x^2*exp(exp(3)/x) + x^3) - 5*x^3*log(6*x)^2), x)
Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {e^3}{x}} x+x^2+e^{\frac {e^3}{x}} \left (e^3+x\right ) \log (6 x)}{\left (-e^{\frac {e^3}{x}} x^2-x^3\right ) \log (6 x)+5 x^3 \log ^2(6 x)} \, dx=-\mathrm {log}\left (\mathrm {log}\left (6 x \right )\right )+\mathrm {log}\left (\frac {e^{\frac {e^{3}}{x}} e^{3}-5 \,\mathrm {log}\left (6 x \right ) e^{3} x +e^{3} x}{x}\right ) \] Input:
int(((exp(3)+x)*exp(exp(3)/x)*log(6*x)+x*exp(exp(3)/x)+x^2)/(5*x^3*log(6*x )^2+(-x^2*exp(exp(3)/x)-x^3)*log(6*x)),x)
Output:
- log(log(6*x)) + log((e**(e**3/x)*e**3 - 5*log(6*x)*e**3*x + e**3*x)/x)