Integrand size = 126, antiderivative size = 21 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=x+\frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{\log (x)} \]
Time = 0.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=x+\frac {\log \left (e^{e^{3 e^{-x}}}+x\right )}{\log (x)} \]
Integrate[(E^x*x*Log[x] + E^x*x^2*Log[x]^2 + E^E^(3/E^x)*(-3*E^(3/E^x)*x*L og[x] + E^x*x*Log[x]^2) + (-E^(E^(3/E^x) + x) - E^x*x)*Log[E^E^(3/E^x) + x ])/(E^(E^(3/E^x) + x)*x*Log[x]^2 + E^x*x^2*Log[x]^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.
\(\displaystyle \int \frac {e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (e^x x \log ^2(x)-3 e^{3 e^{-x}} x \log (x)\right )+e^x x \log (x)+\left (-e^x x-e^{x+e^{3 e^{-x}}}\right ) \log \left (x+e^{e^{3 e^{-x}}}\right )}{e^x x^2 \log ^2(x)+e^{x+e^{3 e^{-x}}} x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (-\frac {\log \left (x+e^{e^{3 e^{-x}}}\right )}{x \log ^2(x)}+\frac {1-3 e^{-x+e^{3 e^{-x}}+3 e^{-x}}}{\left (x+e^{e^{3 e^{-x}}}\right ) \log (x)}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {\log \left (x+e^{e^{3 e^{-x}}}\right )}{x \log ^2(x)}dx+\int \frac {1}{\left (x+e^{e^{3 e^{-x}}}\right ) \log (x)}dx-3 \int \frac {e^{-x+e^{3 e^{-x}}+3 e^{-x}}}{\left (x+e^{e^{3 e^{-x}}}\right ) \log (x)}dx+x\) |
Int[(E^x*x*Log[x] + E^x*x^2*Log[x]^2 + E^E^(3/E^x)*(-3*E^(3/E^x)*x*Log[x] + E^x*x*Log[x]^2) + (-E^(E^(3/E^x) + x) - E^x*x)*Log[E^E^(3/E^x) + x])/(E^ (E^(3/E^x) + x)*x*Log[x]^2 + E^x*x^2*Log[x]^2),x]
3.24.57.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 48.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{3 \,{\mathrm e}^{-x}}}+x \right )}{\ln \left (x \right )}+x\) | \(19\) |
parallelrisch | \(-\frac {-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (x \right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{3 \,{\mathrm e}^{-x}}}+x \right )}{2 \ln \left (x \right )}\) | \(28\) |
int(((-exp(x)*exp(exp(3/exp(x)))-exp(x)*x)*ln(exp(exp(3/exp(x)))+x)+(-3*x* ln(x)*exp(3/exp(x))+x*exp(x)*ln(x)^2)*exp(exp(3/exp(x)))+x^2*exp(x)*ln(x)^ 2+x*exp(x)*ln(x))/(x*exp(x)*ln(x)^2*exp(exp(3/exp(x)))+x^2*exp(x)*ln(x)^2) ,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\frac {x \log \left (x\right ) + \log \left ({\left (x e^{x} + e^{\left (x + e^{\left (3 \, e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (-x\right )}\right )}{\log \left (x\right )} \]
integrate(((-exp(x)*exp(exp(3/exp(x)))-exp(x)*x)*log(exp(exp(3/exp(x)))+x) +(-3*x*log(x)*exp(3/exp(x))+x*exp(x)*log(x)^2)*exp(exp(3/exp(x)))+x^2*exp( x)*log(x)^2+x*exp(x)*log(x))/(x*exp(x)*log(x)^2*exp(exp(3/exp(x)))+x^2*exp (x)*log(x)^2),x, algorithm=\
Exception generated. \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
integrate(((-exp(x)*exp(exp(3/exp(x)))-exp(x)*x)*ln(exp(exp(3/exp(x)))+x)+ (-3*x*ln(x)*exp(3/exp(x))+x*exp(x)*ln(x)**2)*exp(exp(3/exp(x)))+x**2*exp(x )*ln(x)**2+x*exp(x)*ln(x))/(x*exp(x)*ln(x)**2*exp(exp(3/exp(x)))+x**2*exp( x)*ln(x)**2),x)
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\frac {x \log \left (x\right ) + \log \left (x + e^{\left (e^{\left (3 \, e^{\left (-x\right )}\right )}\right )}\right )}{\log \left (x\right )} \]
integrate(((-exp(x)*exp(exp(3/exp(x)))-exp(x)*x)*log(exp(exp(3/exp(x)))+x) +(-3*x*log(x)*exp(3/exp(x))+x*exp(x)*log(x)^2)*exp(exp(3/exp(x)))+x^2*exp( x)*log(x)^2+x*exp(x)*log(x))/(x*exp(x)*log(x)^2*exp(exp(3/exp(x)))+x^2*exp (x)*log(x)^2),x, algorithm=\
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=\frac {x \log \left (x\right ) + \log \left ({\left (x e^{x} + e^{\left (x + e^{\left (3 \, e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (-x\right )}\right )}{\log \left (x\right )} \]
integrate(((-exp(x)*exp(exp(3/exp(x)))-exp(x)*x)*log(exp(exp(3/exp(x)))+x) +(-3*x*log(x)*exp(3/exp(x))+x*exp(x)*log(x)^2)*exp(exp(3/exp(x)))+x^2*exp( x)*log(x)^2+x*exp(x)*log(x))/(x*exp(x)*log(x)^2*exp(exp(3/exp(x)))+x^2*exp (x)*log(x)^2),x, algorithm=\
Timed out. \[ \int \frac {e^x x \log (x)+e^x x^2 \log ^2(x)+e^{e^{3 e^{-x}}} \left (-3 e^{3 e^{-x}} x \log (x)+e^x x \log ^2(x)\right )+\left (-e^{e^{3 e^{-x}}+x}-e^x x\right ) \log \left (e^{e^{3 e^{-x}}}+x\right )}{e^{e^{3 e^{-x}}+x} x \log ^2(x)+e^x x^2 \log ^2(x)} \, dx=-\int \frac {\ln \left (x+{\mathrm {e}}^{{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}\right )\,\left ({\mathrm {e}}^{x+{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}+x\,{\mathrm {e}}^x\right )+{\mathrm {e}}^{{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}\,\left (3\,x\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}\,\ln \left (x\right )-x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2\right )-x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2-x\,{\mathrm {e}}^x\,\ln \left (x\right )}{x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2+x\,{\mathrm {e}}^{x+{\mathrm {e}}^{3\,{\mathrm {e}}^{-x}}}\,{\ln \left (x\right )}^2} \,d x \]
int(-(log(x + exp(exp(3*exp(-x))))*(exp(exp(3*exp(-x)))*exp(x) + x*exp(x)) + exp(exp(3*exp(-x)))*(3*x*exp(3*exp(-x))*log(x) - x*exp(x)*log(x)^2) - x ^2*exp(x)*log(x)^2 - x*exp(x)*log(x))/(x^2*exp(x)*log(x)^2 + x*exp(exp(3*e xp(-x)))*exp(x)*log(x)^2),x)