Integrand size = 68, antiderivative size = 25 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}} \]
Time = 5.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{2^x e^{1-e^x+x} \left (e^{2 x} x^2\right )^x} \]
Integrate[E^(1 - E^x + E^(1 - E^x + x + x*Log[2*E^(2*x)*x^2]) + x + x*Log[ 2*E^(2*x)*x^2])*(3 - E^x + 2*x + Log[2*E^(2*x)*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.
\(\displaystyle \int \left (\log \left (2 e^{2 x} x^2\right )+2 x-e^x+3\right ) \exp \left (x \log \left (2 e^{2 x} x^2\right )+e^{x \log \left (2 e^{2 x} x^2\right )+x-e^x+1}+x-e^x+1\right ) \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (2 x \exp \left (x \log \left (2 e^{2 x} x^2\right )+e^{x \log \left (2 e^{2 x} x^2\right )+x-e^x+1}+x-e^x+1\right )+3 \exp \left (x \log \left (2 e^{2 x} x^2\right )+e^{x \log \left (2 e^{2 x} x^2\right )+x-e^x+1}+x-e^x+1\right )-\exp \left (x \log \left (2 e^{2 x} x^2\right )+e^{x \log \left (2 e^{2 x} x^2\right )+x-e^x+1}+2 x-e^x+1\right )+\log \left (2 e^{2 x} x^2\right ) \exp \left (x \log \left (2 e^{2 x} x^2\right )+e^{x \log \left (2 e^{2 x} x^2\right )+x-e^x+1}+x-e^x+1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \exp \left (\log \left (2 e^{2 x} x^2\right ) x+x-e^x+e^{\log \left (2 e^{2 x} x^2\right ) x+x-e^x+1}+1\right )dx-\int \exp \left (\log \left (2 e^{2 x} x^2\right ) x+2 x-e^x+e^{\log \left (2 e^{2 x} x^2\right ) x+x-e^x+1}+1\right )dx+2 \int \exp \left (\log \left (2 e^{2 x} x^2\right ) x+x-e^x+e^{\log \left (2 e^{2 x} x^2\right ) x+x-e^x+1}+1\right ) xdx+\int \exp \left (\log \left (2 e^{2 x} x^2\right ) x+x-e^x+e^{\log \left (2 e^{2 x} x^2\right ) x+x-e^x+1}+1\right ) \log \left (2 e^{2 x} x^2\right )dx\) |
Int[E^(1 - E^x + E^(1 - E^x + x + x*Log[2*E^(2*x)*x^2]) + x + x*Log[2*E^(2 *x)*x^2])*(3 - E^x + 2*x + Log[2*E^(2*x)*x^2]),x]
3.18.44.3.1 Defintions of rubi rules used
Time = 0.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \({\mathrm e}^{{\mathrm e}^{x \ln \left (2 \,{\mathrm e}^{2 x} x^{2}\right )+1-{\mathrm e}^{x}+x}}\) | \(22\) |
default | \({\mathrm e}^{{\mathrm e}^{x \ln \left (2 \,{\mathrm e}^{2 x} x^{2}\right )+1-{\mathrm e}^{x}+x}}\) | \(22\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{x \ln \left (2 \,{\mathrm e}^{2 x} x^{2}\right )+1-{\mathrm e}^{x}+x}}\) | \(22\) |
risch | \({\mathrm e}^{2^{x} x^{2 x} \left ({\mathrm e}^{x}\right )^{2 x} {\mathrm e}^{1-\frac {i x \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}}{2}+\frac {i x \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \pi }{2}-\frac {i x \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}}{2}-\frac {i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i x \,\operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i x^{2}\right ) \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )}{2}-\frac {i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}+i x \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-\frac {i x \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{3} \pi }{2}+i x \operatorname {csgn}\left (i x^{2}\right )^{2} \pi \,\operatorname {csgn}\left (i x \right )+\frac {i x \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )}{2}-{\mathrm e}^{x}+x}}\) | \(234\) |
int((ln(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*ln(2*exp(2*x)*x^2)+1-exp(x)+x) *exp(exp(x*ln(2*exp(2*x)*x^2)+1-exp(x)+x)),x,method=_RETURNVERBOSE)
Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{\left (e^{\left (x \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + x - e^{x} + 1\right )}\right )} \]
integrate((log(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*log(2*exp(2*x)*x^2)+1-e xp(x)+x)*exp(exp(x*log(2*exp(2*x)*x^2)+1-exp(x)+x)),x, algorithm=\
Timed out. \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=\text {Timed out} \]
integrate((ln(2*exp(2*x)*x**2)-exp(x)+2*x+3)*exp(x*ln(2*exp(2*x)*x**2)+1-e xp(x)+x)*exp(exp(x*ln(2*exp(2*x)*x**2)+1-exp(x)+x)),x)
Time = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{\left (e^{\left (2 \, x^{2} + x \log \left (2\right ) + 2 \, x \log \left (x\right ) + x - e^{x} + 1\right )}\right )} \]
integrate((log(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*log(2*exp(2*x)*x^2)+1-e xp(x)+x)*exp(exp(x*log(2*exp(2*x)*x^2)+1-exp(x)+x)),x, algorithm=\
\[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=\int { {\left (2 \, x - e^{x} + \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + 3\right )} e^{\left (x \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + x + e^{\left (x \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + x - e^{x} + 1\right )} - e^{x} + 1\right )} \,d x } \]
integrate((log(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*log(2*exp(2*x)*x^2)+1-e xp(x)+x)*exp(exp(x*log(2*exp(2*x)*x^2)+1-exp(x)+x)),x, algorithm=\
integrate((2*x - e^x + log(2*x^2*e^(2*x)) + 3)*e^(x*log(2*x^2*e^(2*x)) + x + e^(x*log(2*x^2*e^(2*x)) + x - e^x + 1) - e^x + 1), x)
Time = 12.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx={\mathrm {e}}^{\mathrm {e}\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{-{\mathrm {e}}^x}\,{\mathrm {e}}^x\,{\left (2\,x^2\right )}^x} \]