Integrand size = 137, antiderivative size = 27 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {1}{2} x \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \]
\[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx \]
Integrate[((2*E^x*x^2 + E^(5/(E^x*x))*(-10 - 10*x)*Log[18])*Log[(x + E^(5/ (E^x*x))*Log[18])/Log[18]] + (E^x*x^2 + E^(5/(E^x*x) + x)*x*Log[18])*Log[( x + E^(5/(E^x*x))*Log[18])/Log[18]]^2)/(2*E^x*x^2 + 2*E^(5/(E^x*x) + x)*x* Log[18]),x]
Integrate[((2*E^x*x^2 + E^(5/(E^x*x))*(-10 - 10*x)*Log[18])*Log[(x + E^(5/ (E^x*x))*Log[18])/Log[18]] + (E^x*x^2 + E^(5/(E^x*x) + x)*x*Log[18])*Log[( x + E^(5/(E^x*x))*Log[18])/Log[18]]^2)/(2*E^x*x^2 + 2*E^(5/(E^x*x) + x)*x* Log[18]), 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.
| \(\displaystyle \int \frac {\left (e^x x^2+e^{x+\frac {5 e^{-x}}{x}} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10 x-10) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{x+\frac {5 e^{-x}}{x}} x \log (18)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{-x} \left (\left (e^x x^2+e^{x+\frac {5 e^{-x}}{x}} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10 x-10) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )\right )}{2 x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{-x} \left (\left (e^x x^2+e^{x+\frac {5 e^{-x}}{x}} \log (18) x\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+2 \left (e^x x^2-5 e^{\frac {5 e^{-x}}{x}} (x+1) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right ) \left (\log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right ) x+2 x+e^{\frac {5 e^{-x}}{x}} \log (18) \log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right )\right )}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}-\frac {10 e^{\frac {5 e^{-x}}{x}-x} (x+1) \log (18) \log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right ) \left (\log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right ) x+2 x+e^{\frac {5 e^{-x}}{x}} \log (18) \log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right )\right )}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}-\frac {10 e^{\frac {5 e^{-x}}{x}-x} (x+1) \log (18) \log \left (\frac {x}{\log (18)}+e^{\frac {5 e^{-x}}{x}}\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right )dx\) |
Int[((2*E^x*x^2 + E^(5/(E^x*x))*(-10 - 10*x)*Log[18])*Log[(x + E^(5/(E^x*x ))*Log[18])/Log[18]] + (E^x*x^2 + E^(5/(E^x*x) + x)*x*Log[18])*Log[(x + E^ (5/(E^x*x))*Log[18])/Log[18]]^2)/(2*E^x*x^2 + 2*E^(5/(E^x*x) + x)*x*Log[18 ]),x]
3.30.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 27.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(\frac {\ln \left (\frac {\ln \left (18\right ) {\mathrm e}^{\frac {5 \,{\mathrm e}^{-x}}{x}}+x}{\ln \left (18\right )}\right )^{2} x}{2}\) | \(27\) |
| risch | \(\frac {x \ln \left (\frac {\left (\ln \left (2\right )+2 \ln \left (3\right )\right ) {\mathrm e}^{\frac {5 \,{\mathrm e}^{-x}}{x}}+x}{\ln \left (2\right )+2 \ln \left (3\right )}\right )^{2}}{2}\) | \(37\) |
int(((x*ln(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*ln((ln(18)*exp(5/exp(x)/ x)+x)/ln(18))^2+((-10*x-10)*ln(18)*exp(5/exp(x)/x)+2*exp(x)*x^2)*ln((ln(18 )*exp(5/exp(x)/x)+x)/ln(18)))/(2*x*ln(18)*exp(x)*exp(5/exp(x)/x)+2*exp(x)* x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {1}{2} \, x \log \left (\frac {{\left (x e^{x} + e^{\left (\frac {{\left (x^{2} e^{x} + 5\right )} e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )} e^{\left (-x\right )}}{\log \left (18\right )}\right )^{2} \]
integrate(((x*log(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*log((log(18)*exp( 5/exp(x)/x)+x)/log(18))^2+((-10*x-10)*log(18)*exp(5/exp(x)/x)+2*exp(x)*x^2 )*log((log(18)*exp(5/exp(x)/x)+x)/log(18)))/(2*x*log(18)*exp(x)*exp(5/exp( x)/x)+2*exp(x)*x^2),x, algorithm=\
Time = 0.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {x \log {\left (\frac {x + e^{\frac {5 e^{- x}}{x}} \log {\left (18 \right )}}{\log {\left (18 \right )}} \right )}^{2}}{2} \]
integrate(((x*ln(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x**2)*ln((ln(18)*exp(5/ exp(x)/x)+x)/ln(18))**2+((-10*x-10)*ln(18)*exp(5/exp(x)/x)+2*exp(x)*x**2)* ln((ln(18)*exp(5/exp(x)/x)+x)/ln(18)))/(2*x*ln(18)*exp(x)*exp(5/exp(x)/x)+ 2*exp(x)*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).
Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {1}{2} \, x \log \left ({\left (2 \, \log \left (3\right ) + \log \left (2\right )\right )} e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} + x\right )^{2} - x \log \left ({\left (2 \, \log \left (3\right ) + \log \left (2\right )\right )} e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} + x\right ) \log \left (2 \, \log \left (3\right ) + \log \left (2\right )\right ) + \frac {1}{2} \, x \log \left (2 \, \log \left (3\right ) + \log \left (2\right )\right )^{2} \]
integrate(((x*log(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*log((log(18)*exp( 5/exp(x)/x)+x)/log(18))^2+((-10*x-10)*log(18)*exp(5/exp(x)/x)+2*exp(x)*x^2 )*log((log(18)*exp(5/exp(x)/x)+x)/log(18)))/(2*x*log(18)*exp(x)*exp(5/exp( x)/x)+2*exp(x)*x^2),x, algorithm=\
1/2*x*log((2*log(3) + log(2))*e^(5*e^(-x)/x) + x)^2 - x*log((2*log(3) + lo g(2))*e^(5*e^(-x)/x) + x)*log(2*log(3) + log(2)) + 1/2*x*log(2*log(3) + lo g(2))^2
\[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\int { \frac {{\left (x^{2} e^{x} + x e^{\left (x + \frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )} \log \left (\frac {e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right ) + x}{\log \left (18\right )}\right )^{2} + 2 \, {\left (x^{2} e^{x} - 5 \, {\left (x + 1\right )} e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )} \log \left (\frac {e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right ) + x}{\log \left (18\right )}\right )}{2 \, {\left (x^{2} e^{x} + x e^{\left (x + \frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )}} \,d x } \]
integrate(((x*log(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*log((log(18)*exp( 5/exp(x)/x)+x)/log(18))^2+((-10*x-10)*log(18)*exp(5/exp(x)/x)+2*exp(x)*x^2 )*log((log(18)*exp(5/exp(x)/x)+x)/log(18)))/(2*x*log(18)*exp(x)*exp(5/exp( x)/x)+2*exp(x)*x^2),x, algorithm=\
integrate(1/2*((x^2*e^x + x*e^(x + 5*e^(-x)/x)*log(18))*log((e^(5*e^(-x)/x )*log(18) + x)/log(18))^2 + 2*(x^2*e^x - 5*(x + 1)*e^(5*e^(-x)/x)*log(18)) *log((e^(5*e^(-x)/x)*log(18) + x)/log(18)))/(x^2*e^x + x*e^(x + 5*e^(-x)/x )*log(18)), x)
Timed out. \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\int \frac {\left (x^2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,{\mathrm {e}}^x\,\ln \left (18\right )\right )\,{\ln \left (\frac {x+{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,\ln \left (18\right )}{\ln \left (18\right )}\right )}^2+\left (2\,x^2\,{\mathrm {e}}^x-{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,\ln \left (18\right )\,\left (10\,x+10\right )\right )\,\ln \left (\frac {x+{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,\ln \left (18\right )}{\ln \left (18\right )}\right )}{2\,x^2\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,{\mathrm {e}}^x\,\ln \left (18\right )} \,d x \]
int((log((x + exp((5*exp(-x))/x)*log(18))/log(18))^2*(x^2*exp(x) + x*exp(( 5*exp(-x))/x)*exp(x)*log(18)) + log((x + exp((5*exp(-x))/x)*log(18))/log(1 8))*(2*x^2*exp(x) - exp((5*exp(-x))/x)*log(18)*(10*x + 10)))/(2*x^2*exp(x) + 2*x*exp((5*exp(-x))/x)*exp(x)*log(18)),x)