Integrand size = 139, antiderivative size = 32 \[ \int \frac {\left (-100 e^{5/x}+e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} \left (10 x+e^{5/x} \left (-10 x-10 x^2\right )+50 \log (x)\right )\right ) \log \left (\frac {10+2 e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x}{x^2}\right )}{5 e^{5/x} x+e^{\frac {5}{x}+e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x^2} \, dx=5 \log ^2\left (\frac {2 \left (e^{-x+e^{-5/x} \log (x)}+\frac {5}{x}\right )}{x}\right ) \]
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-100 e^{5/x}+e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} \left (10 x+e^{5/x} \left (-10 x-10 x^2\right )+50 \log (x)\right )\right ) \log \left (\frac {10+2 e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x}{x^2}\right )}{5 e^{5/x} x+e^{\frac {5}{x}+e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x^2} \, dx=5 \log ^2\left (\frac {2 \left (5+e^{-x} x^{1+e^{-5/x}}\right )}{x^2}\right ) \]
Integrate[((-100*E^(5/x) + E^((-(E^(5/x)*x) + Log[x])/E^(5/x))*(10*x + E^( 5/x)*(-10*x - 10*x^2) + 50*Log[x]))*Log[(10 + 2*E^((-(E^(5/x)*x) + Log[x]) /E^(5/x))*x)/x^2])/(5*E^(5/x)*x + E^(5/x + (-(E^(5/x)*x) + Log[x])/E^(5/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.
| \(\displaystyle \int \frac {\left (e^{e^{-5/x} \left (\log (x)-e^{5/x} x\right )} \left (e^{5/x} \left (-10 x^2-10 x\right )+10 x+50 \log (x)\right )-100 e^{5/x}\right ) \log \left (\frac {2 x e^{e^{-5/x} \left (\log (x)-e^{5/x} x\right )}+10}{x^2}\right )}{x^2 e^{\frac {5}{x}+e^{-5/x} \left (\log (x)-e^{5/x} x\right )}+5 e^{5/x} x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{x-\frac {5}{x}} \left (e^{e^{-5/x} \left (\log (x)-e^{5/x} x\right )} \left (e^{5/x} \left (-10 x^2-10 x\right )+10 x+50 \log (x)\right )-100 e^{5/x}\right ) \log \left (\frac {2 e^{-x} \left (x^{e^{-5/x}+1}+5 e^x\right )}{x^2}\right )}{x \left (x^{e^{-5/x}+1}+5 e^x\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {50 e^{-5/x} x^{e^{-5/x}-1} \log (x) \log \left (\frac {2 \left (e^{-x} x^{e^{-5/x}+1}+5\right )}{x^2}\right )}{x^{e^{-5/x}+1}+5 e^x}-\frac {10 x^{e^{-5/x}+1} \log \left (\frac {2 \left (e^{-x} x^{e^{-5/x}+1}+5\right )}{x^2}\right )}{x^{e^{-5/x}+1}+5 e^x}-\frac {100 e^x \log \left (\frac {2 \left (e^{-x} x^{e^{-5/x}+1}+5\right )}{x^2}\right )}{\left (x^{e^{-5/x}+1}+5 e^x\right ) x}+\frac {10 e^{-5/x} x^{e^{-5/x}} \log \left (\frac {2 \left (e^{-x} x^{e^{-5/x}+1}+5\right )}{x^2}\right )}{x^{e^{-5/x}+1}+5 e^x}-\frac {10 x^{e^{-5/x}} \log \left (\frac {2 \left (e^{-x} x^{e^{-5/x}+1}+5\right )}{x^2}\right )}{x^{e^{-5/x}+1}+5 e^x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {10 e^{-5/x} \left (-e^{5/x} (x+1) x^{e^{-5/x}+1}+x^{e^{-5/x}+1}+5 x^{e^{-5/x}} \log (x)-10 e^{x+\frac {5}{x}}\right ) \log \left (\frac {2 \left (e^{-x} x^{e^{-5/x}+1}+5\right )}{x^2}\right )}{x^{e^{-5/x}+2}+5 e^x x}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 10 \int -\frac {e^{-5/x} \left (-5 \log (x) x^{e^{-5/x}}+e^{5/x} (x+1) x^{1+e^{-5/x}}-x^{1+e^{-5/x}}+10 e^{x+\frac {5}{x}}\right ) \log \left (\frac {2 \left (e^{-x} x^{1+e^{-5/x}}+5\right )}{x^2}\right )}{x^{2+e^{-5/x}}+5 e^x x}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -10 \int \frac {e^{-5/x} \left (-5 \log (x) x^{e^{-5/x}}+e^{5/x} (x+1) x^{1+e^{-5/x}}-x^{1+e^{-5/x}}+10 e^{x+\frac {5}{x}}\right ) \log \left (\frac {2 \left (e^{-x} x^{1+e^{-5/x}}+5\right )}{x^2}\right )}{x^{2+e^{-5/x}}+5 e^x x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -10 \int \left (\frac {e^{-5/x} \left (e^{5/x} x^2+e^{5/x} x-x-5 \log (x)\right ) \log \left (\frac {2 \left (e^{-x} x^{1+e^{-5/x}}+5\right )}{x^2}\right )}{x^2}-\frac {5 e^{x-\frac {5}{x}} \left (e^{5/x} x^2-e^{5/x} x-x-5 \log (x)\right ) \log \left (\frac {2 \left (e^{-x} x^{1+e^{-5/x}}+5\right )}{x^2}\right )}{x^2 \left (x^{1+e^{-5/x}}+5 e^x\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -10 \int \left (\frac {e^{-5/x} \left (e^{5/x} x^2+e^{5/x} x-x-5 \log (x)\right ) \log \left (\frac {2 \left (e^{-x} x^{1+e^{-5/x}}+5\right )}{x^2}\right )}{x^2}-\frac {5 e^{x-\frac {5}{x}} \left (e^{5/x} x^2-e^{5/x} x-x-5 \log (x)\right ) \log \left (\frac {2 \left (e^{-x} x^{1+e^{-5/x}}+5\right )}{x^2}\right )}{x^2 \left (x^{1+e^{-5/x}}+5 e^x\right )}\right )dx\) |
Int[((-100*E^(5/x) + E^((-(E^(5/x)*x) + Log[x])/E^(5/x))*(10*x + E^(5/x)*( -10*x - 10*x^2) + 50*Log[x]))*Log[(10 + 2*E^((-(E^(5/x)*x) + Log[x])/E^(5/ x))*x)/x^2])/(5*E^(5/x)*x + E^(5/x + (-(E^(5/x)*x) + Log[x])/E^(5/x))*x^2) ,x]
3.26.98.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 44.37 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16
| method | result | size |
| parallelrisch | \(5 \ln \left (\frac {2 \,{\mathrm e}^{\left (\ln \left (x \right )-x \,{\mathrm e}^{\frac {5}{x}}\right ) {\mathrm e}^{-\frac {5}{x}}} x +10}{x^{2}}\right )^{2}\) | \(37\) |
int(((50*ln(x)+(-10*x^2-10*x)*exp(5/x)+10*x)*exp((ln(x)-x*exp(5/x))/exp(5/ x))-100*exp(5/x))*ln((2*x*exp((ln(x)-x*exp(5/x))/exp(5/x))+10)/x^2)/(x^2*e xp(5/x)*exp((ln(x)-x*exp(5/x))/exp(5/x))+5*x*exp(5/x)),x,method=_RETURNVER BOSE)
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {\left (-100 e^{5/x}+e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} \left (10 x+e^{5/x} \left (-10 x-10 x^2\right )+50 \log (x)\right )\right ) \log \left (\frac {10+2 e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x}{x^2}\right )}{5 e^{5/x} x+e^{\frac {5}{x}+e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x^2} \, dx=5 \, \log \left (\frac {2 \, {\left (x e^{\left (-\frac {{\left ({\left (x^{2} - 5\right )} e^{\frac {5}{x}} - x \log \left (x\right )\right )} e^{\left (-\frac {5}{x}\right )}}{x}\right )} + 5 \, e^{\frac {5}{x}}\right )} e^{\left (-\frac {5}{x}\right )}}{x^{2}}\right )^{2} \]
integrate(((50*log(x)+(-10*x^2-10*x)*exp(5/x)+10*x)*exp((log(x)-x*exp(5/x) )/exp(5/x))-100*exp(5/x))*log((2*x*exp((log(x)-x*exp(5/x))/exp(5/x))+10)/x ^2)/(x^2*exp(5/x)*exp((log(x)-x*exp(5/x))/exp(5/x))+5*x*exp(5/x)),x, algor ithm=\
Time = 76.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-100 e^{5/x}+e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} \left (10 x+e^{5/x} \left (-10 x-10 x^2\right )+50 \log (x)\right )\right ) \log \left (\frac {10+2 e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x}{x^2}\right )}{5 e^{5/x} x+e^{\frac {5}{x}+e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x^2} \, dx=5 \log {\left (\frac {2 x e^{\left (- x e^{\frac {5}{x}} + \log {\left (x \right )}\right ) e^{- \frac {5}{x}}} + 10}{x^{2}} \right )}^{2} \]
integrate(((50*ln(x)+(-10*x**2-10*x)*exp(5/x)+10*x)*exp((ln(x)-x*exp(5/x)) /exp(5/x))-100*exp(5/x))*ln((2*x*exp((ln(x)-x*exp(5/x))/exp(5/x))+10)/x**2 )/(x**2*exp(5/x)*exp((ln(x)-x*exp(5/x))/exp(5/x))+5*x*exp(5/x)),x)
\[ \int \frac {\left (-100 e^{5/x}+e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} \left (10 x+e^{5/x} \left (-10 x-10 x^2\right )+50 \log (x)\right )\right ) \log \left (\frac {10+2 e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x}{x^2}\right )}{5 e^{5/x} x+e^{\frac {5}{x}+e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x^2} \, dx=\int { -\frac {10 \, {\left ({\left ({\left (x^{2} + x\right )} e^{\frac {5}{x}} - x - 5 \, \log \left (x\right )\right )} e^{\left (-{\left (x e^{\frac {5}{x}} - \log \left (x\right )\right )} e^{\left (-\frac {5}{x}\right )}\right )} + 10 \, e^{\frac {5}{x}}\right )} \log \left (\frac {2 \, {\left (x e^{\left (-{\left (x e^{\frac {5}{x}} - \log \left (x\right )\right )} e^{\left (-\frac {5}{x}\right )}\right )} + 5\right )}}{x^{2}}\right )}{x^{2} e^{\left (-{\left (x e^{\frac {5}{x}} - \log \left (x\right )\right )} e^{\left (-\frac {5}{x}\right )} + \frac {5}{x}\right )} + 5 \, x e^{\frac {5}{x}}} \,d x } \]
integrate(((50*log(x)+(-10*x^2-10*x)*exp(5/x)+10*x)*exp((log(x)-x*exp(5/x) )/exp(5/x))-100*exp(5/x))*log((2*x*exp((log(x)-x*exp(5/x))/exp(5/x))+10)/x ^2)/(x^2*exp(5/x)*exp((log(x)-x*exp(5/x))/exp(5/x))+5*x*exp(5/x)),x, algor ithm=\
-10*integrate((((x^2 + x)*e^(5/x) - x - 5*log(x))*e^(-(x*e^(5/x) - log(x)) *e^(-5/x)) + 10*e^(5/x))*log(2*(x*e^(-(x*e^(5/x) - log(x))*e^(-5/x)) + 5)/ x^2)/(x^2*e^(-(x*e^(5/x) - log(x))*e^(-5/x) + 5/x) + 5*x*e^(5/x)), x)
\[ \int \frac {\left (-100 e^{5/x}+e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} \left (10 x+e^{5/x} \left (-10 x-10 x^2\right )+50 \log (x)\right )\right ) \log \left (\frac {10+2 e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x}{x^2}\right )}{5 e^{5/x} x+e^{\frac {5}{x}+e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x^2} \, dx=\int { -\frac {10 \, {\left ({\left ({\left (x^{2} + x\right )} e^{\frac {5}{x}} - x - 5 \, \log \left (x\right )\right )} e^{\left (-{\left (x e^{\frac {5}{x}} - \log \left (x\right )\right )} e^{\left (-\frac {5}{x}\right )}\right )} + 10 \, e^{\frac {5}{x}}\right )} \log \left (\frac {2 \, {\left (x e^{\left (-{\left (x e^{\frac {5}{x}} - \log \left (x\right )\right )} e^{\left (-\frac {5}{x}\right )}\right )} + 5\right )}}{x^{2}}\right )}{x^{2} e^{\left (-{\left (x e^{\frac {5}{x}} - \log \left (x\right )\right )} e^{\left (-\frac {5}{x}\right )} + \frac {5}{x}\right )} + 5 \, x e^{\frac {5}{x}}} \,d x } \]
integrate(((50*log(x)+(-10*x^2-10*x)*exp(5/x)+10*x)*exp((log(x)-x*exp(5/x) )/exp(5/x))-100*exp(5/x))*log((2*x*exp((log(x)-x*exp(5/x))/exp(5/x))+10)/x ^2)/(x^2*exp(5/x)*exp((log(x)-x*exp(5/x))/exp(5/x))+5*x*exp(5/x)),x, algor ithm=\
integrate(-10*(((x^2 + x)*e^(5/x) - x - 5*log(x))*e^(-(x*e^(5/x) - log(x)) *e^(-5/x)) + 10*e^(5/x))*log(2*(x*e^(-(x*e^(5/x) - log(x))*e^(-5/x)) + 5)/ x^2)/(x^2*e^(-(x*e^(5/x) - log(x))*e^(-5/x) + 5/x) + 5*x*e^(5/x)), x)
Timed out. \[ \int \frac {\left (-100 e^{5/x}+e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} \left (10 x+e^{5/x} \left (-10 x-10 x^2\right )+50 \log (x)\right )\right ) \log \left (\frac {10+2 e^{e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x}{x^2}\right )}{5 e^{5/x} x+e^{\frac {5}{x}+e^{-5/x} \left (-e^{5/x} x+\log (x)\right )} x^2} \, dx=\int -\frac {\ln \left (\frac {2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {5}{x}}\,\left (\ln \left (x\right )-x\,{\mathrm {e}}^{5/x}\right )}+10}{x^2}\right )\,\left (100\,{\mathrm {e}}^{5/x}-{\mathrm {e}}^{{\mathrm {e}}^{-\frac {5}{x}}\,\left (\ln \left (x\right )-x\,{\mathrm {e}}^{5/x}\right )}\,\left (10\,x+50\,\ln \left (x\right )-{\mathrm {e}}^{5/x}\,\left (10\,x^2+10\,x\right )\right )\right )}{5\,x\,{\mathrm {e}}^{5/x}+x^2\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {5}{x}}\,\left (\ln \left (x\right )-x\,{\mathrm {e}}^{5/x}\right )}\,{\mathrm {e}}^{5/x}} \,d x \]
int(-(log((2*x*exp(exp(-5/x)*(log(x) - x*exp(5/x))) + 10)/x^2)*(100*exp(5/ x) - exp(exp(-5/x)*(log(x) - x*exp(5/x)))*(10*x + 50*log(x) - exp(5/x)*(10 *x + 10*x^2))))/(5*x*exp(5/x) + x^2*exp(exp(-5/x)*(log(x) - x*exp(5/x)))*e xp(5/x)),x)