\(\int \frac {5^{(2-e^{2 x}) x^x} (2-e^{2 x}) x^{x+(2-e^{2 x}) x^x} (-2+e^{2 x}+(-2 x+3 e^{2 x} x+(-2 x+e^{2 x} x) \log (x)) \log (5 x))}{-2 x+e^{2 x} x} \, dx\) [2633]

Optimal result

Integrand size = 93, antiderivative size = 31 \[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx=5^{\left (2-e^{2 x}\right ) x^x} x^{\left (2-e^{2 x}\right ) x^x} \] Output:

exp(ln(5*x)*exp(ln(-exp(x)^2+2)+x*ln(x)))
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx=\int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx \] Input:

Integrate[(5^((2 - E^(2*x))*x^x)*(2 - E^(2*x))*x^(x + (2 - E^(2*x))*x^x)*( 
-2 + E^(2*x) + (-2*x + 3*E^(2*x)*x + (-2*x + E^(2*x)*x)*Log[x])*Log[5*x])) 
/(-2*x + E^(2*x)*x),x]
 

Output:

Integrate[(5^((2 - E^(2*x))*x^x)*(2 - E^(2*x))*x^(x + (2 - E^(2*x))*x^x)*( 
-2 + E^(2*x) + (-2*x + 3*E^(2*x)*x + (-2*x + E^(2*x)*x)*Log[x])*Log[5*x])) 
/(-2*x + E^(2*x)*x), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{\left (2-e^{2 x}\right ) x^x+x} \left (e^{2 x}+\left (3 e^{2 x} x-2 x+\left (e^{2 x} x-2 x\right ) \log (x)\right ) \log (5 x)-2\right )}{e^{2 x} x-2 x} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (2\ 5^{\left (2-e^{2 x}\right ) x^x} x^{\left (2-e^{2 x}\right ) x^x+x-1} (x \log (5 x)+x \log (x) \log (5 x)+1)-5^{\left (2-e^{2 x}\right ) x^x} e^{2 x} x^{\left (2-e^{2 x}\right ) x^x+x-1} (3 x \log (5 x)+x \log (x) \log (5 x)+1)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int 5^{\left (2-e^{2 x}\right ) x^x} x^{\left (2-e^{2 x}\right ) x^x+x-1}dx-\int 5^{\left (2-e^{2 x}\right ) x^x} e^{2 x} x^{\left (2-e^{2 x}\right ) x^x+x-1}dx-2 \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx+3 \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx+4 \int \frac {\int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx}{x}dx-2 \int \frac {\int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx}{x}dx+2 \log (x) \log (5 x) \int 5^{\left (2-e^{2 x}\right ) x^x} x^{\left (2-e^{2 x}\right ) x^x+x}dx+2 \log (5 x) \int 5^{\left (2-e^{2 x}\right ) x^x} x^{\left (2-e^{2 x}\right ) x^x+x}dx-\log (x) \log (5 x) \int 5^{\left (2-e^{2 x}\right ) x^x} e^{2 x} x^{\left (2-e^{2 x}\right ) x^x+x}dx-3 \log (5 x) \int 5^{\left (2-e^{2 x}\right ) x^x} e^{2 x} x^{\left (2-e^{2 x}\right ) x^x+x}dx-2 \log (x) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx-2 \log (5 x) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx+\log (x) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx+\log (5 x) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x}dx}{x}dx\)

Input:

Int[(5^((2 - E^(2*x))*x^x)*(2 - E^(2*x))*x^(x + (2 - E^(2*x))*x^x)*(-2 + E 
^(2*x) + (-2*x + 3*E^(2*x)*x + (-2*x + E^(2*x)*x)*Log[x])*Log[5*x]))/(-2*x 
 + E^(2*x)*x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58

\[{\mathrm e}^{-\left (\ln \left (5\right )+\ln \left (x \right )\right ) \left ({\mathrm e}^{2 x}-2\right ) x^{x}}\]

Input:

int((((x*exp(x)^2-2*x)*ln(x)+3*x*exp(x)^2-2*x)*ln(5*x)+exp(x)^2-2)*exp(ln( 
-exp(x)^2+2)+x*ln(x))*exp(ln(5*x)*exp(ln(-exp(x)^2+2)+x*ln(x)))/(x*exp(x)^ 
2-2*x),x)
 

Output:

exp(-(ln(5)+ln(x))*(exp(2*x)-2)*x^x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx=e^{\left ({\left (\log \left (5\right ) + \log \left (x\right )\right )} e^{\left (x \log \left (x\right ) + \log \left (-e^{\left (2 \, x\right )} + 2\right )\right )}\right )} \] Input:

integrate((((x*exp(x)^2-2*x)*log(x)+3*x*exp(x)^2-2*x)*log(5*x)+exp(x)^2-2) 
*exp(log(-exp(x)^2+2)+x*log(x))*exp(log(5*x)*exp(log(-exp(x)^2+2)+x*log(x) 
))/(x*exp(x)^2-2*x),x, algorithm="fricas")
 

Output:

e^((log(5) + log(x))*e^(x*log(x) + log(-e^(2*x) + 2)))
 

Sympy [A] (verification not implemented)

Time = 9.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx=e^{\left (2 - e^{2 x}\right ) \left (\log {\left (x \right )} + \log {\left (5 \right )}\right ) e^{x \log {\left (x \right )}}} \] Input:

integrate((((x*exp(x)**2-2*x)*ln(x)+3*x*exp(x)**2-2*x)*ln(5*x)+exp(x)**2-2 
)*exp(ln(-exp(x)**2+2)+x*ln(x))*exp(ln(5*x)*exp(ln(-exp(x)**2+2)+x*ln(x))) 
/(x*exp(x)**2-2*x),x)
 

Output:

exp((2 - exp(2*x))*(log(x) + log(5))*exp(x*log(x)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).

Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx=e^{\left (2 \, x^{x} \log \left (5\right ) - e^{\left (x \log \left (x\right ) + 2 \, x\right )} \log \left (5\right ) + 2 \, x^{x} \log \left (x\right ) - e^{\left (x \log \left (x\right ) + 2 \, x\right )} \log \left (x\right )\right )} \] Input:

integrate((((x*exp(x)^2-2*x)*log(x)+3*x*exp(x)^2-2*x)*log(5*x)+exp(x)^2-2) 
*exp(log(-exp(x)^2+2)+x*log(x))*exp(log(5*x)*exp(log(-exp(x)^2+2)+x*log(x) 
))/(x*exp(x)^2-2*x),x, algorithm="maxima")
 

Output:

e^(2*x^x*log(5) - e^(x*log(x) + 2*x)*log(5) + 2*x^x*log(x) - e^(x*log(x) + 
 2*x)*log(x))
 

Giac [F]

\[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx=\int { \frac {{\left ({\left (3 \, x e^{\left (2 \, x\right )} + {\left (x e^{\left (2 \, x\right )} - 2 \, x\right )} \log \left (x\right ) - 2 \, x\right )} \log \left (5 \, x\right ) + e^{\left (2 \, x\right )} - 2\right )} \left (5 \, x\right )^{e^{\left (x \log \left (x\right ) + \log \left (-e^{\left (2 \, x\right )} + 2\right )\right )}} e^{\left (x \log \left (x\right ) + \log \left (-e^{\left (2 \, x\right )} + 2\right )\right )}}{x e^{\left (2 \, x\right )} - 2 \, x} \,d x } \] Input:

integrate((((x*exp(x)^2-2*x)*log(x)+3*x*exp(x)^2-2*x)*log(5*x)+exp(x)^2-2) 
*exp(log(-exp(x)^2+2)+x*log(x))*exp(log(5*x)*exp(log(-exp(x)^2+2)+x*log(x) 
))/(x*exp(x)^2-2*x),x, algorithm="giac")
 

Output:

integrate(((3*x*e^(2*x) + (x*e^(2*x) - 2*x)*log(x) - 2*x)*log(5*x) + e^(2* 
x) - 2)*(5*x)^e^(x*log(x) + log(-e^(2*x) + 2))*e^(x*log(x) + log(-e^(2*x) 
+ 2))/(x*e^(2*x) - 2*x), x)
 

Mupad [B] (verification not implemented)

Time = 4.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61 \[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx={\left (5\,x\right )}^{2\,x^x-x^x\,{\mathrm {e}}^{2\,x}} \] Input:

int((exp(log(2 - exp(2*x)) + x*log(x))*exp(log(5*x)*exp(log(2 - exp(2*x)) 
+ x*log(x)))*(log(5*x)*(2*x - 3*x*exp(2*x) + log(x)*(2*x - x*exp(2*x))) - 
exp(2*x) + 2))/(2*x - x*exp(2*x)),x)
 

Output:

(5*x)^(2*x^x - x^x*exp(2*x))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx=\frac {e^{2 x^{x} \mathrm {log}\left (5 x \right )}}{e^{x^{x} e^{2 x} \mathrm {log}\left (5 x \right )}} \] Input:

int((((x*exp(x)^2-2*x)*log(x)+3*x*exp(x)^2-2*x)*log(5*x)+exp(x)^2-2)*exp(l 
og(-exp(x)^2+2)+x*log(x))*exp(log(5*x)*exp(log(-exp(x)^2+2)+x*log(x)))/(x* 
exp(x)^2-2*x),x)
 

Output:

e**(2*x**x*log(5*x))/e**(x**x*e**(2*x)*log(5*x))