\(\int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x)))}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx\) [1172]

Optimal result

Integrand size = 105, antiderivative size = 25 \[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=\frac {x^2}{\left (e^{e^{x^2}+x^{\frac {1}{x}}}-\frac {14 x}{5}\right )^2} \] Output:

x^2/(exp(exp(ln(x)/x)+exp(x^2))-14/5*x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=\frac {25 x^2}{\left (5 e^{e^{x^2}+x^{\frac {1}{x}}}-14 x\right )^2} \] Input:

Integrate[(E^(E^x^2 + x^x^(-1))*(250*x - 500*E^x^2*x^3 + x^x^(-1)*(-250 + 
250*Log[x])))/(125*E^(3*E^x^2 + 3*x^x^(-1)) - 1050*E^(2*E^x^2 + 2*x^x^(-1) 
)*x + 2940*E^(E^x^2 + x^x^(-1))*x^2 - 2744*x^3),x]
 

Output:

(25*x^2)/(5*E^(E^x^2 + x^x^(-1)) - 14*x)^2
 

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.

\(\displaystyle \int \frac {e^{x^{\frac {1}{x}}+e^{x^2}} \left (x^{\frac {1}{x}} (250 \log (x)-250)-500 e^{x^2} x^3+250 x\right )}{-2744 x^3+2940 e^{x^{\frac {1}{x}}+e^{x^2}} x^2-1050 e^{2 x^{\frac {1}{x}}+2 e^{x^2}} x+125 e^{3 x^{\frac {1}{x}}+3 e^{x^2}}} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {250 e^{x^{\frac {1}{x}}+e^{x^2}} \left (-x^{\frac {1}{x}}+x^{\frac {1}{x}} \log (x)-2 e^{x^2} x^3+x\right )}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 250 \int \frac {e^{x^{\frac {1}{x}}+e^{x^2}} \left (\log (x) x^{\frac {1}{x}}-x^{\frac {1}{x}}-2 e^{x^2} x^3+x\right )}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 250 \int \left (\frac {e^{x^{\frac {1}{x}}+e^{x^2}} x^{\frac {1}{x}} (\log (x)-1)}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}-\frac {e^{x^{\frac {1}{x}}+e^{x^2}} x \left (2 e^{x^2} x^2-1\right )}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 250 \left (\int \frac {e^{x^{\frac {1}{x}}+e^{x^2}} x}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}dx-\int \frac {e^{x^{\frac {1}{x}}+e^{x^2}} x^{\frac {1}{x}}}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}dx-\int \frac {\int \frac {e^{x^{\frac {1}{x}}+e^{x^2}} x^{\frac {1}{x}}}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}dx}{x}dx+\log (x) \int \frac {e^{x^{\frac {1}{x}}+e^{x^2}} x^{\frac {1}{x}}}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}dx-2 \int \frac {e^{x^{\frac {1}{x}}+x^2+e^{x^2}} x^3}{\left (5 e^{x^{\frac {1}{x}}+e^{x^2}}-14 x\right )^3}dx\right )\)

Input:

Int[(E^(E^x^2 + x^x^(-1))*(250*x - 500*E^x^2*x^3 + x^x^(-1)*(-250 + 250*Lo 
g[x])))/(125*E^(3*E^x^2 + 3*x^x^(-1)) - 1050*E^(2*E^x^2 + 2*x^x^(-1))*x + 
2940*E^(E^x^2 + x^x^(-1))*x^2 - 2744*x^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 29.93 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {25 x^{2}}{\left (14 x -5 \,{\mathrm e}^{x^{\frac {1}{x}}+{\mathrm e}^{x^{2}}}\right )^{2}}\)	\(25\)
parallelrisch	\(\frac {25 x^{2}}{196 x^{2}-140 x \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \left (x \right )}{x}}+{\mathrm e}^{x^{2}}}+25 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {\ln \left (x \right )}{x}}+2 \,{\mathrm e}^{x^{2}}}}\)	\(47\)

Input:

int(((250*ln(x)-250)*exp(ln(x)/x)-500*x^3*exp(x^2)+250*x)*exp(exp(ln(x)/x) 
+exp(x^2))/(125*exp(exp(ln(x)/x)+exp(x^2))^3-1050*x*exp(exp(ln(x)/x)+exp(x 
^2))^2+2940*x^2*exp(exp(ln(x)/x)+exp(x^2))-2744*x^3),x,method=_RETURNVERBO 
SE)
 

Output:

25*x^2/(14*x-5*exp(x^(1/x)+exp(x^2)))^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=\frac {25 \, x^{2}}{196 \, x^{2} - 140 \, x e^{\left (x^{\left (\frac {1}{x}\right )} + e^{\left (x^{2}\right )}\right )} + 25 \, e^{\left (2 \, x^{\left (\frac {1}{x}\right )} + 2 \, e^{\left (x^{2}\right )}\right )}} \] Input:

integrate(((250*log(x)-250)*exp(log(x)/x)-500*x^3*exp(x^2)+250*x)*exp(exp( 
log(x)/x)+exp(x^2))/(125*exp(exp(log(x)/x)+exp(x^2))^3-1050*x*exp(exp(log( 
x)/x)+exp(x^2))^2+2940*x^2*exp(exp(log(x)/x)+exp(x^2))-2744*x^3),x, algori 
thm="fricas")
 

Output:

25*x^2/(196*x^2 - 140*x*e^(x^(1/x) + e^(x^2)) + 25*e^(2*x^(1/x) + 2*e^(x^2 
)))
 

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=\frac {x^{2}}{\frac {196 x^{2}}{25} - \frac {28 x e^{e^{x^{2}} + e^{\frac {\log {\left (x \right )}}{x}}}}{5} + e^{2 e^{x^{2}} + 2 e^{\frac {\log {\left (x \right )}}{x}}}} \] Input:

integrate(((250*ln(x)-250)*exp(ln(x)/x)-500*exp(x**2)*x**3+250*x)*exp(exp( 
ln(x)/x)+exp(x**2))/(125*exp(exp(ln(x)/x)+exp(x**2))**3-1050*x*exp(exp(ln( 
x)/x)+exp(x**2))**2+2940*x**2*exp(exp(ln(x)/x)+exp(x**2))-2744*x**3),x)
 

Output:

x**2/(196*x**2/25 - 28*x*exp(exp(x**2) + exp(log(x)/x))/5 + exp(2*exp(x**2 
) + 2*exp(log(x)/x)))
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=\frac {25 \, x^{2}}{196 \, x^{2} - 140 \, x e^{\left (x^{\left (\frac {1}{x}\right )} + e^{\left (x^{2}\right )}\right )} + 25 \, e^{\left (2 \, x^{\left (\frac {1}{x}\right )} + 2 \, e^{\left (x^{2}\right )}\right )}} \] Input:

integrate(((250*log(x)-250)*exp(log(x)/x)-500*x^3*exp(x^2)+250*x)*exp(exp( 
log(x)/x)+exp(x^2))/(125*exp(exp(log(x)/x)+exp(x^2))^3-1050*x*exp(exp(log( 
x)/x)+exp(x^2))^2+2940*x^2*exp(exp(log(x)/x)+exp(x^2))-2744*x^3),x, algori 
thm="maxima")
 

Output:

25*x^2/(196*x^2 - 140*x*e^(x^(1/x) + e^(x^2)) + 25*e^(2*x^(1/x) + 2*e^(x^2 
)))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=\text {Exception raised: NotImplementedError} \] Input:

integrate(((250*log(x)-250)*exp(log(x)/x)-500*x^3*exp(x^2)+250*x)*exp(exp( 
log(x)/x)+exp(x^2))/(125*exp(exp(log(x)/x)+exp(x^2))^3-1050*x*exp(exp(log( 
x)/x)+exp(x^2))^2+2940*x^2*exp(exp(log(x)/x)+exp(x^2))-2744*x^3),x, algori 
thm="giac")
 

Output:

Exception raised: NotImplementedError >> unable to parse Giac output: Exch 
ange 57736848 31 Vector [13,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 
1,1,1,1,1,1,4] 70 Vector [13,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 
,1,1,1,1,1,1,4] 70/
 

Mupad [B] (verification not implemented)

Time = 2.90 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=-\frac {25\,x^5\,\left (x-2\,x^3\,{\mathrm {e}}^{x^2}-x^{1/x}+x^{1/x}\,\ln \left (x\right )\right )}{\left (25\,{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2}+2\,x^{1/x}}-140\,x\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}+x^{1/x}}+196\,x^2\right )\,\left (2\,x^6\,{\mathrm {e}}^{x^2}-x^4+x^{1/x}\,x^3-x^{1/x}\,x^3\,\ln \left (x\right )\right )} \] Input:

int((exp(exp(x^2) + exp(log(x)/x))*(250*x - 500*x^3*exp(x^2) + exp(log(x)/ 
x)*(250*log(x) - 250)))/(125*exp(3*exp(x^2) + 3*exp(log(x)/x)) - 1050*x*ex 
p(2*exp(x^2) + 2*exp(log(x)/x)) + 2940*x^2*exp(exp(x^2) + exp(log(x)/x)) - 
 2744*x^3),x)
 

Output:

-(25*x^5*(x - 2*x^3*exp(x^2) - x^(1/x) + x^(1/x)*log(x)))/((25*exp(2*exp(x 
^2) + 2*x^(1/x)) - 140*x*exp(exp(x^2) + x^(1/x)) + 196*x^2)*(2*x^6*exp(x^2 
) - x^4 + x^(1/x)*x^3 - x^(1/x)*x^3*log(x)))
 

Reduce [F]

\[ \int \frac {e^{e^{x^2}+x^{\frac {1}{x}}} \left (250 x-500 e^{x^2} x^3+x^{\frac {1}{x}} (-250+250 \log (x))\right )}{125 e^{3 e^{x^2}+3 x^{\frac {1}{x}}}-1050 e^{2 e^{x^2}+2 x^{\frac {1}{x}}} x+2940 e^{e^{x^2}+x^{\frac {1}{x}}} x^2-2744 x^3} \, dx=\int \frac {\left (\left (250 \,\mathrm {log}\left (x \right )-250\right ) {\mathrm e}^{\frac {\mathrm {log}\left (x \right )}{x}}-500 \,{\mathrm e}^{x^{2}} x^{3}+250 x \right ) {\mathrm e}^{{\mathrm e}^{\frac {\mathrm {log}\left (x \right )}{x}}+{\mathrm e}^{x^{2}}}}{125 \left ({\mathrm e}^{{\mathrm e}^{\frac {\mathrm {log}\left (x \right )}{x}}+{\mathrm e}^{x^{2}}}\right )^{3}-1050 x \left ({\mathrm e}^{{\mathrm e}^{\frac {\mathrm {log}\left (x \right )}{x}}+{\mathrm e}^{x^{2}}}\right )^{2}+2940 x^{2} {\mathrm e}^{{\mathrm e}^{\frac {\mathrm {log}\left (x \right )}{x}}+{\mathrm e}^{x^{2}}}-2744 x^{3}}d x \] Input:

int(((250*log(x)-250)*exp(log(x)/x)-500*exp(x^2)*x^3+250*x)*exp(exp(log(x) 
/x)+exp(x^2))/(125*exp(exp(log(x)/x)+exp(x^2))^3-1050*x*exp(exp(log(x)/x)+ 
exp(x^2))^2+2940*x^2*exp(exp(log(x)/x)+exp(x^2))-2744*x^3),x)
 

Output:

int(((250*log(x)-250)*exp(log(x)/x)-500*exp(x^2)*x^3+250*x)*exp(exp(log(x) 
/x)+exp(x^2))/(125*exp(exp(log(x)/x)+exp(x^2))^3-1050*x*exp(exp(log(x)/x)+ 
exp(x^2))^2+2940*x^2*exp(exp(log(x)/x)+exp(x^2))-2744*x^3),x)