\(\int \frac {4 x^2+e^4 (-8+2 x)+e^x (2 e^4-x^2+x^3)}{(4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4) \log ^2(\frac {e^{-\frac {e^4}{x^2}} (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x)}{x})} \, dx\) [1015]

Optimal result

Integrand size = 104, antiderivative size = 25 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (-2+\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x\right )}{x}\right )} \] Output:

1/ln((exp(x)+x-4)/x/exp(exp(4)/x^2)-2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \] Input:

Integrate[(4*x^2 + E^4*(-8 + 2*x) + E^x*(2*E^4 - x^2 + x^3))/((4*x^3 - E^x 
*x^3 - x^4 + 2*E^(E^4/x^2)*x^4)*Log[(-4 + E^x + x - 2*E^(E^4/x^2)*x)/(E^(E 
^4/x^2)*x)]^2),x]
 

Output:

Log[(-4 + E^x + x - 2*E^(E^4/x^2)*x)/(E^(E^4/x^2)*x)]^(-1)
 

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.

\(\displaystyle \int \frac {4 x^2+e^x \left (x^3-x^2+2 e^4\right )+e^4 (2 x-8)}{\left (-x^4-e^x x^3+4 x^3+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(4*x^2 + E^4*(-8 + 2*x) + E^x*(2*E^4 - x^2 + x^3))/((4*x^3 - E^x*x^3 - 
 x^4 + 2*E^(E^4/x^2)*x^4)*Log[(-4 + E^x + x - 2*E^(E^4/x^2)*x)/(E^(E^4/x^2 
)*x)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 188.86 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28

method	result	size
parallelrisch	\(\frac {1}{\ln \left (\frac {\left (-2 x \,{\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}+{\mathrm e}^{x}+x -4\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}}{x}\right )}\)	\(32\)
risch	\(\frac {2 i}{\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}} \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}}{x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}}{x}\right )}^{2}+2 \pi {\operatorname {csgn}\left (\frac {i \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}}{x}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}} \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right )+\pi \,\operatorname {csgn}\left (i \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}} \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right )^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}} \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right )^{2}-\pi \operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}} \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right )^{3}+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}} \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}}{x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (-\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x +\frac {{\mathrm e}^{x}}{2}-2\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{x^{2}}}}{x}\right )}^{3}-2 \pi -2 i \ln \left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}\right )+2 i \ln \left (2\right )-2 i \ln \left (x \right )+2 i \ln \left (\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x -\frac {{\mathrm e}^{x}}{2}+2\right )}\)	\(515\)

Input:

int(((2*exp(4)+x^3-x^2)*exp(x)+(2*x-8)*exp(4)+4*x^2)/(2*x^4*exp(exp(4)/x^2 
)-exp(x)*x^3-x^4+4*x^3)/ln((-2*x*exp(exp(4)/x^2)+exp(x)+x-4)/x/exp(exp(4)/ 
x^2))^2,x,method=_RETURNVERBOSE)
 

Output:

1/ln((-2*x*exp(exp(4)/x^2)+exp(x)+x-4)/x/exp(exp(4)/x^2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (-\frac {{\left (2 \, x e^{\left (\frac {e^{4}}{x^{2}}\right )} - x - e^{x} + 4\right )} e^{\left (-\frac {e^{4}}{x^{2}}\right )}}{x}\right )} \] Input:

integrate(((2*exp(4)+x^3-x^2)*exp(x)+(2*x-8)*exp(4)+4*x^2)/(2*x^4*exp(exp( 
4)/x^2)-exp(x)*x^3-x^4+4*x^3)/log((-2*x*exp(exp(4)/x^2)+exp(x)+x-4)/x/exp( 
exp(4)/x^2))^2,x, algorithm="fricas")
 

Output:

1/log(-(2*x*e^(e^4/x^2) - x - e^x + 4)*e^(-e^4/x^2)/x)
 

Sympy [A] (verification not implemented)

Time = 1.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log {\left (\frac {\left (- 2 x e^{\frac {e^{4}}{x^{2}}} + x + e^{x} - 4\right ) e^{- \frac {e^{4}}{x^{2}}}}{x} \right )}} \] Input:

integrate(((2*exp(4)+x**3-x**2)*exp(x)+(2*x-8)*exp(4)+4*x**2)/(2*x**4*exp( 
exp(4)/x**2)-exp(x)*x**3-x**4+4*x**3)/ln((-2*x*exp(exp(4)/x**2)+exp(x)+x-4 
)/x/exp(exp(4)/x**2))**2,x)
 

Output:

1/log((-2*x*exp(exp(4)/x**2) + x + exp(x) - 4)*exp(-exp(4)/x**2)/x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {x^{2}}{x^{2} \log \left (-2 \, x e^{\left (\frac {e^{4}}{x^{2}}\right )} + x + e^{x} - 4\right ) - x^{2} \log \left (x\right ) - e^{4}} \] Input:

integrate(((2*exp(4)+x^3-x^2)*exp(x)+(2*x-8)*exp(4)+4*x^2)/(2*x^4*exp(exp( 
4)/x^2)-exp(x)*x^3-x^4+4*x^3)/log((-2*x*exp(exp(4)/x^2)+exp(x)+x-4)/x/exp( 
exp(4)/x^2))^2,x, algorithm="maxima")
 

Output:

x^2/(x^2*log(-2*x*e^(e^4/x^2) + x + e^x - 4) - x^2*log(x) - e^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (22) = 44\).

Time = 0.51 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (-\frac {{\left (2 \, x e^{x} - x e^{\left (\frac {x^{3} - e^{4}}{x^{2}}\right )} - e^{\left (x + \frac {x^{3} - e^{4}}{x^{2}}\right )} + 4 \, e^{\left (\frac {x^{3} - e^{4}}{x^{2}}\right )}\right )} e^{\left (-x\right )}}{x}\right )} \] Input:

integrate(((2*exp(4)+x^3-x^2)*exp(x)+(2*x-8)*exp(4)+4*x^2)/(2*x^4*exp(exp( 
4)/x^2)-exp(x)*x^3-x^4+4*x^3)/log((-2*x*exp(exp(4)/x^2)+exp(x)+x-4)/x/exp( 
exp(4)/x^2))^2,x, algorithm="giac")
 

Output:

1/log(-(2*x*e^x - x*e^((x^3 - e^4)/x^2) - e^(x + (x^3 - e^4)/x^2) + 4*e^(( 
x^3 - e^4)/x^2))*e^(-x)/x)
 

Mupad [B] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\ln \left (\frac {x+{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^2}}-4}{x}\right )-\frac {{\mathrm {e}}^4}{x^2}} \] Input:

int(-(exp(x)*(2*exp(4) - x^2 + x^3) + 4*x^2 + exp(4)*(2*x - 8))/(log((exp( 
-exp(4)/x^2)*(x + exp(x) - 2*x*exp(exp(4)/x^2) - 4))/x)^2*(x^3*exp(x) - 2* 
x^4*exp(exp(4)/x^2) - 4*x^3 + x^4)),x)
 

Output:

1/(log((x + exp(x) - 2*x*exp(exp(4)/x^2) - 4)/x) - exp(4)/x^2)
 

Reduce [F]

\[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\int \frac {\left (2 \,{\mathrm e}^{4}+x^{3}-x^{2}\right ) {\mathrm e}^{x}+\left (2 x -8\right ) {\mathrm e}^{4}+4 x^{2}}{\left (2 x^{4} {\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-{\mathrm e}^{x} x^{3}-x^{4}+4 x^{3}\right ) \mathrm {log}\left (\frac {-2 x \,{\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}+{\mathrm e}^{x}+x -4}{x \,{\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}}\right )^{2}}d x \] Input:

int(((2*exp(4)+x^3-x^2)*exp(x)+(2*x-8)*exp(4)+4*x^2)/(2*x^4*exp(exp(4)/x^2 
)-exp(x)*x^3-x^4+4*x^3)/log((-2*x*exp(exp(4)/x^2)+exp(x)+x-4)/x/exp(exp(4) 
/x^2))^2,x)
 

Output:

int(((2*exp(4)+x^3-x^2)*exp(x)+(2*x-8)*exp(4)+4*x^2)/(2*x^4*exp(exp(4)/x^2 
)-exp(x)*x^3-x^4+4*x^3)/log((-2*x*exp(exp(4)/x^2)+exp(x)+x-4)/x/exp(exp(4) 
/x^2))^2,x)