Integrand size = 104, antiderivative size = 25 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (-2+\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x\right )}{x}\right )} \]
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \]
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]
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 |
\(\displaystyle -\int \frac {1}{\log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx+\int \frac {1}{x \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx-5 \int \frac {1}{\left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx-2 \int \frac {e^{\frac {e^4}{x^2}}}{\left (2 e^{\frac {e^4}{x^2}} x-x-e^x+4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx+4 \int \frac {e^{4+\frac {e^4}{x^2}}}{x^2 \left (2 e^{\frac {e^4}{x^2}} x-x-e^x+4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx-\int \frac {x}{\left (2 e^{\frac {e^4}{x^2}} x-x-e^x+4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx+2 \int \frac {e^{\frac {e^4}{x^2}} x}{\left (2 e^{\frac {e^4}{x^2}} x-x-e^x+4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx-2 e^4 \int \frac {1}{x^3 \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-2 e^{\frac {e^4}{x^2}} x+x+e^x-4\right )}{x}\right )}dx\) |
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]
3.11.15.3.1 Defintions of rubi rules used
Time = 162.71 (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}{-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 (\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}-\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 (\left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}-\frac {1}{2}\right ) x -\frac {{\mathrm e}^{x}}{2}+2\right )-2 i \ln \left (2\right )+2 i \ln \left (x \right )+2 i \ln \left ({\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{2}}}\right )}\) | \(517\) |
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)
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (-\frac {{\left (2 \, x e^{\left (\frac {e^{4}}{x^{2}}\right )} - x - e^{x} + 4\right )} e^{\left (-\frac {e^{4}}{x^{2}}\right )}}{x}\right )} \]
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=\
Time = 1.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log {\left (\frac {\left (- 2 x e^{\frac {e^{4}}{x^{2}}} + x + e^{x} - 4\right ) e^{- \frac {e^{4}}{x^{2}}}}{x} \right )}} \]
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)
Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {x^{2}}{x^{2} \log \left (-2 \, x e^{\left (\frac {e^{4}}{x^{2}}\right )} + x + e^{x} - 4\right ) - x^{2} \log \left (x\right ) - e^{4}} \]
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=\
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (22) = 44\).
Time = 0.68 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\log \left (-\frac {{\left (2 \, x e^{x} - x e^{\left (\frac {x^{3} - e^{4}}{x^{2}}\right )} - e^{\left (x + \frac {x^{3} - e^{4}}{x^{2}}\right )} + 4 \, e^{\left (\frac {x^{3} - e^{4}}{x^{2}}\right )}\right )} e^{\left (-x\right )}}{x}\right )} \]
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=\
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)
Time = 10.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {4 x^2+e^4 (-8+2 x)+e^x \left (2 e^4-x^2+x^3\right )}{\left (4 x^3-e^x x^3-x^4+2 e^{\frac {e^4}{x^2}} x^4\right ) \log ^2\left (\frac {e^{-\frac {e^4}{x^2}} \left (-4+e^x+x-2 e^{\frac {e^4}{x^2}} x\right )}{x}\right )} \, dx=\frac {1}{\ln \left (\frac {x+{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^2}}-4}{x}\right )-\frac {{\mathrm {e}}^4}{x^2}} \]