Integrand size = 80, antiderivative size = 29 \[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\frac {60 e^{2 e^{-e^{2/x}} \left (2-e^{2 x}\right )}}{x} \] Output:
60/x*exp(2*(2-exp(2*x))/exp(exp(2/x)))
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\frac {60 e^{-2 e^{-e^{2/x}} \left (-2+e^{2 x}\right )}}{x} \] Input:
Integrate[(E^(-E^(2/x) + (4 - 2*E^(2*x))/E^E^(2/x))*(480*E^(2/x) - 60*E^E^ (2/x)*x + E^(2*x)*(-240*E^(2/x) - 240*x^2)))/x^3,x]
Output:
60/(E^((2*(-2 + E^(2*x)))/E^E^(2/x))*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.
\(\displaystyle \int \frac {e^{e^{-e^{2/x}} \left (4-2 e^{2 x}\right )-e^{2/x}} \left (e^{2 x} \left (-240 x^2-240 e^{2/x}\right )-60 e^{e^{2/x}} x+480 e^{2/x}\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {60 e^{e^{-e^{2/x}} \left (4-2 e^{2 x}\right )-e^{2/x}} \left (e^{e^{2/x}} x-8 e^{2/x}\right )}{x^3}-\frac {240 e^{e^{-e^{2/x}} \left (4-2 e^{2 x}\right )-e^{2/x}+2 x} \left (x^2+e^{2/x}\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -240 \int \frac {\exp \left (e^{-e^{2/x}} \left (4-2 e^{2 x}\right )-e^{2/x}+2 x+\frac {2}{x}\right )}{x^3}dx+480 \int \frac {e^{e^{-e^{2/x}} \left (4-2 e^{2 x}\right )-e^{2/x}+\frac {2}{x}}}{x^3}dx-60 \int \frac {e^{-2 e^{-e^{2/x}} \left (-2+e^{2 x}\right )}}{x^2}dx-240 \int \frac {e^{e^{-e^{2/x}} \left (4-2 e^{2 x}\right )-e^{2/x}+2 x}}{x}dx\) |
Input:
Int[(E^(-E^(2/x) + (4 - 2*E^(2*x))/E^E^(2/x))*(480*E^(2/x) - 60*E^E^(2/x)* x + E^(2*x)*(-240*E^(2/x) - 240*x^2)))/x^3,x]
Output:
$Aborted
Time = 1.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {60 \,{\mathrm e}^{-2 \left ({\mathrm e}^{2 x}-2\right ) {\mathrm e}^{-{\mathrm e}^{\frac {2}{x}}}}}{x}\) | \(24\) |
parallelrisch | \(\frac {60 \,{\mathrm e}^{-2 \left ({\mathrm e}^{2 x}-2\right ) {\mathrm e}^{-{\mathrm e}^{\frac {2}{x}}}}}{x}\) | \(24\) |
Input:
int((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/x))*ex p((-2*exp(2*x)+4)/exp(exp(2/x)))/x^3/exp(exp(2/x)),x,method=_RETURNVERBOSE )
Output:
60/x*exp(-2*(exp(2*x)-2)*exp(-exp(2/x)))
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\frac {60 \, e^{\left (-{\left (2 \, e^{\left (2 \, x\right )} + e^{\left (\frac {2}{x} + e^{\frac {2}{x}}\right )} - 4\right )} e^{\left (-e^{\frac {2}{x}}\right )} + e^{\frac {2}{x}}\right )}}{x} \] Input:
integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/ x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)))/x^3/exp(exp(2/x)),x, algorithm="fri cas")
Output:
60*e^(-(2*e^(2*x) + e^(2/x + e^(2/x)) - 4)*e^(-e^(2/x)) + e^(2/x))/x
Time = 0.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\frac {60 e^{\left (4 - 2 e^{2 x}\right ) e^{- e^{\frac {2}{x}}}}}{x} \] Input:
integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x**2)*exp(2*x)+480*exp(2 /x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)))/x**3/exp(exp(2/x)),x)
Output:
60*exp((4 - 2*exp(2*x))*exp(-exp(2/x)))/x
\[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\int { -\frac {60 \, {\left (4 \, {\left (x^{2} + e^{\frac {2}{x}}\right )} e^{\left (2 \, x\right )} + x e^{\left (e^{\frac {2}{x}}\right )} - 8 \, e^{\frac {2}{x}}\right )} e^{\left (-2 \, {\left (e^{\left (2 \, x\right )} - 2\right )} e^{\left (-e^{\frac {2}{x}}\right )} - e^{\frac {2}{x}}\right )}}{x^{3}} \,d x } \] Input:
integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/ x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)))/x^3/exp(exp(2/x)),x, algorithm="max ima")
Output:
-60*integrate((4*(x^2 + e^(2/x))*e^(2*x) + x*e^(e^(2/x)) - 8*e^(2/x))*e^(- 2*(e^(2*x) - 2)*e^(-e^(2/x)) - e^(2/x))/x^3, x)
\[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\int { -\frac {60 \, {\left (4 \, {\left (x^{2} + e^{\frac {2}{x}}\right )} e^{\left (2 \, x\right )} + x e^{\left (e^{\frac {2}{x}}\right )} - 8 \, e^{\frac {2}{x}}\right )} e^{\left (-2 \, {\left (e^{\left (2 \, x\right )} - 2\right )} e^{\left (-e^{\frac {2}{x}}\right )} - e^{\frac {2}{x}}\right )}}{x^{3}} \,d x } \] Input:
integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/ x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)))/x^3/exp(exp(2/x)),x, algorithm="gia c")
Output:
integrate(-60*(4*(x^2 + e^(2/x))*e^(2*x) + x*e^(e^(2/x)) - 8*e^(2/x))*e^(- 2*(e^(2*x) - 2)*e^(-e^(2/x)) - e^(2/x))/x^3, x)
Time = 1.69 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\frac {60\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{-{\mathrm {e}}^{2/x}}\,\left ({\mathrm {e}}^{2\,x}-2\right )}}{x} \] Input:
int(-(exp(-exp(-exp(2/x))*(2*exp(2*x) - 4))*exp(-exp(2/x))*(exp(2*x)*(240* exp(2/x) + 240*x^2) - 480*exp(2/x) + 60*x*exp(exp(2/x))))/x^3,x)
Output:
(60*exp(-2*exp(-exp(2/x))*(exp(2*x) - 2)))/x
\[ \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx=\int \frac {\left (-60 x \,{\mathrm e}^{{\mathrm e}^{\frac {2}{x}}}+\left (-240 \,{\mathrm e}^{\frac {2}{x}}-240 x^{2}\right ) {\mathrm e}^{2 x}+480 \,{\mathrm e}^{\frac {2}{x}}\right ) {\mathrm e}^{\frac {-2 \,{\mathrm e}^{2 x}+4}{{\mathrm e}^{{\mathrm e}^{\frac {2}{x}}}}}}{x^{3} {\mathrm e}^{{\mathrm e}^{\frac {2}{x}}}}d x \] Input:
int((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/x))*ex p((-2*exp(2*x)+4)/exp(exp(2/x)))/x^3/exp(exp(2/x)),x)
Output:
int((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/x))*ex p((-2*exp(2*x)+4)/exp(exp(2/x)))/x^3/exp(exp(2/x)),x)