Integrand size = 114, antiderivative size = 23 \[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=\left (x-e^{-5+\frac {e^x}{4}-\frac {15}{x^4}} x\right )^2 \] Output:
(x-x/exp(15/x^4-1/4*exp(x)+5))^2
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=e^{-10-\frac {30}{x^4}} \left (e^{\frac {e^x}{4}}-e^{5+\frac {15}{x^4}}\right )^2 x^2 \] Input:
Integrate[(240 + 4*x^4 + 4*E^((60 + 20*x^4 - E^x*x^4)/(2*x^4))*x^4 + E^x*x ^5 + E^((60 + 20*x^4 - E^x*x^4)/(4*x^4))*(-240 - 8*x^4 - E^x*x^5))/(2*E^(( 60 + 20*x^4 - E^x*x^4)/(2*x^4))*x^3),x]
Output:
E^(-10 - 30/x^4)*(E^(E^x/4) - E^(5 + 15/x^4))^2*x^2
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.
\(\displaystyle \int \frac {e^{-\frac {-e^x x^4+20 x^4+60}{2 x^4}} \left (e^x x^5+4 e^{\frac {-e^x x^4+20 x^4+60}{2 x^4}} x^4+4 x^4+e^{\frac {-e^x x^4+20 x^4+60}{4 x^4}} \left (-e^x x^5-8 x^4-240\right )+240\right )}{2 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {e^{-\frac {-e^x x^4+20 x^4+60}{2 x^4}} \left (e^x x^5+4 e^{\frac {-e^x x^4+20 x^4+60}{2 x^4}} x^4+4 x^4-e^{\frac {-e^x x^4+20 x^4+60}{4 x^4}} \left (e^x x^5+8 x^4+240\right )+240\right )}{x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (e^{x-\frac {e^x}{4}-\frac {-e^x x^4+20 x^4+60}{2 x^4}} \left (e^{\frac {e^x}{4}}-e^{5+\frac {15}{x^4}}\right ) x^2+\frac {4 e^{-\frac {-e^x x^4+20 x^4+60}{2 x^4}-\frac {e^x}{2}} \left (e^{\frac {e^x}{4}}-e^{5+\frac {15}{x^4}}\right ) \left (e^{\frac {e^x}{4}} x^4-e^{5+\frac {15}{x^4}} x^4+60 e^{\frac {e^x}{4}}\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (4 \int e^{\frac {e^x}{2}-10-\frac {30}{x^4}} xdx-8 \int e^{\frac {e^x}{4}-5-\frac {15}{x^4}} xdx+240 \int \frac {e^{\frac {e^x}{2}-10-\frac {30}{x^4}}}{x^3}dx-240 \int \frac {e^{\frac {e^x}{4}-5-\frac {15}{x^4}}}{x^3}dx+\int e^{x+\frac {e^x}{2}-10-\frac {30}{x^4}} x^2dx-\int e^{x+\frac {e^x}{4}-5-\frac {15}{x^4}} x^2dx+2 x^2\right )\) |
Input:
Int[(240 + 4*x^4 + 4*E^((60 + 20*x^4 - E^x*x^4)/(2*x^4))*x^4 + E^x*x^5 + E ^((60 + 20*x^4 - E^x*x^4)/(4*x^4))*(-240 - 8*x^4 - E^x*x^5))/(2*E^((60 + 2 0*x^4 - E^x*x^4)/(2*x^4))*x^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(21)=42\).
Time = 0.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.26
method | result | size |
risch | \(x^{2}-2 x^{2} {\mathrm e}^{\frac {{\mathrm e}^{x} x^{4}-20 x^{4}-60}{4 x^{4}}}+x^{2} {\mathrm e}^{\frac {{\mathrm e}^{x} x^{4}-20 x^{4}-60}{2 x^{4}}}\) | \(52\) |
parallelrisch | \(\frac {\left (8 \,{\mathrm e}^{-\frac {{\mathrm e}^{x} x^{4}-20 x^{4}-60}{2 x^{4}}} x^{6}-16 \,{\mathrm e}^{-\frac {{\mathrm e}^{x} x^{4}-20 x^{4}-60}{4 x^{4}}} x^{6}+8 x^{6}\right ) {\mathrm e}^{\frac {{\mathrm e}^{x} x^{4}-20 x^{4}-60}{2 x^{4}}}}{8 x^{4}}\) | \(83\) |
Input:
int(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8*x^4-2 40)*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/exp(1/4 *(-exp(x)*x^4+20*x^4+60)/x^4)^2,x,method=_RETURNVERBOSE)
Output:
x^2-2*x^2*exp(1/4*(exp(x)*x^4-20*x^4-60)/x^4)+x^2*exp(1/2*(exp(x)*x^4-20*x ^4-60)/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26 \[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=-{\left (2 \, x^{2} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{4 \, x^{4}}\right )} - x^{2} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )} - x^{2}\right )} e^{\left (\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )} \] Input:
integrate(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8 *x^4-240)*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/e xp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2,x, algorithm="fricas")
Output:
-(2*x^2*e^(-1/4*(x^4*e^x - 20*x^4 - 60)/x^4) - x^2*e^(-1/2*(x^4*e^x - 20*x ^4 - 60)/x^4) - x^2)*e^(1/2*(x^4*e^x - 20*x^4 - 60)/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (17) = 34\).
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=x^{2} - 2 x^{2} e^{- \frac {- \frac {x^{4} e^{x}}{4} + 5 x^{4} + 15}{x^{4}}} + x^{2} e^{- \frac {2 \left (- \frac {x^{4} e^{x}}{4} + 5 x^{4} + 15\right )}{x^{4}}} \] Input:
integrate(1/2*(4*x**4*exp(1/4*(-exp(x)*x**4+20*x**4+60)/x**4)**2+(-x**5*ex p(x)-8*x**4-240)*exp(1/4*(-exp(x)*x**4+20*x**4+60)/x**4)+x**5*exp(x)+4*x** 4+240)/x**3/exp(1/4*(-exp(x)*x**4+20*x**4+60)/x**4)**2,x)
Output:
x**2 - 2*x**2*exp(-(-x**4*exp(x)/4 + 5*x**4 + 15)/x**4) + x**2*exp(-2*(-x* *4*exp(x)/4 + 5*x**4 + 15)/x**4)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=x^{2} + {\left (x^{2} e^{\left (\frac {1}{2} \, e^{x}\right )} - 2 \, x^{2} e^{\left (\frac {15}{x^{4}} + \frac {1}{4} \, e^{x} + 5\right )}\right )} e^{\left (-\frac {30}{x^{4}} - 10\right )} \] Input:
integrate(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8 *x^4-240)*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/e xp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2,x, algorithm="maxima")
Output:
x^2 + (x^2*e^(1/2*e^x) - 2*x^2*e^(15/x^4 + 1/4*e^x + 5))*e^(-30/x^4 - 10)
\[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=\int { \frac {{\left (x^{5} e^{x} + 4 \, x^{4} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )} + 4 \, x^{4} - {\left (x^{5} e^{x} + 8 \, x^{4} + 240\right )} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{4 \, x^{4}}\right )} + 240\right )} e^{\left (\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )}}{2 \, x^{3}} \,d x } \] Input:
integrate(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8 *x^4-240)*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/e xp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2,x, algorithm="giac")
Output:
integrate(1/2*(x^5*e^x + 4*x^4*e^(-1/2*(x^4*e^x - 20*x^4 - 60)/x^4) + 4*x^ 4 - (x^5*e^x + 8*x^4 + 240)*e^(-1/4*(x^4*e^x - 20*x^4 - 60)/x^4) + 240)*e^ (1/2*(x^4*e^x - 20*x^4 - 60)/x^4)/x^3, x)
Time = 2.71 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}-\frac {30}{x^4}-10}\,{\left ({\mathrm {e}}^{\frac {15}{x^4}-\frac {{\mathrm {e}}^x}{4}+5}-1\right )}^2 \] Input:
int((exp(-(2*(5*x^4 - (x^4*exp(x))/4 + 15))/x^4)*((x^5*exp(x))/2 - (exp((5 *x^4 - (x^4*exp(x))/4 + 15)/x^4)*(x^5*exp(x) + 8*x^4 + 240))/2 + 2*x^4*exp ((2*(5*x^4 - (x^4*exp(x))/4 + 15))/x^4) + 2*x^4 + 120))/x^3,x)
Output:
x^2*exp(exp(x)/2 - 30/x^4 - 10)*(exp(15/x^4 - exp(x)/4 + 5) - 1)^2
Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx=\frac {x^{2} \left (-2 e^{\frac {e^{x} x^{4}+60}{4 x^{4}}} e^{5}+e^{\frac {e^{x}}{2}}+e^{\frac {30}{x^{4}}} e^{10}\right )}{e^{\frac {30}{x^{4}}} e^{10}} \] Input:
int(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8*x^4-2 40)*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/exp(1/4 *(-exp(x)*x^4+20*x^4+60)/x^4)^2,x)
Output:
(x**2*( - 2*e**((e**x*x**4 + 60)/(4*x**4))*e**5 + e**(e**x/2) + e**(30/x** 4)*e**10))/(e**(30/x**4)*e**10)