Integrand size = 154, antiderivative size = 30 \begin {dmath*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\frac {1}{\frac {2}{-5+\frac {e^{e^{-x} x (-2+2 x)}}{x}}+x^2} \end {dmath*}
\begin {dmath*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx \end {dmath*}
Integrate[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2 *x^2)/E^x)*(-4*x + 12*x^2 - 4*x^3 + E^x*(-2 + 20*x^2)))/(E^(x + (2*(-2*x + 2*x^2))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4*x ^2 - 20*x^4 + 25*x^6)),x]
Integrate[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2 *x^2)/E^x)*(-4*x + 12*x^2 - 4*x^3 + E^x*(-2 + 20*x^2)))/(E^(x + (2*(-2*x + 2*x^2))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4*x ^2 - 20*x^4 + 25*x^6)), 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.
\(\displaystyle \int \frac {-50 e^x x^3-2 e^{2 e^{-x} \left (2 x^2-2 x\right )+x} x+e^{e^{-x} \left (2 x^2-2 x\right )} \left (-4 x^3+12 x^2+e^x \left (20 x^2-2\right )-4 x\right )}{e^{2 e^{-x} \left (2 x^2-2 x\right )+x} x^4+e^x \left (25 x^6-20 x^4+4 x^2\right )+e^{e^{-x} \left (2 x^2-2 x\right )+x} \left (4 x^3-10 x^5\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-e^{-x} \left (e^x-4\right ) x} \left (-50 e^x x^3-2 e^{2 e^{-x} \left (2 x^2-2 x\right )+x} x+e^{e^{-x} \left (2 x^2-2 x\right )} \left (-4 x^3+12 x^2+e^x \left (20 x^2-2\right )-4 x\right )\right )}{x^2 \left (-5 e^{2 e^{-x} x} x^2+e^{2 e^{-x} x^2} x+2 e^{2 e^{-x} x}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2}{x^3}+\frac {2 e^{-2 e^{-x} x-e^{-x} \left (e^x-4\right ) x} \left (2 x^3-6 x^2+2 x-3 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )}-\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (10 x^5-30 x^4+6 x^3+5 e^x x^2+12 x^2-4 x+2 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )^2}\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \left (-\frac {2}{x^3}+\frac {2 e^{-e^{-x} \left (e^x-2\right ) x} \left (2 x^3-6 x^2+2 x-3 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )}-\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (10 x^5-30 x^4+6 x^3+5 e^x x^2+12 x^2-4 x+2 e^x\right )}{x^3 \left (5 e^{2 e^{-x} x} x^2-e^{2 e^{-x} x^2} x-2 e^{2 e^{-x} x}\right )^2}\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2}{x^3}-\frac {2 e^{-x} \left (-2 x^3+6 x^2-2 x+3 e^x\right )}{x^3 \left (5 x^2-e^{2 e^{-x} (x-1) x} x-2\right )}+\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (-10 x^5+30 x^4-6 x^3-5 e^x x^2-12 x^2+4 x-2 e^x\right )}{x^3 \left (-5 e^{2 e^{-x} x} x^2+e^{2 e^{-x} x^2} x+2 e^{2 e^{-x} x}\right )^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (-\frac {2}{x^3}-\frac {2 e^{-x} \left (-2 x^3+6 x^2-2 x+3 e^x\right )}{x^3 \left (5 x^2-e^{2 e^{-x} (x-1) x} x-2\right )}+\frac {2 e^{-e^{-x} \left (e^x-4\right ) x} \left (-10 x^5+30 x^4-6 x^3-5 e^x x^2-12 x^2+4 x-2 e^x\right )}{x^3 \left (-5 e^{2 e^{-x} x} x^2+e^{2 e^{-x} x^2} x+2 e^{2 e^{-x} x}\right )^2}\right )dx\) |
Int[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2*x^2)/ E^x)*(-4*x + 12*x^2 - 4*x^3 + E^x*(-2 + 20*x^2)))/(E^(x + (2*(-2*x + 2*x^2 ))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4*x^2 - 2 0*x^4 + 25*x^6)),x]
3.12.12.3.1 Defintions of rubi rules used
Time = 0.61 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
method | result | size |
risch | \(\frac {1}{x^{2}}+\frac {2}{x^{2} \left (5 x^{2}-{\mathrm e}^{2 x \left (-1+x \right ) {\mathrm e}^{-x}} x -2\right )}\) | \(33\) |
parallelrisch | \(\frac {5 x -{\mathrm e}^{\left (2 x^{2}-2 x \right ) {\mathrm e}^{-x}}}{x \left (5 x^{2}-x \,{\mathrm e}^{\left (2 x^{2}-2 x \right ) {\mathrm e}^{-x}}-2\right )}\) | \(53\) |
int((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+12*x^2 -4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2-2*x)/e xp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x^4+4*x ^2)*exp(x)),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \begin {dmath*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=-\frac {5 \, x e^{x} - e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )}}{x^{2} e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{x}} \end {dmath*}
integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+ 12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2- 2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x ^4+4*x^2)*exp(x)),x, algorithm=\
-(5*x*e^x - e^((2*x^2 + x*e^x - 2*x)*e^(-x)))/(x^2*e^((2*x^2 + x*e^x - 2*x )*e^(-x)) - (5*x^3 - 2*x)*e^x)
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \begin {dmath*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=- \frac {2}{- 5 x^{4} + x^{3} e^{\left (2 x^{2} - 2 x\right ) e^{- x}} + 2 x^{2}} + \frac {1}{x^{2}} \end {dmath*}
integrate((-2*x*exp(x)*exp((2*x**2-2*x)/exp(x))**2+((20*x**2-2)*exp(x)-4*x **3+12*x**2-4*x)*exp((2*x**2-2*x)/exp(x))-50*exp(x)*x**3)/(x**4*exp(x)*exp ((2*x**2-2*x)/exp(x))**2+(-10*x**5+4*x**3)*exp(x)*exp((2*x**2-2*x)/exp(x)) +(25*x**6-20*x**4+4*x**2)*exp(x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \begin {dmath*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=-\frac {5 \, x e^{\left (2 \, x e^{\left (-x\right )}\right )} - e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )}}{x^{2} e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )}} \end {dmath*}
integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+ 12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2- 2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x ^4+4*x^2)*exp(x)),x, algorithm=\
-(5*x*e^(2*x*e^(-x)) - e^(2*x^2*e^(-x)))/(x^2*e^(2*x^2*e^(-x)) - (5*x^3 - 2*x)*e^(2*x*e^(-x)))
Leaf count of result is larger than twice the leaf count of optimal. 4292 vs. \(2 (27) = 54\).
Time = 0.45 (sec) , antiderivative size = 4292, normalized size of antiderivative = 143.07 \begin {dmath*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\text {Too large to display} \end {dmath*}
integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+ 12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2- 2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x ^4+4*x^2)*exp(x)),x, algorithm=\
(12500*x^15*e^(4*x*e^(-x) + 3/2*x) - 7500*x^14*e^(2*x^2*e^(-x) + 2*x*e^(-x ) + 3/2*x) - 75000*x^14*e^(4*x*e^(-x) + 3/2*x) + 1500*x^13*e^(4*x^2*e^(-x) + 3/2*x) + 45000*x^13*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 117500*x^13 *e^(4*x*e^(-x) + 3/2*x) - 100*x^12*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) - 9000*x^12*e^(4*x^2*e^(-x) + 3/2*x) - 72500*x^12*e^(2*x^2*e^(-x) + 2*x*e^( -x) + 3/2*x) + 12500*x^12*e^(4*x*e^(-x) + 5/2*x) + 45000*x^12*e^(4*x*e^(-x ) + 3/2*x) + 600*x^11*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) + 14900*x^11*e ^(4*x^2*e^(-x) + 3/2*x) - 7500*x^11*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 15000*x^11*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 37500*x^11*e^(4*x*e^( -x) + 5/2*x) - 195500*x^11*e^(4*x*e^(-x) + 3/2*x) - 1020*x^10*e^(6*x^2*e^( -x) - 2*x*e^(-x) + 3/2*x) + 1500*x^10*e^(4*x^2*e^(-x) + 5/2*x) + 600*x^10* e^(4*x^2*e^(-x) + 3/2*x) + 22500*x^10*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x ) + 97700*x^10*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 2500*x^10*e^(4*x*e^ (-x) + 5/2*x) + 48000*x^10*e^(4*x*e^(-x) + 3/2*x) - 100*x^9*e^(6*x^2*e^(-x ) - 2*x*e^(-x) + 5/2*x) + 120*x^9*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) - 4500*x^9*e^(4*x^2*e^(-x) + 5/2*x) - 15540*x^9*e^(4*x^2*e^(-x) + 3/2*x) - 3 500*x^9*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 31200*x^9*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 3125*x^9*e^(4*x*e^(-x) + 7/2*x) + 30000*x^9*e^(4*x *e^(-x) + 5/2*x) + 108800*x^9*e^(4*x*e^(-x) + 3/2*x) + 300*x^8*e^(6*x^2*e^ (-x) - 2*x*e^(-x) + 5/2*x) + 764*x^8*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2...
Time = 14.84 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \begin {dmath*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx=\frac {1}{x^2}-\frac {2}{2\,x^2-5\,x^4+x^3\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{-x}}} \end {dmath*}