Integrand size = 89, antiderivative size = 28 \begin {dmath*} \int \frac {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx=3+\frac {4}{2+2 e^{-5+e^x-x} (4-x)}-x \end {dmath*}
Time = 2.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \begin {dmath*} \int \frac {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx=-\frac {2 e^{5+x}}{-e^{5+x}+e^{e^x} (-4+x)}-x \end {dmath*}
Integrate[(-1 + E^(-10 + 2*E^x - 2*x)*(-16 + 8*x - x^2) + E^(-5 + E^x - x) *(2 + E^x*(-8 + 2*x)))/(1 + E^(-5 + E^x - x)*(8 - 2*x) + E^(-10 + 2*E^x - 2*x)*(16 - 8*x + 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.
\(\displaystyle \int \frac {e^{-2 x+2 e^x-10} \left (-x^2+8 x-16\right )+e^{-x+e^x-5} \left (e^x (2 x-8)+2\right )-1}{e^{-2 x+2 e^x-10} \left (x^2-8 x+16\right )+e^{-x+e^x-5} (8-2 x)+1} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 x+10} \left (e^{-2 x+2 e^x-10} \left (-x^2+8 x-16\right )+e^{-x+e^x-5} \left (e^x (2 x-8)+2\right )-1\right )}{\left (-e^{e^x} x+4 e^{e^x}+e^{x+5}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 e^{2 x+5} \left (-e^{e^x} x^2+8 e^{e^x} x+e^5 x-16 e^{e^x}-5 e^5\right )}{(x-4) \left (-e^{e^x} x+4 e^{e^x}+e^{x+5}\right )^2}-\frac {2 e^{2 x-e^x+10} (x-5)}{(x-4)^2 \left (e^{e^x} x-4 e^{e^x}-e^{x+5}\right )}-\frac {2 e^{x-e^x+5} (x-5)}{(x-4)^2}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {e^{x-e^x+5}}{(x-4)^2}dx-2 \int \frac {e^{x-e^x+5}}{x-4}dx+10 \int \frac {e^{2 x+10}}{(x-4) \left (-e^{e^x} x+4 e^{e^x}+e^{x+5}\right )^2}dx-2 \int \frac {e^{2 x+10}}{\left (e^{e^x} x-4 e^{e^x}-e^{x+5}\right )^2}dx-8 \int \frac {e^{2 x+e^x+5}}{\left (e^{e^x} x-4 e^{e^x}-e^{x+5}\right )^2}dx-8 \int \frac {e^{2 x+10}}{(x-4) \left (e^{e^x} x-4 e^{e^x}-e^{x+5}\right )^2}dx+2 \int \frac {e^{2 x+e^x+5} x}{\left (e^{e^x} x-4 e^{e^x}-e^{x+5}\right )^2}dx+2 \int \frac {e^{2 x-e^x+10}}{(x-4)^2 \left (e^{e^x} x-4 e^{e^x}-e^{x+5}\right )}dx-2 \int \frac {e^{2 x-e^x+10}}{(x-4) \left (e^{e^x} x-4 e^{e^x}-e^{x+5}\right )}dx-x\) |
Int[(-1 + E^(-10 + 2*E^x - 2*x)*(-16 + 8*x - x^2) + E^(-5 + E^x - x)*(2 + E^x*(-8 + 2*x)))/(1 + E^(-5 + E^x - x)*(8 - 2*x) + E^(-10 + 2*E^x - 2*x)*( 16 - 8*x + x^2)),x]
3.6.50.3.1 Defintions of rubi rules used
Time = 0.97 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11
method | result | size |
risch | \(-x -\frac {2}{{\mathrm e}^{{\mathrm e}^{x}-5-x} x -4 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}-1}\) | \(31\) |
norman | \(\frac {x +16 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}-x^{2} {\mathrm e}^{{\mathrm e}^{x}-5-x}+2}{{\mathrm e}^{{\mathrm e}^{x}-5-x} x -4 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}-1}\) | \(52\) |
parallelrisch | \(-\frac {x^{2} {\mathrm e}^{{\mathrm e}^{x}-5-x}-2-x -16 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}}{{\mathrm e}^{{\mathrm e}^{x}-5-x} x -4 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}-1}\) | \(54\) |
int(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5-x)-1) /((x^2-8*x+16)*exp(exp(x)-5-x)^2+(-2*x+8)*exp(exp(x)-5-x)+1),x,method=_RET URNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \begin {dmath*} \int \frac {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx=-\frac {{\left (x^{2} - 4 \, x\right )} e^{\left (-x + e^{x} - 5\right )} - x + 2}{{\left (x - 4\right )} e^{\left (-x + e^{x} - 5\right )} - 1} \end {dmath*}
integrate(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5 -x)-1)/((x^2-8*x+16)*exp(exp(x)-5-x)^2+(-2*x+8)*exp(exp(x)-5-x)+1),x, algo rithm=\
Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \begin {dmath*} \int \frac {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx=- x - \frac {2}{\left (x - 4\right ) e^{- x + e^{x} - 5} - 1} \end {dmath*}
integrate(((-x**2+8*x-16)*exp(exp(x)-5-x)**2+((2*x-8)*exp(x)+2)*exp(exp(x) -5-x)-1)/((x**2-8*x+16)*exp(exp(x)-5-x)**2+(-2*x+8)*exp(exp(x)-5-x)+1),x)
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \begin {dmath*} \int \frac {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx=\frac {{\left (x e^{5} - 2 \, e^{5}\right )} e^{x} - {\left (x^{2} - 4 \, x\right )} e^{\left (e^{x}\right )}}{{\left (x - 4\right )} e^{\left (e^{x}\right )} - e^{\left (x + 5\right )}} \end {dmath*}
integrate(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5 -x)-1)/((x^2-8*x+16)*exp(exp(x)-5-x)^2+(-2*x+8)*exp(exp(x)-5-x)+1),x, algo rithm=\
Leaf count of result is larger than twice the leaf count of optimal. 1136 vs. \(2 (23) = 46\).
Time = 0.32 (sec) , antiderivative size = 1136, normalized size of antiderivative = 40.57 \begin {dmath*} \int \frac {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx=\text {Too large to display} \end {dmath*}
integrate(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5 -x)-1)/((x^2-8*x+16)*exp(exp(x)-5-x)^2+(-2*x+8)*exp(exp(x)-5-x)+1),x, algo rithm=\
-(x^6*e^(1/2*x + 4*e^x) - x^6*e^(-1/2*x + 4*e^x) - 4*x^5*e^(3/2*x + 3*e^x + 5) - 20*x^5*e^(1/2*x + 4*e^x) + 4*x^5*e^(1/2*x + 3*e^x + 5) + 21*x^5*e^( -1/2*x + 4*e^x) + 6*x^4*e^(5/2*x + 2*e^x + 10) + 66*x^4*e^(3/2*x + 3*e^x + 5) - 6*x^4*e^(3/2*x + 2*e^x + 10) + 160*x^4*e^(1/2*x + 4*e^x) - 70*x^4*e^ (1/2*x + 3*e^x + 5) - 176*x^4*e^(-1/2*x + 4*e^x) - 4*x^3*e^(7/2*x + e^x + 15) - 78*x^3*e^(5/2*x + 2*e^x + 10) + 4*x^3*e^(5/2*x + e^x + 15) - 416*x^3 *e^(3/2*x + 3*e^x + 5) + 84*x^3*e^(3/2*x + 2*e^x + 10) - 640*x^3*e^(1/2*x + 4*e^x) + 466*x^3*e^(1/2*x + 3*e^x + 5) + 736*x^3*e^(-1/2*x + 4*e^x) + x^ 2*e^(9/2*x + 20) + 38*x^2*e^(7/2*x + e^x + 15) - x^2*e^(7/2*x + 20) + 360* x^2*e^(5/2*x + 2*e^x + 10) - 42*x^2*e^(5/2*x + e^x + 15) + 1216*x^2*e^(3/2 *x + 3*e^x + 5) - 414*x^2*e^(3/2*x + 2*e^x + 10) + 1280*x^2*e^(1/2*x + 4*e ^x) - 1432*x^2*e^(1/2*x + 3*e^x + 5) - 1536*x^2*e^(-1/2*x + 4*e^x) - 6*x*e ^(9/2*x + 20) - 112*x*e^(7/2*x + e^x + 15) + 7*x*e^(7/2*x + 20) - 672*x*e^ (5/2*x + 2*e^x + 10) + 134*x*e^(5/2*x + e^x + 15) - 1536*x*e^(3/2*x + 3*e^ x + 5) + 816*x*e^(3/2*x + 2*e^x + 10) - 1024*x*e^(1/2*x + 4*e^x) + 1888*x* e^(1/2*x + 3*e^x + 5) + 1280*x*e^(-1/2*x + 4*e^x) + 8*e^(9/2*x + 20) + 96* e^(7/2*x + e^x + 15) - 10*e^(7/2*x + 20) + 384*e^(5/2*x + 2*e^x + 10) - 12 0*e^(5/2*x + e^x + 15) + 512*e^(3/2*x + 3*e^x + 5) - 480*e^(3/2*x + 2*e^x + 10) - 640*e^(1/2*x + 3*e^x + 5))/(x^5*e^(1/2*x + 4*e^x) - x^5*e^(-1/2*x + 4*e^x) - 4*x^4*e^(3/2*x + 3*e^x + 5) - 20*x^4*e^(1/2*x + 4*e^x) + 4*x...
Timed out. \begin {dmath*} \int \frac {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx=\int -\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^x-2\,x-10}\,\left (x^2-8\,x+16\right )-{\mathrm {e}}^{{\mathrm {e}}^x-x-5}\,\left ({\mathrm {e}}^x\,\left (2\,x-8\right )+2\right )+1}{{\mathrm {e}}^{2\,{\mathrm {e}}^x-2\,x-10}\,\left (x^2-8\,x+16\right )-{\mathrm {e}}^{{\mathrm {e}}^x-x-5}\,\left (2\,x-8\right )+1} \,d x \end {dmath*}
int(-(exp(2*exp(x) - 2*x - 10)*(x^2 - 8*x + 16) - exp(exp(x) - x - 5)*(exp (x)*(2*x - 8) + 2) + 1)/(exp(2*exp(x) - 2*x - 10)*(x^2 - 8*x + 16) - exp(e xp(x) - x - 5)*(2*x - 8) + 1),x)