Integrand size = 49, antiderivative size = 20 \[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx=e^{e^{-\frac {2 x}{1+e^8}} (12-x)} \] Output:
exp((12-x)/exp(1/2*x/(1/4*exp(2)^4+1/4)))
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx=-\frac {e^{-e^{-\frac {2 x}{1+e^8}} (-12+x)} \left (-1-e^8\right )}{1+e^8} \] Input:
Integrate[(E^((12 - x)/E^((2*x)/(1 + E^8)) - (2*x)/(1 + E^8))*(-25 - E^8 + 2*x))/(1 + E^8),x]
Output:
-((-1 - E^8)/(E^((-12 + x)/E^((2*x)/(1 + E^8)))*(1 + E^8)))
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.
\(\displaystyle \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (2 x-e^8-25\right )}{1+e^8} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-2 x+e^8+25\right )dx}{1+e^8}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-2 x+e^8+25\right )dx}{1+e^8}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (25 e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (1+\frac {e^8}{25}\right )-2 e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} x\right )dx}{1+e^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (25+e^8\right ) \int e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}}dx-2 \int e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} xdx}{1+e^8}\) |
Input:
Int[(E^((12 - x)/E^((2*x)/(1 + E^8)) - (2*x)/(1 + E^8))*(-25 - E^8 + 2*x)) /(1 + E^8),x]
Output:
$Aborted
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
risch | \({\mathrm e}^{-\left (x -12\right ) {\mathrm e}^{-\frac {2 x}{{\mathrm e}^{8}+1}}}\) | \(17\) |
norman | \({\mathrm e}^{\left (12-x \right ) {\mathrm e}^{-\frac {2 x}{{\mathrm e}^{8}+1}}}\) | \(22\) |
parallelrisch | \(\frac {{\mathrm e}^{8} {\mathrm e}^{-\left (x -12\right ) {\mathrm e}^{-\frac {2 x}{{\mathrm e}^{8}+1}}}+{\mathrm e}^{-\left (x -12\right ) {\mathrm e}^{-\frac {2 x}{{\mathrm e}^{8}+1}}}}{{\mathrm e}^{8}+1}\) | \(56\) |
Input:
int((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1)/exp( 2*x/(exp(2)^4+1)),x,method=_RETURNVERBOSE)
Output:
exp(-(x-12)*exp(-2*x/(exp(8)+1)))
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (16) = 32\).
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.15 \[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx=e^{\left (-\frac {{\left ({\left (x - 12\right )} e^{8} + x - 12\right )} e^{\left (-\frac {2 \, x}{e^{8} + 1}\right )} + 2 \, x}{e^{8} + 1} + \frac {2 \, x}{e^{8} + 1}\right )} \] Input:
integrate((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1 )/exp(2*x/(exp(2)^4+1)),x, algorithm="fricas")
Output:
e^(-(((x - 12)*e^8 + x - 12)*e^(-2*x/(e^8 + 1)) + 2*x)/(e^8 + 1) + 2*x/(e^ 8 + 1))
Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx=e^{\left (12 - x\right ) e^{- \frac {2 x}{1 + e^{8}}}} \] Input:
integrate((-exp(2)**4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)**4+1)))/(exp(2)** 4+1)/exp(2*x/(exp(2)**4+1)),x)
Output:
exp((12 - x)*exp(-2*x/(1 + exp(8))))
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx=e^{\left (-x e^{\left (-\frac {2 \, x}{e^{8} + 1}\right )} + 12 \, e^{\left (-\frac {2 \, x}{e^{8} + 1}\right )}\right )} \] Input:
integrate((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1 )/exp(2*x/(exp(2)^4+1)),x, algorithm="maxima")
Output:
e^(-x*e^(-2*x/(e^8 + 1)) + 12*e^(-2*x/(e^8 + 1)))
\[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx=\int { \frac {{\left (2 \, x - e^{8} - 25\right )} e^{\left (-{\left (x - 12\right )} e^{\left (-\frac {2 \, x}{e^{8} + 1}\right )} - \frac {2 \, x}{e^{8} + 1}\right )}}{e^{8} + 1} \,d x } \] Input:
integrate((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1 )/exp(2*x/(exp(2)^4+1)),x, algorithm="giac")
Output:
integrate((2*x - e^8 - 25)*e^(-(x - 12)*e^(-2*x/(e^8 + 1)) - 2*x/(e^8 + 1) )/(e^8 + 1), x)
Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx={\mathrm {e}}^{12\,{\mathrm {e}}^{-\frac {2\,x}{{\mathrm {e}}^8+1}}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-\frac {2\,x}{{\mathrm {e}}^8+1}}} \] Input:
int(-(exp(-(2*x)/(exp(8) + 1))*exp(-exp(-(2*x)/(exp(8) + 1))*(x - 12))*(ex p(8) - 2*x + 25))/(exp(8) + 1),x)
Output:
exp(12*exp(-(2*x)/(exp(8) + 1)))*exp(-x*exp(-(2*x)/(exp(8) + 1)))
\[ \int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx=\frac {-\left (\int \frac {e^{\frac {12}{e^{\frac {2 x}{e^{8}+1}}}}}{e^{\frac {2 e^{\frac {2 x}{e^{8}+1}} x +e^{8} x +x}{e^{\frac {2 x}{e^{8}+1}} e^{8}+e^{\frac {2 x}{e^{8}+1}}}}}d x \right ) e^{8}-25 \left (\int \frac {e^{\frac {12}{e^{\frac {2 x}{e^{8}+1}}}}}{e^{\frac {2 e^{\frac {2 x}{e^{8}+1}} x +e^{8} x +x}{e^{\frac {2 x}{e^{8}+1}} e^{8}+e^{\frac {2 x}{e^{8}+1}}}}}d x \right )+2 \left (\int \frac {e^{\frac {12}{e^{\frac {2 x}{e^{8}+1}}}} x}{e^{\frac {2 e^{\frac {2 x}{e^{8}+1}} x +e^{8} x +x}{e^{\frac {2 x}{e^{8}+1}} e^{8}+e^{\frac {2 x}{e^{8}+1}}}}}d x \right )}{e^{8}+1} \] Input:
int((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1)/exp( 2*x/(exp(2)^4+1)),x)
Output:
( - int(e**(12/e**((2*x)/(e**8 + 1)))/e**((2*e**((2*x)/(e**8 + 1))*x + e** 8*x + x)/(e**((2*x)/(e**8 + 1))*e**8 + e**((2*x)/(e**8 + 1)))),x)*e**8 - 2 5*int(e**(12/e**((2*x)/(e**8 + 1)))/e**((2*e**((2*x)/(e**8 + 1))*x + e**8* x + x)/(e**((2*x)/(e**8 + 1))*e**8 + e**((2*x)/(e**8 + 1)))),x) + 2*int((e **(12/e**((2*x)/(e**8 + 1)))*x)/e**((2*e**((2*x)/(e**8 + 1))*x + e**8*x + x)/(e**((2*x)/(e**8 + 1))*e**8 + e**((2*x)/(e**8 + 1)))),x))/(e**8 + 1)