Integrand size = 45, antiderivative size = 13 \[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx=\left (-3+e^{-1+x} x\right )^{e^3} \] Output:
exp(exp(3)*ln(x/exp(1-x)-3))
Time = 0.12 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx=\left (-3+e^{-1+x} x\right )^{e^3} \] Input:
Integrate[(E^3*(-1 - x)*(E^(-1 + x)*(-3*E^(1 - x) + x))^E^3)/(3*E^(1 - x) - x),x]
Output:
(-3 + E^(-1 + x)*x)^E^3
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.
\(\displaystyle \int \frac {e^3 (-x-1) \left (e^{x-1} \left (x-3 e^{1-x}\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^3 \int -\frac {\left (-e^{x-1} \left (3 e^{1-x}-x\right )\right )^{e^3} (x+1)}{3 e^{1-x}-x}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -e^3 \int \frac {\left (-e^{x-1} \left (3 e^{1-x}-x\right )\right )^{e^3} (x+1)}{3 e^{1-x}-x}dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle -e^3 \left (e^{x-1}\right )^{-e^3} \left (3 e^{1-x}-x\right )^{-e^3} \left (-e^{x-1} \left (3 e^{1-x}-x\right )\right )^{e^3} \int \left (e^{x-1}\right )^{e^3} \left (3 e^{1-x}-x\right )^{-1+e^3} (x+1)dx\) |
\(\Big \downarrow \) 2717 |
\(\displaystyle -e^{e^3 (1-x)+3} \left (3 e^{1-x}-x\right )^{-e^3} \left (-e^{x-1} \left (3 e^{1-x}-x\right )\right )^{e^3} \int e^{-e^3 (1-x)} \left (3 e^{1-x}-x\right )^{-1+e^3} (x+1)dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -e^{e^3 (1-x)+3} \left (3 e^{1-x}-x\right )^{-e^3} \left (-e^{x-1} \left (3 e^{1-x}-x\right )\right )^{e^3} \int e^{e^3 x-e^3} \left (3 e^{1-x}-x\right )^{-1+e^3} (x+1)dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -e^{e^3 (1-x)+3} \left (3 e^{1-x}-x\right )^{-e^3} \left (-e^{x-1} \left (3 e^{1-x}-x\right )\right )^{e^3} \int \left (e^{e^3 x-e^3} \left (3 e^{1-x}-x\right )^{-1+e^3}+e^{e^3 x-e^3} x \left (3 e^{1-x}-x\right )^{-1+e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -e^{e^3 (1-x)+3} \left (3 e^{1-x}-x\right )^{-e^3} \left (-e^{x-1} \left (3 e^{1-x}-x\right )\right )^{e^3} \left (\int e^{e^3 x-e^3} \left (3 e^{1-x}-x\right )^{-1+e^3}dx+\int e^{e^3 x-e^3} \left (3 e^{1-x}-x\right )^{-1+e^3} xdx\right )\) |
Input:
Int[(E^3*(-1 - x)*(E^(-1 + x)*(-3*E^(1 - x) + x))^E^3)/(3*E^(1 - x) - x),x ]
Output:
$Aborted
Time = 0.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.92
method | result | size |
norman | \({\mathrm e}^{{\mathrm e}^{3} \ln \left (\left (-3 \,{\mathrm e}^{1-x}+x \right ) {\mathrm e}^{-1+x}\right )}\) | \(25\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{3} \ln \left (-\left (3 \,{\mathrm e}^{1-x}-x \right ) {\mathrm e}^{-1+x}\right )}\) | \(28\) |
risch | \(\left ({\mathrm e}^{1-x}\right )^{-{\mathrm e}^{3}} \left (-3 \,{\mathrm e}^{1-x}+x \right )^{{\mathrm e}^{3}} {\mathrm e}^{-\frac {i {\mathrm e}^{3} \operatorname {csgn}\left (i \left (-3 \,{\mathrm e}^{1-x}+x \right ) {\mathrm e}^{-1+x}\right ) \pi \left (-\operatorname {csgn}\left (i \left (-3 \,{\mathrm e}^{1-x}+x \right ) {\mathrm e}^{-1+x}\right )+\operatorname {csgn}\left (i \left (-3 \,{\mathrm e}^{1-x}+x \right )\right )\right ) \left (-\operatorname {csgn}\left (i \left (-3 \,{\mathrm e}^{1-x}+x \right ) {\mathrm e}^{-1+x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{-1+x}\right )\right )}{2}}\) | \(115\) |
Input:
int((-1-x)*exp(3)*exp(exp(3)*ln((-3*exp(1-x)+x)/exp(1-x)))/(3*exp(1-x)-x), x,method=_RETURNVERBOSE)
Output:
exp(exp(3)*ln((-3*exp(1-x)+x)/exp(1-x)))
Time = 0.11 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx={\left (x e^{\left (x - 1\right )} - 3\right )}^{e^{3}} \] Input:
integrate((-1-x)*exp(3)*exp(exp(3)*log((-3*exp(1-x)+x)/exp(1-x)))/(3*exp(1 -x)-x),x, algorithm="fricas")
Output:
(x*e^(x - 1) - 3)^e^3
Time = 7.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx=\left (\left (x - 3 e^{1 - x}\right ) e^{x - 1}\right )^{e^{3}} \] Input:
integrate((-1-x)*exp(3)*exp(exp(3)*ln((-3*exp(1-x)+x)/exp(1-x)))/(3*exp(1- x)-x),x)
Output:
((x - 3*exp(1 - x))*exp(x - 1))**exp(3)
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx=e^{\left (e^{3} \log \left (x e^{x} - 3 \, e\right ) - e^{3}\right )} \] Input:
integrate((-1-x)*exp(3)*exp(exp(3)*log((-3*exp(1-x)+x)/exp(1-x)))/(3*exp(1 -x)-x),x, algorithm="maxima")
Output:
e^(e^3*log(x*e^x - 3*e) - e^3)
\[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx=\int { \frac {\left ({\left (x - 3 \, e^{\left (-x + 1\right )}\right )} e^{\left (x - 1\right )}\right )^{e^{3}} {\left (x + 1\right )} e^{3}}{x - 3 \, e^{\left (-x + 1\right )}} \,d x } \] Input:
integrate((-1-x)*exp(3)*exp(exp(3)*log((-3*exp(1-x)+x)/exp(1-x)))/(3*exp(1 -x)-x),x, algorithm="giac")
Output:
integrate(((x - 3*e^(-x + 1))*e^(x - 1))^e^3*(x + 1)*e^3/(x - 3*e^(-x + 1) ), x)
Time = 8.72 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx={\left (x\,{\mathrm {e}}^{x-1}-3\right )}^{{\mathrm {e}}^3} \] Input:
int((exp(3)*(exp(x - 1)*(x - 3*exp(1 - x)))^exp(3)*(x + 1))/(x - 3*exp(1 - x)),x)
Output:
(x*exp(x - 1) - 3)^exp(3)
Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int \frac {e^3 (-1-x) \left (e^{-1+x} \left (-3 e^{1-x}+x\right )\right )^{e^3}}{3 e^{1-x}-x} \, dx=\frac {\left (e^{x} x -3 e \right )^{e^{3}}}{e^{e^{3}}} \] Input:
int((-1-x)*exp(3)*exp(exp(3)*log((-3*exp(1-x)+x)/exp(1-x)))/(3*exp(1-x)-x) ,x)
Output:
(e**x*x - 3*e)**(e**3)/e**(e**3)