Integrand size = 235, antiderivative size = 30 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{1-x-x^2+\frac {x}{\log \left (\left (e^x+e^{2 x}\right )^2+x\right )}} \]
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{1-x-x^2+\frac {x}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \]
Integrate[(E^((x + (1 - x - x^2)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])/L og[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])*(-x - 2*E^(2*x)*x - 6*E^(3*x)*x - 4 *E^(4*x)*x + (E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x] + (E^(3*x)*(-2 - 4*x) + E^(2*x)*(-1 - 2*x) + E^(4*x)*(-1 - 2 *x) - x - 2*x^2)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2))/((E^(2*x) + 2* E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4*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.
| \(\displaystyle \int \frac {\left (\left (-2 x^2-x+e^{3 x} (-4 x-2)+e^{2 x} (-2 x-1)+e^{4 x} (-2 x-1)\right ) \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x-x+\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )\right ) \exp \left (\frac {\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )+x}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right )}{\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right ) \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x \left (4 x+2 e^{2 x}+2 e^{3 x}-1\right ) \exp \left (\frac {\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )+x}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right )}{\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right ) \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}+\frac {\left (-4 x-2 x \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )-\log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )+\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )\right ) \exp \left (\frac {\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )+x}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right )}{\log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \int \frac {\exp \left (\frac {x+\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right ) x}{\log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}dx+2 \int \frac {\exp \left (-x^2+\frac {x}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}+x+1\right ) x}{\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right ) \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}dx+2 \int \frac {\exp \left (-x^2+\frac {x}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}+2 x+1\right ) x}{\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right ) \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}dx-\int \frac {\exp \left (\frac {x+\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right ) x}{\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right ) \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}dx+4 \int \frac {\exp \left (\frac {x+\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right ) x^2}{\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right ) \log ^2\left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}dx-\int \exp \left (\frac {x+\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right )dx-2 \int \exp \left (\frac {x+\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right ) xdx+\int \frac {\exp \left (\frac {x+\left (-x^2-x+1\right ) \log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}\right )}{\log \left (x+e^{2 x}+2 e^{3 x}+e^{4 x}\right )}dx\) |
Int[(E^((x + (1 - x - x^2)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x])/Log[E^( 2*x) + 2*E^(3*x) + E^(4*x) + x])*(-x - 2*E^(2*x)*x - 6*E^(3*x)*x - 4*E^(4* x)*x + (E^(2*x) + 2*E^(3*x) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4* x) + x] + (E^(3*x)*(-2 - 4*x) + E^(2*x)*(-1 - 2*x) + E^(4*x)*(-1 - 2*x) - x - 2*x^2)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2))/((E^(2*x) + 2*E^(3*x ) + E^(4*x) + x)*Log[E^(2*x) + 2*E^(3*x) + E^(4*x) + x]^2),x]
3.16.57.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(27)=54\).
Time = 84.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (-x^{2}-x +1\right ) \ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{x} {\mathrm e}^{2 x}+{\mathrm e}^{2 x}+x \right )+x}{\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{x} {\mathrm e}^{2 x}+{\mathrm e}^{2 x}+x \right )}}\) | \(60\) |
| risch | \({\mathrm e}^{-\frac {\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right ) x^{2}+\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right ) x -\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right )-x}{\ln \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+{\mathrm e}^{2 x}+x \right )}}\) | \(86\) |
int((((-1-2*x)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-1-2*x)*exp(x)^2-2*x^2 -x)*ln(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)^2+(exp(2*x)^2+2*exp(x)*exp (2*x)+exp(x)^2+x)*ln(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*x*exp(2*x) ^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*ln(exp(2*x)^2+2*exp (x)*exp(2*x)+exp(x)^2+x)+x)/ln(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x))/( exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/ln(exp(2*x)^2+2*exp(x)*exp(2*x)+e xp(x)^2+x)^2,x,method=_RETURNVERBOSE)
exp(((-x^2-x+1)*ln(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+x)/ln(exp(2*x) ^2+2*exp(x)*exp(2*x)+exp(x)^2+x))
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{\left (-\frac {{\left (x^{2} + x - 1\right )} \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right ) - x}{\log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right )}\right )} \]
integrate((((-1-2*x)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-1-2*x)*exp(x)^2 -2*x^2-x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)^2+(exp(2*x)^2+2*exp (x)*exp(2*x)+exp(x)^2+x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*x* exp(2*x)^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*log(exp(2*x )^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+x)/log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp( x)^2+x))/(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/log(exp(2*x)^2+2*exp(x) *exp(2*x)+exp(x)^2+x)^2,x, algorithm=\
e^(-((x^2 + x - 1)*log(x + e^(4*x) + 2*e^(3*x) + e^(2*x)) - x)/log(x + e^( 4*x) + 2*e^(3*x) + e^(2*x)))
Timed out. \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=\text {Timed out} \]
integrate((((-1-2*x)*exp(2*x)**2+(-4*x-2)*exp(x)*exp(2*x)+(-1-2*x)*exp(x)* *2-2*x**2-x)*ln(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x)**2+(exp(2*x)**2 +2*exp(x)*exp(2*x)+exp(x)**2+x)*ln(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2 +x)-4*x*exp(2*x)**2-6*x*exp(x)*exp(2*x)-2*x*exp(x)**2-x)*exp(((-x**2-x+1)* ln(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x)+x)/ln(exp(2*x)**2+2*exp(x)*e xp(2*x)+exp(x)**2+x))/(exp(2*x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x)/ln(exp(2 *x)**2+2*exp(x)*exp(2*x)+exp(x)**2+x)**2,x)
Exception generated. \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=\text {Exception raised: RuntimeError} \]
integrate((((-1-2*x)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-1-2*x)*exp(x)^2 -2*x^2-x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)^2+(exp(2*x)^2+2*exp (x)*exp(2*x)+exp(x)^2+x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*x* exp(2*x)^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*log(exp(2*x )^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+x)/log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp( x)^2+x))/(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/log(exp(2*x)^2+2*exp(x) *exp(2*x)+exp(x)^2+x)^2,x, algorithm=\
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.65 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.83 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx=e^{\left (-\frac {x^{2} \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right ) + x \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right ) - x - \log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right )}{\log \left (x + e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )}\right )}\right )} \]
integrate((((-1-2*x)*exp(2*x)^2+(-4*x-2)*exp(x)*exp(2*x)+(-1-2*x)*exp(x)^2 -2*x^2-x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)^2+(exp(2*x)^2+2*exp (x)*exp(2*x)+exp(x)^2+x)*log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)-4*x* exp(2*x)^2-6*x*exp(x)*exp(2*x)-2*x*exp(x)^2-x)*exp(((-x^2-x+1)*log(exp(2*x )^2+2*exp(x)*exp(2*x)+exp(x)^2+x)+x)/log(exp(2*x)^2+2*exp(x)*exp(2*x)+exp( x)^2+x))/(exp(2*x)^2+2*exp(x)*exp(2*x)+exp(x)^2+x)/log(exp(2*x)^2+2*exp(x) *exp(2*x)+exp(x)^2+x)^2,x, algorithm=\
e^(-(x^2*log(x + e^(4*x) + 2*e^(3*x) + e^(2*x)) + x*log(x + e^(4*x) + 2*e^ (3*x) + e^(2*x)) - x - log(x + e^(4*x) + 2*e^(3*x) + e^(2*x)))/log(x + e^( 4*x) + 2*e^(3*x) + e^(2*x)))
Time = 13.78 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {x+\left (1-x-x^2\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}{\log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )}} \left (-x-2 e^{2 x} x-6 e^{3 x} x-4 e^{4 x} x+\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log \left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )+\left (e^{3 x} (-2-4 x)+e^{2 x} (-1-2 x)+e^{4 x} (-1-2 x)-x-2 x^2\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )\right )}{\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right ) \log ^2\left (e^{2 x}+2 e^{3 x}+e^{4 x}+x\right )} \, dx={\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{\frac {x}{\ln \left (x+{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\right )}} \]
int(-(exp((x - log(x + exp(2*x) + 2*exp(3*x) + exp(4*x))*(x + x^2 - 1))/lo g(x + exp(2*x) + 2*exp(3*x) + exp(4*x)))*(x + 2*x*exp(2*x) + 6*x*exp(3*x) + 4*x*exp(4*x) + log(x + exp(2*x) + 2*exp(3*x) + exp(4*x))^2*(x + exp(2*x) *(2*x + 1) + exp(4*x)*(2*x + 1) + exp(3*x)*(4*x + 2) + 2*x^2) - log(x + ex p(2*x) + 2*exp(3*x) + exp(4*x))*(x + exp(2*x) + 2*exp(3*x) + exp(4*x))))/( log(x + exp(2*x) + 2*exp(3*x) + exp(4*x))^2*(x + exp(2*x) + 2*exp(3*x) + e xp(4*x))),x)