Integrand size = 100, antiderivative size = 33 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=\log \left (\frac {e^x}{5 \left (5+e^x+\frac {2}{(3-x) x^2}-x\right ) x}\right ) \] Output:
ln(2/5*exp(x)/x/(2*exp(x)-2*x+4/(3-x)/x^2+10))
Time = 8.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=x+\log (3-x)+\log (x)-\log \left (2+15 x^2+3 e^x x^2-8 x^3-e^x x^3+x^4\right ) \] Input:
Integrate[(6 + 2*x - 47*x^2 + 93*x^3 - 56*x^4 + 13*x^5 - x^6 + E^x*(-9*x^2 + 6*x^3 - x^4))/(6*x - 2*x^2 + 45*x^3 - 39*x^4 + 11*x^5 - x^6 + E^x*(9*x^ 3 - 6*x^4 + x^5)),x]
Output:
x + Log[3 - x] + Log[x] - Log[2 + 15*x^2 + 3*E^x*x^2 - 8*x^3 - E^x*x^3 + x ^4]
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.
\(\displaystyle \int \frac {-x^6+13 x^5-56 x^4+93 x^3-47 x^2+e^x \left (-x^4+6 x^3-9 x^2\right )+2 x+6}{-x^6+11 x^5-39 x^4+45 x^3-2 x^2+e^x \left (x^5-6 x^4+9 x^3\right )+6 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-x^6+13 x^5-56 x^4+93 x^3-47 x^2+e^x \left (-x^4+6 x^3-9 x^2\right )+2 x+6}{(3-x) x \left (x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^6-12 x^5+45 x^4-54 x^3+2 x^2-12}{(x-3) x \left (x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2\right )}-\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {1}{x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2}dx+2 \int \frac {1}{(x-3) \left (x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2\right )}dx+4 \int \frac {1}{x \left (x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2\right )}dx+18 \int \frac {x^2}{x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2}dx-9 \int \frac {x^3}{x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2}dx+\int \frac {x^4}{x^4-e^x x^3-8 x^3+3 e^x x^2+15 x^2+2}dx-\log (x)\) |
Input:
Int[(6 + 2*x - 47*x^2 + 93*x^3 - 56*x^4 + 13*x^5 - x^6 + E^x*(-9*x^2 + 6*x ^3 - x^4))/(6*x - 2*x^2 + 45*x^3 - 39*x^4 + 11*x^5 - x^6 + E^x*(9*x^3 - 6* x^4 + x^5)),x]
Output:
$Aborted
Time = 0.50 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\ln \left (x \right )+x -\ln \left ({\mathrm e}^{x}-\frac {x^{4}-8 x^{3}+15 x^{2}+2}{x^{2} \left (-3+x \right )}\right )\) | \(38\) |
norman | \(x -\ln \left (x^{4}-{\mathrm e}^{x} x^{3}-8 x^{3}+3 \,{\mathrm e}^{x} x^{2}+15 x^{2}+2\right )+\ln \left (x \right )+\ln \left (-3+x \right )\) | \(41\) |
parallelrisch | \(x -\ln \left (x^{4}-{\mathrm e}^{x} x^{3}-8 x^{3}+3 \,{\mathrm e}^{x} x^{2}+15 x^{2}+2\right )+\ln \left (x \right )+\ln \left (-3+x \right )\) | \(41\) |
Input:
int(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6)/((x^ 5-6*x^4+9*x^3)*exp(x)-x^6+11*x^5-39*x^4+45*x^3-2*x^2+6*x),x,method=_RETURN VERBOSE)
Output:
-ln(x)+x-ln(exp(x)-(x^4-8*x^3+15*x^2+2)/x^2/(-3+x))
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=x - \log \left (x\right ) - \log \left (-\frac {x^{4} - 8 \, x^{3} + 15 \, x^{2} - {\left (x^{3} - 3 \, x^{2}\right )} e^{x} + 2}{x^{3} - 3 \, x^{2}}\right ) \] Input:
integrate(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6 )/((x^5-6*x^4+9*x^3)*exp(x)-x^6+11*x^5-39*x^4+45*x^3-2*x^2+6*x),x, algorit hm="fricas")
Output:
x - log(x) - log(-(x^4 - 8*x^3 + 15*x^2 - (x^3 - 3*x^2)*e^x + 2)/(x^3 - 3* x^2))
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=x - \log {\left (x \right )} - \log {\left (e^{x} + \frac {- x^{4} + 8 x^{3} - 15 x^{2} - 2}{x^{3} - 3 x^{2}} \right )} \] Input:
integrate(((-x**4+6*x**3-9*x**2)*exp(x)-x**6+13*x**5-56*x**4+93*x**3-47*x* *2+2*x+6)/((x**5-6*x**4+9*x**3)*exp(x)-x**6+11*x**5-39*x**4+45*x**3-2*x**2 +6*x),x)
Output:
x - log(x) - log(exp(x) + (-x**4 + 8*x**3 - 15*x**2 - 2)/(x**3 - 3*x**2))
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=x - \log \left (x\right ) - \log \left (-\frac {x^{4} - 8 \, x^{3} + 15 \, x^{2} - {\left (x^{3} - 3 \, x^{2}\right )} e^{x} + 2}{x^{3} - 3 \, x^{2}}\right ) \] Input:
integrate(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6 )/((x^5-6*x^4+9*x^3)*exp(x)-x^6+11*x^5-39*x^4+45*x^3-2*x^2+6*x),x, algorit hm="maxima")
Output:
x - log(x) - log(-(x^4 - 8*x^3 + 15*x^2 - (x^3 - 3*x^2)*e^x + 2)/(x^3 - 3* x^2))
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=x - \log \left (-x^{4} + x^{3} e^{x} + 8 \, x^{3} - 3 \, x^{2} e^{x} - 15 \, x^{2} - 2\right ) + \log \left (x - 3\right ) + \log \left (x\right ) \] Input:
integrate(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6 )/((x^5-6*x^4+9*x^3)*exp(x)-x^6+11*x^5-39*x^4+45*x^3-2*x^2+6*x),x, algorit hm="giac")
Output:
x - log(-x^4 + x^3*e^x + 8*x^3 - 3*x^2*e^x - 15*x^2 - 2) + log(x - 3) + lo g(x)
Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=x-\ln \left (3\,x^2\,{\mathrm {e}}^x-x^3\,{\mathrm {e}}^x+15\,x^2-8\,x^3+x^4+2\right )+\ln \left (x-3\right )+\ln \left (x\right ) \] Input:
int((2*x - 47*x^2 + 93*x^3 - 56*x^4 + 13*x^5 - x^6 - exp(x)*(9*x^2 - 6*x^3 + x^4) + 6)/(6*x - 2*x^2 + 45*x^3 - 39*x^4 + 11*x^5 - x^6 + exp(x)*(9*x^3 - 6*x^4 + x^5)),x)
Output:
x - log(3*x^2*exp(x) - x^3*exp(x) + 15*x^2 - 8*x^3 + x^4 + 2) + log(x - 3) + log(x)
Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {6+2 x-47 x^2+93 x^3-56 x^4+13 x^5-x^6+e^x \left (-9 x^2+6 x^3-x^4\right )}{6 x-2 x^2+45 x^3-39 x^4+11 x^5-x^6+e^x \left (9 x^3-6 x^4+x^5\right )} \, dx=-\mathrm {log}\left (e^{x} x^{3}-3 e^{x} x^{2}-x^{4}+8 x^{3}-15 x^{2}-2\right )+\mathrm {log}\left (x -3\right )+\mathrm {log}\left (x \right )+x \] Input:
int(((-x^4+6*x^3-9*x^2)*exp(x)-x^6+13*x^5-56*x^4+93*x^3-47*x^2+2*x+6)/((x^ 5-6*x^4+9*x^3)*exp(x)-x^6+11*x^5-39*x^4+45*x^3-2*x^2+6*x),x)
Output:
- log(e**x*x**3 - 3*e**x*x**2 - x**4 + 8*x**3 - 15*x**2 - 2) + log(x - 3) + log(x) + x