Integrand size = 125, antiderivative size = 34 \[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx=-\frac {e^{\left (-e^4+x\right ) \left (-2+\log ^4(x)\right )}}{5 (2-x)}+\frac {e}{x}+x \] Output:
exp(1)/x+x-exp((x-exp(4))*(ln(x)^4-2))/(-5*x+10)
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx=\frac {1}{5} \left (\frac {e^{-\left (\left (e^4-x\right ) \left (-2+\log ^4(x)\right )\right )}}{-2+x}+\frac {5 e}{x}+5 x\right ) \] Input:
Integrate[(20*x^2 - 20*x^3 + 5*x^4 + E*(-20 + 20*x - 5*x^2) + E^(2*E^4 - 2 *x + (-E^4 + x)*Log[x]^4)*(3*x^2 - 2*x^3 + (-8*x^2 + 4*x^3 + E^4*(8*x - 4* x^2))*Log[x]^3 + (-2*x^2 + x^3)*Log[x]^4))/(20*x^2 - 20*x^3 + 5*x^4),x]
Output:
(1/(E^((E^4 - x)*(-2 + Log[x]^4))*(-2 + x)) + (5*E)/x + 5*x)/5
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.
| \(\displaystyle \int \frac {5 x^4-20 x^3+20 x^2+e \left (-5 x^2+20 x-20\right )+e^{-2 x+\left (x-e^4\right ) \log ^4(x)+2 e^4} \left (-2 x^3+3 x^2+\left (x^3-2 x^2\right ) \log ^4(x)+\left (4 x^3-8 x^2+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)\right )}{5 x^4-20 x^3+20 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {5 x^4-20 x^3+20 x^2+e \left (-5 x^2+20 x-20\right )+e^{-2 x+\left (x-e^4\right ) \log ^4(x)+2 e^4} \left (-2 x^3+3 x^2+\left (x^3-2 x^2\right ) \log ^4(x)+\left (4 x^3-8 x^2+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)\right )}{x^2 \left (5 x^2-20 x+20\right )}dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 20 \int \frac {5 x^4-20 x^3+20 x^2-5 e \left (x^2-4 x+4\right )+e^{-\left (\left (e^4-x\right ) \log ^4(x)\right )-2 x+2 e^4} \left (-\left (\left (2 x^2-x^3\right ) \log ^4(x)\right )-4 \left (-x^3+2 x^2-e^4 \left (2 x-x^2\right )\right ) \log ^3(x)-2 x^3+3 x^2\right )}{100 (2-x)^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {5 x^4-20 x^3+20 x^2-5 e \left (x^2-4 x+4\right )+e^{-\left (\left (e^4-x\right ) \log ^4(x)\right )-2 x+2 e^4} \left (-\left (\left (2 x^2-x^3\right ) \log ^4(x)\right )-4 \left (-x^3+2 x^2-e^4 \left (2 x-x^2\right )\right ) \log ^3(x)-2 x^3+3 x^2\right )}{(2-x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^{-\left (\left (e^4-x\right ) \left (\log ^4(x)-2\right )\right )} \left (x^2 \log ^4(x)-2 x \log ^4(x)+4 x^2 \log ^3(x)-8 \left (1+\frac {e^4}{2}\right ) x \log ^3(x)+8 e^4 \log ^3(x)-2 x^2+3 x\right )}{(2-x)^2 x}-\frac {5 \left (e-x^2\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (-\int \frac {e^{-\left (\left (e^4-x\right ) \left (\log ^4(x)-2\right )\right )}}{(x-2)^2}dx-2 \int \frac {e^{-\left (\left (e^4-x\right ) \left (\log ^4(x)-2\right )\right )}}{x-2}dx+\int \frac {e^{-\left (\left (e^4-x\right ) \left (\log ^4(x)-2\right )\right )} \log ^4(x)}{x-2}dx+2 \left (2-e^4\right ) \int \frac {e^{-\left (\left (e^4-x\right ) \left (\log ^4(x)-2\right )\right )} \log ^3(x)}{x-2}dx+2 \int \frac {e^{4-\left (e^4-x\right ) \left (\log ^4(x)-2\right )} \log ^3(x)}{x}dx+5 x+\frac {5 e}{x}\right )\) |
Input:
Int[(20*x^2 - 20*x^3 + 5*x^4 + E*(-20 + 20*x - 5*x^2) + E^(2*E^4 - 2*x + ( -E^4 + x)*Log[x]^4)*(3*x^2 - 2*x^3 + (-8*x^2 + 4*x^3 + E^4*(8*x - 4*x^2))* Log[x]^3 + (-2*x^2 + x^3)*Log[x]^4))/(20*x^2 - 20*x^3 + 5*x^4),x]
Output:
$Aborted
Time = 14.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(x +\frac {{\mathrm e}}{x}+\frac {{\mathrm e}^{-\left (\ln \left (x \right )^{4}-2\right ) \left ({\mathrm e}^{4}-x \right )}}{5 x -10}\) | \(31\) |
| parallelrisch | \(\frac {5 x^{3}+5 x \,{\mathrm e}+{\mathrm e}^{\left (x -{\mathrm e}^{4}\right ) \ln \left (x \right )^{4}+2 \,{\mathrm e}^{4}-2 x} x -10 \,{\mathrm e}-20 x}{5 x \left (-2+x \right )}\) | \(51\) |
Input:
int((((x^3-2*x^2)*ln(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*ln(x)^3-2*x^3+ 3*x^2)*exp((x-exp(4))*ln(x)^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp(1)+5*x^4- 20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x,method=_RETURNVERBOSE)
Output:
x+exp(1)/x+1/5/(-2+x)*exp(-(ln(x)^4-2)*(exp(4)-x))
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx=\frac {5 \, x^{3} - 10 \, x^{2} + 5 \, {\left (x - 2\right )} e + x e^{\left ({\left (x - e^{4}\right )} \log \left (x\right )^{4} - 2 \, x + 2 \, e^{4}\right )}}{5 \, {\left (x^{2} - 2 \, x\right )}} \] Input:
integrate((((x^3-2*x^2)*log(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*log(x)^ 3-2*x^3+3*x^2)*exp((x-exp(4))*log(x)^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp( 1)+5*x^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x, algorithm="fricas")
Output:
1/5*(5*x^3 - 10*x^2 + 5*(x - 2)*e + x*e^((x - e^4)*log(x)^4 - 2*x + 2*e^4) )/(x^2 - 2*x)
Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx=x + \frac {e^{- 2 x + \left (x - e^{4}\right ) \log {\left (x \right )}^{4} + 2 e^{4}}}{5 x - 10} + \frac {e}{x} \] Input:
integrate((((x**3-2*x**2)*ln(x)**4+((-4*x**2+8*x)*exp(4)+4*x**3-8*x**2)*ln (x)**3-2*x**3+3*x**2)*exp((x-exp(4))*ln(x)**4+2*exp(4)-2*x)+(-5*x**2+20*x- 20)*exp(1)+5*x**4-20*x**3+20*x**2)/(5*x**4-20*x**3+20*x**2),x)
Output:
x + exp(-2*x + (x - exp(4))*log(x)**4 + 2*exp(4))/(5*x - 10) + E/x
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (29) = 58\).
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.53 \[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx={\left (\frac {2 \, {\left (x - 1\right )}}{x^{2} - 2 \, x} + \log \left (x - 2\right ) - \log \left (x\right )\right )} e - {\left (\frac {2}{x - 2} + \log \left (x - 2\right ) - \log \left (x\right )\right )} e + x + \frac {e}{x - 2} + \frac {e^{\left (x \log \left (x\right )^{4} - e^{4} \log \left (x\right )^{4} - 2 \, x + 2 \, e^{4}\right )}}{5 \, {\left (x - 2\right )}} \] Input:
integrate((((x^3-2*x^2)*log(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*log(x)^ 3-2*x^3+3*x^2)*exp((x-exp(4))*log(x)^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp( 1)+5*x^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x, algorithm="maxima")
Output:
(2*(x - 1)/(x^2 - 2*x) + log(x - 2) - log(x))*e - (2/(x - 2) + log(x - 2) - log(x))*e + x + e/(x - 2) + 1/5*e^(x*log(x)^4 - e^4*log(x)^4 - 2*x + 2*e ^4)/(x - 2)
\[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx=\int { \frac {5 \, x^{4} - 20 \, x^{3} + 20 \, x^{2} - 5 \, {\left (x^{2} - 4 \, x + 4\right )} e + {\left ({\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right )^{4} + 4 \, {\left (x^{3} - 2 \, x^{2} - {\left (x^{2} - 2 \, x\right )} e^{4}\right )} \log \left (x\right )^{3} - 2 \, x^{3} + 3 \, x^{2}\right )} e^{\left ({\left (x - e^{4}\right )} \log \left (x\right )^{4} - 2 \, x + 2 \, e^{4}\right )}}{5 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}} \,d x } \] Input:
integrate((((x^3-2*x^2)*log(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*log(x)^ 3-2*x^3+3*x^2)*exp((x-exp(4))*log(x)^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp( 1)+5*x^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x, algorithm="giac")
Output:
integrate(1/5*(5*x^4 - 20*x^3 + 20*x^2 - 5*(x^2 - 4*x + 4)*e + ((x^3 - 2*x ^2)*log(x)^4 + 4*(x^3 - 2*x^2 - (x^2 - 2*x)*e^4)*log(x)^3 - 2*x^3 + 3*x^2) *e^((x - e^4)*log(x)^4 - 2*x + 2*e^4))/(x^4 - 4*x^3 + 4*x^2), x)
Time = 4.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx=x+\frac {\mathrm {e}}{x}+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-{\mathrm {e}}^4\,{\ln \left (x\right )}^4}\,{\mathrm {e}}^{x\,{\ln \left (x\right )}^4}}{5\,\left (x-2\right )} \] Input:
int(-(exp(1)*(5*x^2 - 20*x + 20) - 20*x^2 + 20*x^3 - 5*x^4 + exp(2*exp(4) - 2*x + log(x)^4*(x - exp(4)))*(log(x)^4*(2*x^2 - x^3) - log(x)^3*(exp(4)* (8*x - 4*x^2) - 8*x^2 + 4*x^3) - 3*x^2 + 2*x^3))/(20*x^2 - 20*x^3 + 5*x^4) ,x)
Output:
x + exp(1)/x + (exp(2*exp(4))*exp(-2*x)*exp(-exp(4)*log(x)^4)*exp(x*log(x) ^4))/(5*(x - 2))
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.41 \[ \int \frac {20 x^2-20 x^3+5 x^4+e \left (-20+20 x-5 x^2\right )+e^{2 e^4-2 x+\left (-e^4+x\right ) \log ^4(x)} \left (3 x^2-2 x^3+\left (-8 x^2+4 x^3+e^4 \left (8 x-4 x^2\right )\right ) \log ^3(x)+\left (-2 x^2+x^3\right ) \log ^4(x)\right )}{20 x^2-20 x^3+5 x^4} \, dx=\frac {e^{\mathrm {log}\left (x \right )^{4} x +2 e^{4}} x +5 e^{\mathrm {log}\left (x \right )^{4} e^{4}+2 x} e x -10 e^{\mathrm {log}\left (x \right )^{4} e^{4}+2 x} e +5 e^{\mathrm {log}\left (x \right )^{4} e^{4}+2 x} x^{3}-10 e^{\mathrm {log}\left (x \right )^{4} e^{4}+2 x} x^{2}}{5 e^{\mathrm {log}\left (x \right )^{4} e^{4}+2 x} x \left (x -2\right )} \] Input:
int((((x^3-2*x^2)*log(x)^4+((-4*x^2+8*x)*exp(4)+4*x^3-8*x^2)*log(x)^3-2*x^ 3+3*x^2)*exp((x-exp(4))*log(x)^4+2*exp(4)-2*x)+(-5*x^2+20*x-20)*exp(1)+5*x ^4-20*x^3+20*x^2)/(5*x^4-20*x^3+20*x^2),x)
Output:
(e**(log(x)**4*x + 2*e**4)*x + 5*e**(log(x)**4*e**4 + 2*x)*e*x - 10*e**(lo g(x)**4*e**4 + 2*x)*e + 5*e**(log(x)**4*e**4 + 2*x)*x**3 - 10*e**(log(x)** 4*e**4 + 2*x)*x**2)/(5*e**(log(x)**4*e**4 + 2*x)*x*(x - 2))