Integrand size = 266, antiderivative size = 30 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx=\left (x-x \left (-x+\frac {e^{25}+x}{x}\right ) \log \left (e^4-x+\log (x)\right )\right )^2 \] Output:
(x-ln(ln(x)+exp(4)-x)*((exp(25)+x)/x-x)*x)^2
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx=\left (x-\left (e^{25}+x-x^2\right ) \log \left (e^4-x+\log (x)\right )\right )^2 \] Input:
Integrate[(-2*x^2 + 2*E^4*x^2 + 2*x^3 - 2*x^4 + E^25*(-2*x + 2*x^2) + 2*x^ 2*Log[x] + (E^50*(2 - 2*x) + 2*x^2 - 2*x^3 - 2*x^5 + E^25*(4*x - 2*E^4*x - 6*x^2 + 4*x^3) + E^4*(-4*x^2 + 6*x^3) + (-2*E^25*x - 4*x^2 + 6*x^3)*Log[x ])*Log[E^4 - x + Log[x]] + (-2*x^3 + 6*x^4 - 4*x^5 + E^4*(2*x^2 - 6*x^3 + 4*x^4) + E^25*(-2*x^2 + 4*x^3 + E^4*(2*x - 4*x^2)) + (2*x^2 - 6*x^3 + 4*x^ 4 + E^25*(2*x - 4*x^2))*Log[x])*Log[E^4 - x + Log[x]]^2)/(E^4*x - x^2 + x* Log[x]),x]
Output:
(x - (E^25 + x - x^2)*Log[E^4 - x + Log[x]])^2
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.
| \(\displaystyle \int \frac {-2 x^4+2 x^3+2 e^4 x^2-2 x^2+e^{25} \left (2 x^2-2 x\right )+2 x^2 \log (x)+\left (-2 x^5-2 x^3+2 x^2+e^{25} \left (4 x^3-6 x^2-2 e^4 x+4 x\right )+e^4 \left (6 x^3-4 x^2\right )+\left (6 x^3-4 x^2-2 e^{25} x\right ) \log (x)+e^{50} (2-2 x)\right ) \log \left (-x+\log (x)+e^4\right )+\left (-4 x^5+6 x^4-2 x^3+e^{25} \left (4 x^3-2 x^2+e^4 \left (2 x-4 x^2\right )\right )+e^4 \left (4 x^4-6 x^3+2 x^2\right )+\left (4 x^4-6 x^3+2 x^2+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (-x+\log (x)+e^4\right )}{-x^2+e^4 x+x \log (x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-2 x^4+2 x^3+\left (2 e^4-2\right ) x^2+e^{25} \left (2 x^2-2 x\right )+2 x^2 \log (x)+\left (-2 x^5-2 x^3+2 x^2+e^{25} \left (4 x^3-6 x^2-2 e^4 x+4 x\right )+e^4 \left (6 x^3-4 x^2\right )+\left (6 x^3-4 x^2-2 e^{25} x\right ) \log (x)+e^{50} (2-2 x)\right ) \log \left (-x+\log (x)+e^4\right )+\left (-4 x^5+6 x^4-2 x^3+e^{25} \left (4 x^3-2 x^2+e^4 \left (2 x-4 x^2\right )\right )+e^4 \left (4 x^4-6 x^3+2 x^2\right )+\left (4 x^4-6 x^3+2 x^2+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (-x+\log (x)+e^4\right )}{-x^2+e^4 x+x \log (x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x-\left (-x^2+x+e^{25}\right ) \log \left (-x+\log (x)+e^4\right )\right ) \left (-x^3+x^2-\left (1-e^4 \left (1+e^{21}\right )\right ) x+\left (e^4-x\right ) (2 x-1) x \log \left (-x+\log (x)+e^4\right )+\log (x) \left (x+(2 x-1) x \log \left (-x+\log (x)+e^4\right )\right )-e^{25}\right )}{x \left (-x+\log (x)+e^4\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\left (x-\left (-x^2+x+e^{25}\right ) \log \left (-x+\log (x)+e^4\right )\right ) \left (x^3-x^2+(1-2 x) \left (e^4-x\right ) \log \left (-x+\log (x)+e^4\right ) x+\left (1-e^4-e^{25}\right ) x-\log (x) \left (x-(1-2 x) x \log \left (-x+\log (x)+e^4\right )\right )+e^{25}\right )}{x \left (-x+\log (x)+e^4\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\left (x-\left (-x^2+x+e^{25}\right ) \log \left (-x+\log (x)+e^4\right )\right ) \left (x^3-x^2+(1-2 x) \left (e^4-x\right ) \log \left (-x+\log (x)+e^4\right ) x+\left (1-e^4-e^{25}\right ) x-\log (x) \left (x-(1-2 x) x \log \left (-x+\log (x)+e^4\right )\right )+e^{25}\right )}{x \left (-x+\log (x)+e^4\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\left ((2 x-1) \left (x^2-x-e^{25}\right ) \log ^2\left (-x+\log (x)+e^4\right )\right )+\frac {\left (x^5-3 \log (x) x^3+\left (1-e^4 \left (3+2 e^{21}\right )\right ) x^3+2 \log (x) x^2-\left (1-e^4 \left (2+3 e^{21}\right )\right ) x^2+e^{25} \log (x) x-2 e^{25} \left (1-\frac {1}{2} e^4 \left (1+e^{21}\right )\right ) x-e^{50}\right ) \log \left (-x+\log (x)+e^4\right )}{x \left (-x+\log (x)+e^4\right )}+\frac {x^3-x^2-\log (x) x+\left (1-e^4 \left (1+e^{21}\right )\right ) x+e^{25}}{-x+\log (x)+e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\int \frac {x^4 \log \left (-x+\log (x)+e^4\right )}{-x+\log (x)+e^4}dx-2 \int x^3 \log ^2\left (-x+\log (x)+e^4\right )dx+\int \frac {x^3}{-x+\log (x)+e^4}dx+3 \int x^2 \log ^2\left (-x+\log (x)+e^4\right )dx-2 \int \frac {x^2}{-x+\log (x)+e^4}dx+\left (1-3 e^4-2 e^{25}\right ) \int \frac {x^2 \log \left (-x+\log (x)+e^4\right )}{-x+\log (x)+e^4}dx-3 \int \frac {x^2 \log (x) \log \left (-x+\log (x)+e^4\right )}{-x+\log (x)+e^4}dx-e^{25} \int \log ^2\left (-x+\log (x)+e^4\right )dx-\left (1-2 e^{25}\right ) \int x \log ^2\left (-x+\log (x)+e^4\right )dx+e^{25} \int \frac {1}{-x+\log (x)+e^4}dx+\left (1-e^{25}\right ) \int \frac {x}{-x+\log (x)+e^4}dx+e^{50} \int \frac {\log \left (-x+\log (x)+e^4\right )}{x \left (x-\log (x)-e^4\right )}dx-e^{25} \left (2-e^4-e^{25}\right ) \int \frac {\log \left (-x+\log (x)+e^4\right )}{-x+\log (x)+e^4}dx-\left (1-2 e^4-3 e^{25}\right ) \int \frac {x \log \left (-x+\log (x)+e^4\right )}{-x+\log (x)+e^4}dx+e^{25} \int \frac {\log (x) \log \left (-x+\log (x)+e^4\right )}{-x+\log (x)+e^4}dx+2 \int \frac {x \log (x) \log \left (-x+\log (x)+e^4\right )}{-x+\log (x)+e^4}dx-\frac {x^2}{2}\right )\) |
Input:
Int[(-2*x^2 + 2*E^4*x^2 + 2*x^3 - 2*x^4 + E^25*(-2*x + 2*x^2) + 2*x^2*Log[ x] + (E^50*(2 - 2*x) + 2*x^2 - 2*x^3 - 2*x^5 + E^25*(4*x - 2*E^4*x - 6*x^2 + 4*x^3) + E^4*(-4*x^2 + 6*x^3) + (-2*E^25*x - 4*x^2 + 6*x^3)*Log[x])*Log [E^4 - x + Log[x]] + (-2*x^3 + 6*x^4 - 4*x^5 + E^4*(2*x^2 - 6*x^3 + 4*x^4) + E^25*(-2*x^2 + 4*x^3 + E^4*(2*x - 4*x^2)) + (2*x^2 - 6*x^3 + 4*x^4 + E^ 25*(2*x - 4*x^2))*Log[x])*Log[E^4 - x + Log[x]]^2)/(E^4*x - x^2 + x*Log[x] ),x]
Output:
$Aborted
Time = 4.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80
| method | result | size |
| risch | \(\left ({\mathrm e}^{25}-x^{2}+x \right )^{2} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )^{2}+\left (-2 x \,{\mathrm e}^{25}+2 x^{3}-2 x^{2}\right ) \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )+x^{2}\) | \(54\) |
| parallelrisch | \(\ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )^{2} x^{4}-2 \,{\mathrm e}^{25} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )^{2} x^{2}-2 \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )^{2} x^{3}+{\mathrm e}^{50} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )^{2}+2 x \,{\mathrm e}^{25} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )^{2}+x^{2} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right )^{2}+2 \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right ) x^{3}-2 \,{\mathrm e}^{25} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right ) x -2 \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-x \right ) x^{2}+x^{2}\) | \(143\) |
Input:
int(((((-4*x^2+2*x)*exp(25)+4*x^4-6*x^3+2*x^2)*ln(x)+((-4*x^2+2*x)*exp(4)+ 4*x^3-2*x^2)*exp(25)+(4*x^4-6*x^3+2*x^2)*exp(4)-4*x^5+6*x^4-2*x^3)*ln(ln(x )+exp(4)-x)^2+((-2*x*exp(25)+6*x^3-4*x^2)*ln(x)+(2-2*x)*exp(25)^2+(-2*x*ex p(4)+4*x^3-6*x^2+4*x)*exp(25)+(6*x^3-4*x^2)*exp(4)-2*x^5-2*x^3+2*x^2)*ln(l n(x)+exp(4)-x)+2*x^2*ln(x)+(2*x^2-2*x)*exp(25)+2*x^2*exp(4)-2*x^4+2*x^3-2* x^2)/(x*ln(x)+x*exp(4)-x^2),x,method=_RETURNVERBOSE)
Output:
(exp(25)-x^2+x)^2*ln(ln(x)+exp(4)-x)^2+(-2*x*exp(25)+2*x^3-2*x^2)*ln(ln(x) +exp(4)-x)+x^2
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx={\left (x^{4} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{2} - x\right )} e^{25} + e^{50}\right )} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} + x^{2} + 2 \, {\left (x^{3} - x^{2} - x e^{25}\right )} \log \left (-x + e^{4} + \log \left (x\right )\right ) \] Input:
integrate(((((-4*x^2+2*x)*exp(25)+4*x^4-6*x^3+2*x^2)*log(x)+((-4*x^2+2*x)* exp(4)+4*x^3-2*x^2)*exp(25)+(4*x^4-6*x^3+2*x^2)*exp(4)-4*x^5+6*x^4-2*x^3)* log(log(x)+exp(4)-x)^2+((-2*x*exp(25)+6*x^3-4*x^2)*log(x)+(2-2*x)*exp(25)^ 2+(-2*x*exp(4)+4*x^3-6*x^2+4*x)*exp(25)+(6*x^3-4*x^2)*exp(4)-2*x^5-2*x^3+2 *x^2)*log(log(x)+exp(4)-x)+2*x^2*log(x)+(2*x^2-2*x)*exp(25)+2*x^2*exp(4)-2 *x^4+2*x^3-2*x^2)/(x*log(x)+x*exp(4)-x^2),x, algorithm="fricas")
Output:
(x^4 - 2*x^3 + x^2 - 2*(x^2 - x)*e^25 + e^50)*log(-x + e^4 + log(x))^2 + x ^2 + 2*(x^3 - x^2 - x*e^25)*log(-x + e^4 + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx=x^{2} + \left (2 x^{3} - 2 x^{2} - 2 x e^{25}\right ) \log {\left (- x + \log {\left (x \right )} + e^{4} \right )} + \left (x^{4} - 2 x^{3} - 2 x^{2} e^{25} + x^{2} + 2 x e^{25} + e^{50}\right ) \log {\left (- x + \log {\left (x \right )} + e^{4} \right )}^{2} \] Input:
integrate(((((-4*x**2+2*x)*exp(25)+4*x**4-6*x**3+2*x**2)*ln(x)+((-4*x**2+2 *x)*exp(4)+4*x**3-2*x**2)*exp(25)+(4*x**4-6*x**3+2*x**2)*exp(4)-4*x**5+6*x **4-2*x**3)*ln(ln(x)+exp(4)-x)**2+((-2*x*exp(25)+6*x**3-4*x**2)*ln(x)+(2-2 *x)*exp(25)**2+(-2*x*exp(4)+4*x**3-6*x**2+4*x)*exp(25)+(6*x**3-4*x**2)*exp (4)-2*x**5-2*x**3+2*x**2)*ln(ln(x)+exp(4)-x)+2*x**2*ln(x)+(2*x**2-2*x)*exp (25)+2*x**2*exp(4)-2*x**4+2*x**3-2*x**2)/(x*ln(x)+x*exp(4)-x**2),x)
Output:
x**2 + (2*x**3 - 2*x**2 - 2*x*exp(25))*log(-x + log(x) + exp(4)) + (x**4 - 2*x**3 - 2*x**2*exp(25) + x**2 + 2*x*exp(25) + exp(50))*log(-x + log(x) + exp(4))**2
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx={\left (x^{4} - 2 \, x^{3} - x^{2} {\left (2 \, e^{25} - 1\right )} + 2 \, x e^{25} + e^{50}\right )} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} + x^{2} + 2 \, {\left (x^{3} - x^{2} - x e^{25}\right )} \log \left (-x + e^{4} + \log \left (x\right )\right ) \] Input:
integrate(((((-4*x^2+2*x)*exp(25)+4*x^4-6*x^3+2*x^2)*log(x)+((-4*x^2+2*x)* exp(4)+4*x^3-2*x^2)*exp(25)+(4*x^4-6*x^3+2*x^2)*exp(4)-4*x^5+6*x^4-2*x^3)* log(log(x)+exp(4)-x)^2+((-2*x*exp(25)+6*x^3-4*x^2)*log(x)+(2-2*x)*exp(25)^ 2+(-2*x*exp(4)+4*x^3-6*x^2+4*x)*exp(25)+(6*x^3-4*x^2)*exp(4)-2*x^5-2*x^3+2 *x^2)*log(log(x)+exp(4)-x)+2*x^2*log(x)+(2*x^2-2*x)*exp(25)+2*x^2*exp(4)-2 *x^4+2*x^3-2*x^2)/(x*log(x)+x*exp(4)-x^2),x, algorithm="maxima")
Output:
(x^4 - 2*x^3 - x^2*(2*e^25 - 1) + 2*x*e^25 + e^50)*log(-x + e^4 + log(x))^ 2 + x^2 + 2*(x^3 - x^2 - x*e^25)*log(-x + e^4 + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (26) = 52\).
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.67 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx=x^{4} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} - 2 \, x^{3} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} - 2 \, x^{2} e^{25} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} + 2 \, x^{3} \log \left (-x + e^{4} + \log \left (x\right )\right ) + x^{2} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} + 2 \, x e^{25} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} - 2 \, x^{2} \log \left (-x + e^{4} + \log \left (x\right )\right ) - 2 \, x e^{25} \log \left (-x + e^{4} + \log \left (x\right )\right ) + e^{50} \log \left (-x + e^{4} + \log \left (x\right )\right )^{2} + x^{2} \] Input:
integrate(((((-4*x^2+2*x)*exp(25)+4*x^4-6*x^3+2*x^2)*log(x)+((-4*x^2+2*x)* exp(4)+4*x^3-2*x^2)*exp(25)+(4*x^4-6*x^3+2*x^2)*exp(4)-4*x^5+6*x^4-2*x^3)* log(log(x)+exp(4)-x)^2+((-2*x*exp(25)+6*x^3-4*x^2)*log(x)+(2-2*x)*exp(25)^ 2+(-2*x*exp(4)+4*x^3-6*x^2+4*x)*exp(25)+(6*x^3-4*x^2)*exp(4)-2*x^5-2*x^3+2 *x^2)*log(log(x)+exp(4)-x)+2*x^2*log(x)+(2*x^2-2*x)*exp(25)+2*x^2*exp(4)-2 *x^4+2*x^3-2*x^2)/(x*log(x)+x*exp(4)-x^2),x, algorithm="giac")
Output:
x^4*log(-x + e^4 + log(x))^2 - 2*x^3*log(-x + e^4 + log(x))^2 - 2*x^2*e^25 *log(-x + e^4 + log(x))^2 + 2*x^3*log(-x + e^4 + log(x)) + x^2*log(-x + e^ 4 + log(x))^2 + 2*x*e^25*log(-x + e^4 + log(x))^2 - 2*x^2*log(-x + e^4 + l og(x)) - 2*x*e^25*log(-x + e^4 + log(x)) + e^50*log(-x + e^4 + log(x))^2 + x^2
Time = 3.76 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx={\ln \left ({\mathrm {e}}^4-x+\ln \left (x\right )\right )}^2\,\left (x^4-2\,x^3+\left (1-2\,{\mathrm {e}}^{25}\right )\,x^2+2\,{\mathrm {e}}^{25}\,x+{\mathrm {e}}^{50}\right )-\ln \left ({\mathrm {e}}^4-x+\ln \left (x\right )\right )\,\left (-2\,x^3+2\,x^2+2\,{\mathrm {e}}^{25}\,x\right )+x^2 \] Input:
int((2*x^2*log(x) - exp(25)*(2*x - 2*x^2) + 2*x^2*exp(4) + log(exp(4) - x + log(x))^2*(log(x)*(exp(25)*(2*x - 4*x^2) + 2*x^2 - 6*x^3 + 4*x^4) + exp( 25)*(exp(4)*(2*x - 4*x^2) - 2*x^2 + 4*x^3) + exp(4)*(2*x^2 - 6*x^3 + 4*x^4 ) - 2*x^3 + 6*x^4 - 4*x^5) - log(exp(4) - x + log(x))*(log(x)*(2*x*exp(25) + 4*x^2 - 6*x^3) + exp(4)*(4*x^2 - 6*x^3) - exp(25)*(4*x - 2*x*exp(4) - 6 *x^2 + 4*x^3) - 2*x^2 + 2*x^3 + 2*x^5 + exp(50)*(2*x - 2)) - 2*x^2 + 2*x^3 - 2*x^4)/(x*exp(4) + x*log(x) - x^2),x)
Output:
log(exp(4) - x + log(x))^2*(exp(50) + 2*x*exp(25) - x^2*(2*exp(25) - 1) - 2*x^3 + x^4) - log(exp(4) - x + log(x))*(2*x*exp(25) + 2*x^2 - 2*x^3) + x^ 2
Time = 0.16 (sec) , antiderivative size = 153, normalized size of antiderivative = 5.10 \[ \int \frac {-2 x^2+2 e^4 x^2+2 x^3-2 x^4+e^{25} \left (-2 x+2 x^2\right )+2 x^2 \log (x)+\left (e^{50} (2-2 x)+2 x^2-2 x^3-2 x^5+e^{25} \left (4 x-2 e^4 x-6 x^2+4 x^3\right )+e^4 \left (-4 x^2+6 x^3\right )+\left (-2 e^{25} x-4 x^2+6 x^3\right ) \log (x)\right ) \log \left (e^4-x+\log (x)\right )+\left (-2 x^3+6 x^4-4 x^5+e^4 \left (2 x^2-6 x^3+4 x^4\right )+e^{25} \left (-2 x^2+4 x^3+e^4 \left (2 x-4 x^2\right )\right )+\left (2 x^2-6 x^3+4 x^4+e^{25} \left (2 x-4 x^2\right )\right ) \log (x)\right ) \log ^2\left (e^4-x+\log (x)\right )}{e^4 x-x^2+x \log (x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right )^{2} e^{50}-2 \mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right )^{2} e^{25} x^{2}+2 \mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right )^{2} e^{25} x +\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right )^{2} x^{4}-2 \mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right )^{2} x^{3}+\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right )^{2} x^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right ) e^{25} x +2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right ) x^{3}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{4}-x \right ) x^{2}+x^{2} \] Input:
int(((((-4*x^2+2*x)*exp(25)+4*x^4-6*x^3+2*x^2)*log(x)+((-4*x^2+2*x)*exp(4) +4*x^3-2*x^2)*exp(25)+(4*x^4-6*x^3+2*x^2)*exp(4)-4*x^5+6*x^4-2*x^3)*log(lo g(x)+exp(4)-x)^2+((-2*x*exp(25)+6*x^3-4*x^2)*log(x)+(2-2*x)*exp(25)^2+(-2* x*exp(4)+4*x^3-6*x^2+4*x)*exp(25)+(6*x^3-4*x^2)*exp(4)-2*x^5-2*x^3+2*x^2)* log(log(x)+exp(4)-x)+2*x^2*log(x)+(2*x^2-2*x)*exp(25)+2*x^2*exp(4)-2*x^4+2 *x^3-2*x^2)/(x*log(x)+x*exp(4)-x^2),x)
Output:
log(log(x) + e**4 - x)**2*e**50 - 2*log(log(x) + e**4 - x)**2*e**25*x**2 + 2*log(log(x) + e**4 - x)**2*e**25*x + log(log(x) + e**4 - x)**2*x**4 - 2* log(log(x) + e**4 - x)**2*x**3 + log(log(x) + e**4 - x)**2*x**2 - 2*log(lo g(x) + e**4 - x)*e**25*x + 2*log(log(x) + e**4 - x)*x**3 - 2*log(log(x) + e**4 - x)*x**2 + x**2