Integrand size = 111, antiderivative size = 32 \[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=\frac {-5+\left (x-x^2\right )^2-\frac {5}{4} e^2 \log ^2(5-x+\log (x))}{x} \] Output:
((-x^2+x)^2-5/4*exp(2)*ln(ln(x)-x+5)^2-5)/x
\[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=\int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx \] Input:
Integrate[(100 - 20*x + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + (20 + 4*x^2 - 16*x^3 + 12*x^4)*Log[x] + E^2*(-10 + 10*x)*Log[5 - x + Log[x]] + (E^2*(25 - 5*x) + 5*E^2*Log[x])*Log[5 - x + Log[x]]^2)/(20*x^2 - 4*x^3 + 4*x^2*Log[ x]),x]
Output:
Integrate[(100 - 20*x + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + (20 + 4*x^2 - 16*x^3 + 12*x^4)*Log[x] + E^2*(-10 + 10*x)*Log[5 - x + Log[x]] + (E^2*(25 - 5*x) + 5*E^2*Log[x])*Log[5 - x + Log[x]]^2)/(20*x^2 - 4*x^3 + 4*x^2*Log[ x]), 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.
\(\displaystyle \int \frac {-12 x^5+76 x^4-84 x^3+20 x^2+\left (12 x^4-16 x^3+4 x^2+20\right ) \log (x)-20 x+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(-x+\log (x)+5)+e^2 (10 x-10) \log (-x+\log (x)+5)+100}{-4 x^3+20 x^2+4 x^2 \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-12 x^5+76 x^4-84 x^3+20 x^2+\left (12 x^4-16 x^3+4 x^2+20\right ) \log (x)-20 x+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(-x+\log (x)+5)+e^2 (10 x-10) \log (-x+\log (x)+5)+100}{4 x^2 (-x+\log (x)+5)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {-12 x^5+76 x^4-84 x^3+20 x^2-20 x+5 \left (e^2 (5-x)+e^2 \log (x)\right ) \log ^2(-x+\log (x)+5)+4 \left (3 x^4-4 x^3+x^2+5\right ) \log (x)-10 e^2 (1-x) \log (-x+\log (x)+5)+100}{x^2 (-x+\log (x)+5)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{4} \int \left (\frac {12 x^3}{x-\log (x)-5}-\frac {76 x^2}{x-\log (x)-5}+\frac {84 x}{x-\log (x)-5}-\frac {20}{x-\log (x)-5}+\frac {20}{(x-\log (x)-5) x}+\frac {5 e^2 \log ^2(-x+\log (x)+5)}{x^2}-\frac {4 \left (3 x^4-4 x^3+x^2+5\right ) \log (x)}{(x-\log (x)-5) x^2}-\frac {10 e^2 (x-1) \log (-x+\log (x)+5)}{(x-\log (x)-5) x^2}-\frac {100}{(x-\log (x)-5) x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (5 e^2 \int \frac {\log ^2(-x+\log (x)+5)}{x^2}dx+10 e^2 \int \frac {\log (-x+\log (x)+5)}{x^2 (x-\log (x)-5)}dx-10 e^2 \int \frac {\log (-x+\log (x)+5)}{x (x-\log (x)-5)}dx+4 x^3-8 x^2+4 x-\frac {20}{x}\right )\) |
Input:
Int[(100 - 20*x + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + (20 + 4*x^2 - 16*x^3 + 12*x^4)*Log[x] + E^2*(-10 + 10*x)*Log[5 - x + Log[x]] + (E^2*(25 - 5*x) + 5*E^2*Log[x])*Log[5 - x + Log[x]]^2)/(20*x^2 - 4*x^3 + 4*x^2*Log[x]),x]
Output:
$Aborted
Time = 1.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {5 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )-x +5\right )^{2}}{4 x}+\frac {x^{4}-2 x^{3}+x^{2}-5}{x}\) | \(36\) |
parallelrisch | \(-\frac {-4 x^{4}+5 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )-x +5\right )^{2}+8 x^{3}-4 x^{2}+20}{4 x}\) | \(37\) |
Input:
int(((5*exp(2)*ln(x)+(-5*x+25)*exp(2))*ln(ln(x)-x+5)^2+(10*x-10)*exp(2)*ln (ln(x)-x+5)+(12*x^4-16*x^3+4*x^2+20)*ln(x)-12*x^5+76*x^4-84*x^3+20*x^2-20* x+100)/(4*x^2*ln(x)-4*x^3+20*x^2),x,method=_RETURNVERBOSE)
Output:
-5/4*exp(2)/x*ln(ln(x)-x+5)^2+(x^4-2*x^3+x^2-5)/x
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=\frac {4 \, x^{4} - 8 \, x^{3} - 5 \, e^{2} \log \left (-x + \log \left (x\right ) + 5\right )^{2} + 4 \, x^{2} - 20}{4 \, x} \] Input:
integrate(((5*exp(2)*log(x)+(-5*x+25)*exp(2))*log(log(x)-x+5)^2+(10*x-10)* exp(2)*log(log(x)-x+5)+(12*x^4-16*x^3+4*x^2+20)*log(x)-12*x^5+76*x^4-84*x^ 3+20*x^2-20*x+100)/(4*x^2*log(x)-4*x^3+20*x^2),x, algorithm="fricas")
Output:
1/4*(4*x^4 - 8*x^3 - 5*e^2*log(-x + log(x) + 5)^2 + 4*x^2 - 20)/x
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=x^{3} - 2 x^{2} + x - \frac {5 e^{2} \log {\left (- x + \log {\left (x \right )} + 5 \right )}^{2}}{4 x} - \frac {5}{x} \] Input:
integrate(((5*exp(2)*ln(x)+(-5*x+25)*exp(2))*ln(ln(x)-x+5)**2+(10*x-10)*ex p(2)*ln(ln(x)-x+5)+(12*x**4-16*x**3+4*x**2+20)*ln(x)-12*x**5+76*x**4-84*x* *3+20*x**2-20*x+100)/(4*x**2*ln(x)-4*x**3+20*x**2),x)
Output:
x**3 - 2*x**2 + x - 5*exp(2)*log(-x + log(x) + 5)**2/(4*x) - 5/x
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=\frac {4 \, x^{4} - 8 \, x^{3} - 5 \, e^{2} \log \left (-x + \log \left (x\right ) + 5\right )^{2} + 4 \, x^{2} - 20}{4 \, x} \] Input:
integrate(((5*exp(2)*log(x)+(-5*x+25)*exp(2))*log(log(x)-x+5)^2+(10*x-10)* exp(2)*log(log(x)-x+5)+(12*x^4-16*x^3+4*x^2+20)*log(x)-12*x^5+76*x^4-84*x^ 3+20*x^2-20*x+100)/(4*x^2*log(x)-4*x^3+20*x^2),x, algorithm="maxima")
Output:
1/4*(4*x^4 - 8*x^3 - 5*e^2*log(-x + log(x) + 5)^2 + 4*x^2 - 20)/x
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=\frac {4 \, x^{4} - 8 \, x^{3} - 5 \, e^{2} \log \left (-x + \log \left (x\right ) + 5\right )^{2} + 4 \, x^{2} - 20}{4 \, x} \] Input:
integrate(((5*exp(2)*log(x)+(-5*x+25)*exp(2))*log(log(x)-x+5)^2+(10*x-10)* exp(2)*log(log(x)-x+5)+(12*x^4-16*x^3+4*x^2+20)*log(x)-12*x^5+76*x^4-84*x^ 3+20*x^2-20*x+100)/(4*x^2*log(x)-4*x^3+20*x^2),x, algorithm="giac")
Output:
1/4*(4*x^4 - 8*x^3 - 5*e^2*log(-x + log(x) + 5)^2 + 4*x^2 - 20)/x
Time = 1.75 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=x-\frac {5}{x}-2\,x^2+x^3-\frac {5\,{\mathrm {e}}^2\,{\ln \left (\ln \left (x\right )-x+5\right )}^2}{4\,x} \] Input:
int((log(x)*(4*x^2 - 16*x^3 + 12*x^4 + 20) - 20*x + log(log(x) - x + 5)^2* (5*exp(2)*log(x) - exp(2)*(5*x - 25)) + 20*x^2 - 84*x^3 + 76*x^4 - 12*x^5 + exp(2)*log(log(x) - x + 5)*(10*x - 10) + 100)/(4*x^2*log(x) + 20*x^2 - 4 *x^3),x)
Output:
x - 5/x - 2*x^2 + x^3 - (5*exp(2)*log(log(x) - x + 5)^2)/(4*x)
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {100-20 x+20 x^2-84 x^3+76 x^4-12 x^5+\left (20+4 x^2-16 x^3+12 x^4\right ) \log (x)+e^2 (-10+10 x) \log (5-x+\log (x))+\left (e^2 (25-5 x)+5 e^2 \log (x)\right ) \log ^2(5-x+\log (x))}{20 x^2-4 x^3+4 x^2 \log (x)} \, dx=\frac {-5 \mathrm {log}\left (\mathrm {log}\left (x \right )-x +5\right )^{2} e^{2}+4 x^{4}-8 x^{3}+4 x^{2}-20}{4 x} \] Input:
int(((5*exp(2)*log(x)+(-5*x+25)*exp(2))*log(log(x)-x+5)^2+(10*x-10)*exp(2) *log(log(x)-x+5)+(12*x^4-16*x^3+4*x^2+20)*log(x)-12*x^5+76*x^4-84*x^3+20*x ^2-20*x+100)/(4*x^2*log(x)-4*x^3+20*x^2),x)
Output:
( - 5*log(log(x) - x + 5)**2*e**2 + 4*x**4 - 8*x**3 + 4*x**2 - 20)/(4*x)