Integrand size = 147, antiderivative size = 27 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=5-\frac {e^4}{x}+x+\frac {5}{9+\log \left (-x+4 x^2\right )} \] Output:
x+5-exp(4)/x+5/(ln(4*x^2-x)+9)
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=-\frac {e^4}{x}+x+\frac {5}{9+\log (x (-1+4 x))} \] Input:
Integrate[(5*x - 121*x^2 + 324*x^3 + E^4*(-81 + 324*x) + (-18*x^2 + 72*x^3 + E^4*(-18 + 72*x))*Log[-x + 4*x^2] + (-x^2 + 4*x^3 + E^4*(-1 + 4*x))*Log [-x + 4*x^2]^2)/(-81*x^2 + 324*x^3 + (-18*x^2 + 72*x^3)*Log[-x + 4*x^2] + (-x^2 + 4*x^3)*Log[-x + 4*x^2]^2),x]
Output:
-(E^4/x) + x + 5/(9 + Log[x*(-1 + 4*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.
\(\displaystyle \int \frac {324 x^3-121 x^2+\left (4 x^3-x^2+e^4 (4 x-1)\right ) \log ^2\left (4 x^2-x\right )+\left (72 x^3-18 x^2+e^4 (72 x-18)\right ) \log \left (4 x^2-x\right )+5 x+e^4 (324 x-81)}{324 x^3-81 x^2+\left (4 x^3-x^2\right ) \log ^2\left (4 x^2-x\right )+\left (72 x^3-18 x^2\right ) \log \left (4 x^2-x\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-324 x^3+121 x^2-\left (4 x^3-x^2+e^4 (4 x-1)\right ) \log ^2\left (4 x^2-x\right )-\left (72 x^3-18 x^2+e^4 (72 x-18)\right ) \log \left (4 x^2-x\right )-5 x-e^4 (324 x-81)}{(1-4 x) x^2 (\log (x (4 x-1))+9)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (x^2+e^4\right ) \log ^2(x (4 x-1))}{x^2 (\log (x (4 x-1))+9)^2}+\frac {18 \left (x^2+e^4\right ) \log (x (4 x-1))}{x^2 (\log (x (4 x-1))+9)^2}+\frac {81 e^4}{x^2 (\log (x (4 x-1))+9)^2}+\frac {324 x}{(4 x-1) (\log (x (4 x-1))+9)^2}+\frac {5}{x (4 x-1) (\log (x (4 x-1))+9)^2}-\frac {121}{(4 x-1) (\log (x (4 x-1))+9)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -5 \int \frac {1}{x (\log (x (4 x-1))+9)^2}dx-20 \int \frac {1}{(4 x-1) (\log (x (4 x-1))+9)^2}dx+x-\frac {e^4}{x}\) |
Input:
Int[(5*x - 121*x^2 + 324*x^3 + E^4*(-81 + 324*x) + (-18*x^2 + 72*x^3 + E^4 *(-18 + 72*x))*Log[-x + 4*x^2] + (-x^2 + 4*x^3 + E^4*(-1 + 4*x))*Log[-x + 4*x^2]^2)/(-81*x^2 + 324*x^3 + (-18*x^2 + 72*x^3)*Log[-x + 4*x^2] + (-x^2 + 4*x^3)*Log[-x + 4*x^2]^2),x]
Output:
$Aborted
Time = 0.80 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {-x^{2}+{\mathrm e}^{4}}{x}+\frac {5}{\ln \left (4 x^{2}-x \right )+9}\) | \(31\) |
norman | \(\frac {5 x +\ln \left (4 x^{2}-x \right ) x^{2}+9 x^{2}-{\mathrm e}^{4} \ln \left (4 x^{2}-x \right )-9 \,{\mathrm e}^{4}}{x \left (\ln \left (4 x^{2}-x \right )+9\right )}\) | \(60\) |
parallelrisch | \(-\frac {-16 \ln \left (4 x^{2}-x \right ) x^{2}+16 \,{\mathrm e}^{4} \ln \left (4 x^{2}-x \right )-144 x^{2}-8 \ln \left (4 x^{2}-x \right ) x +144 \,{\mathrm e}^{4}-152 x}{16 x \left (\ln \left (4 x^{2}-x \right )+9\right )}\) | \(75\) |
Input:
int((((-1+4*x)*exp(4)+4*x^3-x^2)*ln(4*x^2-x)^2+((72*x-18)*exp(4)+72*x^3-18 *x^2)*ln(4*x^2-x)+(324*x-81)*exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x^2)*ln(4 *x^2-x)^2+(72*x^3-18*x^2)*ln(4*x^2-x)+324*x^3-81*x^2),x,method=_RETURNVERB OSE)
Output:
-(-x^2+exp(4))/x+5/(ln(4*x^2-x)+9)
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=\frac {9 \, x^{2} + {\left (x^{2} - e^{4}\right )} \log \left (4 \, x^{2} - x\right ) + 5 \, x - 9 \, e^{4}}{x \log \left (4 \, x^{2} - x\right ) + 9 \, x} \] Input:
integrate((((-1+4*x)*exp(4)+4*x^3-x^2)*log(4*x^2-x)^2+((72*x-18)*exp(4)+72 *x^3-18*x^2)*log(4*x^2-x)+(324*x-81)*exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x ^2)*log(4*x^2-x)^2+(72*x^3-18*x^2)*log(4*x^2-x)+324*x^3-81*x^2),x, algorit hm="fricas")
Output:
(9*x^2 + (x^2 - e^4)*log(4*x^2 - x) + 5*x - 9*e^4)/(x*log(4*x^2 - x) + 9*x )
Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=x + \frac {5}{\log {\left (4 x^{2} - x \right )} + 9} - \frac {e^{4}}{x} \] Input:
integrate((((-1+4*x)*exp(4)+4*x**3-x**2)*ln(4*x**2-x)**2+((72*x-18)*exp(4) +72*x**3-18*x**2)*ln(4*x**2-x)+(324*x-81)*exp(4)+324*x**3-121*x**2+5*x)/(( 4*x**3-x**2)*ln(4*x**2-x)**2+(72*x**3-18*x**2)*ln(4*x**2-x)+324*x**3-81*x* *2),x)
Output:
x + 5/(log(4*x**2 - x) + 9) - exp(4)/x
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=\frac {9 \, x^{2} + {\left (x^{2} - e^{4}\right )} \log \left (4 \, x - 1\right ) + {\left (x^{2} - e^{4}\right )} \log \left (x\right ) + 5 \, x - 9 \, e^{4}}{x \log \left (4 \, x - 1\right ) + x \log \left (x\right ) + 9 \, x} \] Input:
integrate((((-1+4*x)*exp(4)+4*x^3-x^2)*log(4*x^2-x)^2+((72*x-18)*exp(4)+72 *x^3-18*x^2)*log(4*x^2-x)+(324*x-81)*exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x ^2)*log(4*x^2-x)^2+(72*x^3-18*x^2)*log(4*x^2-x)+324*x^3-81*x^2),x, algorit hm="maxima")
Output:
(9*x^2 + (x^2 - e^4)*log(4*x - 1) + (x^2 - e^4)*log(x) + 5*x - 9*e^4)/(x*l og(4*x - 1) + x*log(x) + 9*x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=\frac {x^{2} \log \left (4 \, x^{2} - x\right ) + 9 \, x^{2} - e^{4} \log \left (4 \, x^{2} - x\right ) + 5 \, x - 9 \, e^{4}}{x \log \left (4 \, x^{2} - x\right ) + 9 \, x} \] Input:
integrate((((-1+4*x)*exp(4)+4*x^3-x^2)*log(4*x^2-x)^2+((72*x-18)*exp(4)+72 *x^3-18*x^2)*log(4*x^2-x)+(324*x-81)*exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x ^2)*log(4*x^2-x)^2+(72*x^3-18*x^2)*log(4*x^2-x)+324*x^3-81*x^2),x, algorit hm="giac")
Output:
(x^2*log(4*x^2 - x) + 9*x^2 - e^4*log(4*x^2 - x) + 5*x - 9*e^4)/(x*log(4*x ^2 - x) + 9*x)
Time = 2.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=x+\frac {5}{\ln \left (4\,x^2-x\right )+9}-\frac {{\mathrm {e}}^4}{x} \] Input:
int(-(5*x + log(4*x^2 - x)*(72*x^3 - 18*x^2 + exp(4)*(72*x - 18)) + log(4* x^2 - x)^2*(4*x^3 - x^2 + exp(4)*(4*x - 1)) - 121*x^2 + 324*x^3 + exp(4)*( 324*x - 81))/(log(4*x^2 - x)*(18*x^2 - 72*x^3) + log(4*x^2 - x)^2*(x^2 - 4 *x^3) + 81*x^2 - 324*x^3),x)
Output:
x + 5/(log(4*x^2 - x) + 9) - exp(4)/x
Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx=\frac {-9 \,\mathrm {log}\left (4 x^{2}-x \right ) e^{4}+9 \,\mathrm {log}\left (4 x^{2}-x \right ) x^{2}-5 \,\mathrm {log}\left (4 x^{2}-x \right ) x -81 e^{4}+81 x^{2}}{9 x \left (\mathrm {log}\left (4 x^{2}-x \right )+9\right )} \] Input:
int((((-1+4*x)*exp(4)+4*x^3-x^2)*log(4*x^2-x)^2+((72*x-18)*exp(4)+72*x^3-1 8*x^2)*log(4*x^2-x)+(324*x-81)*exp(4)+324*x^3-121*x^2+5*x)/((4*x^3-x^2)*lo g(4*x^2-x)^2+(72*x^3-18*x^2)*log(4*x^2-x)+324*x^3-81*x^2),x)
Output:
( - 9*log(4*x**2 - x)*e**4 + 9*log(4*x**2 - x)*x**2 - 5*log(4*x**2 - x)*x - 81*e**4 + 81*x**2)/(9*x*(log(4*x**2 - x) + 9))