Integrand size = 78, antiderivative size = 22 \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=\frac {x^2}{\frac {9}{2}-\frac {1}{4} \left (3+e^2\right ) x \log (x)} \] Output:
x^2/(9/2-1/4*(exp(2)+3)*x*ln(x))
Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=-\frac {4 x^2}{-18+\left (3+e^2\right ) x \log (x)} \] Input:
Integrate[(144*x + 12*x^2 + 4*E^2*x^2 + (-12*x^2 - 4*E^2*x^2)*Log[x])/(324 + (-108*x - 36*E^2*x)*Log[x] + (9*x^2 + 6*E^2*x^2 + E^4*x^2)*Log[x]^2),x]
Output:
(-4*x^2)/(-18 + (3 + E^2)*x*Log[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.
\(\displaystyle \int \frac {4 e^2 x^2+12 x^2+\left (-4 e^2 x^2-12 x^2\right ) \log (x)+144 x}{\left (e^4 x^2+6 e^2 x^2+9 x^2\right ) \log ^2(x)+\left (-36 e^2 x-108 x\right ) \log (x)+324} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (12+4 e^2\right ) x^2+\left (-4 e^2 x^2-12 x^2\right ) \log (x)+144 x}{\left (e^4 x^2+6 e^2 x^2+9 x^2\right ) \log ^2(x)+\left (-36 e^2 x-108 x\right ) \log (x)+324}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {4 x \left (\left (3+e^2\right ) x-\left (\left (3+e^2\right ) x \log (x)\right )+36\right )}{\left (18-\left (3+e^2\right ) x \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \frac {x \left (-\left (\left (3+e^2\right ) \log (x) x\right )+\left (3+e^2\right ) x+36\right )}{\left (18-\left (3+e^2\right ) x \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 4 \int \left (\frac {x}{18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)}+\frac {\left (\left (3+e^2\right ) x+18\right ) x}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (\left (3+e^2\right ) \int \frac {x^2}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2}dx+18 \int \frac {x}{\left (18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)\right )^2}dx+\int \frac {x}{18-3 \left (1+\frac {e^2}{3}\right ) x \log (x)}dx\right )\) |
Input:
Int[(144*x + 12*x^2 + 4*E^2*x^2 + (-12*x^2 - 4*E^2*x^2)*Log[x])/(324 + (-1 08*x - 36*E^2*x)*Log[x] + (9*x^2 + 6*E^2*x^2 + E^4*x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {4 x^{2}}{x \,{\mathrm e}^{2} \ln \left (x \right )+3 x \ln \left (x \right )-18}\) | \(21\) |
norman | \(-\frac {4 x^{2}}{x \,{\mathrm e}^{2} \ln \left (x \right )+3 x \ln \left (x \right )-18}\) | \(21\) |
risch | \(-\frac {4 x^{2}}{x \,{\mathrm e}^{2} \ln \left (x \right )+3 x \ln \left (x \right )-18}\) | \(21\) |
parallelrisch | \(-\frac {4 x^{2}}{x \,{\mathrm e}^{2} \ln \left (x \right )+3 x \ln \left (x \right )-18}\) | \(21\) |
Input:
int(((-4*x^2*exp(2)-12*x^2)*ln(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2*exp(2)^ 2+6*x^2*exp(2)+9*x^2)*ln(x)^2+(-36*exp(2)*x-108*x)*ln(x)+324),x,method=_RE TURNVERBOSE)
Output:
-4*x^2/(x*exp(2)*ln(x)+3*x*ln(x)-18)
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=-\frac {4 \, x^{2}}{{\left (x e^{2} + 3 \, x\right )} \log \left (x\right ) - 18} \] Input:
integrate(((-4*x^2*exp(2)-12*x^2)*log(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2* exp(2)^2+6*x^2*exp(2)+9*x^2)*log(x)^2+(-36*exp(2)*x-108*x)*log(x)+324),x, algorithm="fricas")
Output:
-4*x^2/((x*e^2 + 3*x)*log(x) - 18)
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=- \frac {4 x^{2}}{\left (3 x + x e^{2}\right ) \log {\left (x \right )} - 18} \] Input:
integrate(((-4*x**2*exp(2)-12*x**2)*ln(x)+4*x**2*exp(2)+12*x**2+144*x)/((x **2*exp(2)**2+6*x**2*exp(2)+9*x**2)*ln(x)**2+(-36*exp(2)*x-108*x)*ln(x)+32 4),x)
Output:
-4*x**2/((3*x + x*exp(2))*log(x) - 18)
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=-\frac {4 \, x^{2}}{x {\left (e^{2} + 3\right )} \log \left (x\right ) - 18} \] Input:
integrate(((-4*x^2*exp(2)-12*x^2)*log(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2* exp(2)^2+6*x^2*exp(2)+9*x^2)*log(x)^2+(-36*exp(2)*x-108*x)*log(x)+324),x, algorithm="maxima")
Output:
-4*x^2/(x*(e^2 + 3)*log(x) - 18)
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=-\frac {4 \, x^{2}}{x e^{2} \log \left (x\right ) + 3 \, x \log \left (x\right ) - 18} \] Input:
integrate(((-4*x^2*exp(2)-12*x^2)*log(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2* exp(2)^2+6*x^2*exp(2)+9*x^2)*log(x)^2+(-36*exp(2)*x-108*x)*log(x)+324),x, algorithm="giac")
Output:
-4*x^2/(x*e^2*log(x) + 3*x*log(x) - 18)
Timed out. \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=\int \frac {144\,x-\ln \left (x\right )\,\left (4\,x^2\,{\mathrm {e}}^2+12\,x^2\right )+4\,x^2\,{\mathrm {e}}^2+12\,x^2}{\left (6\,x^2\,{\mathrm {e}}^2+x^2\,{\mathrm {e}}^4+9\,x^2\right )\,{\ln \left (x\right )}^2+\left (-108\,x-36\,x\,{\mathrm {e}}^2\right )\,\ln \left (x\right )+324} \,d x \] Input:
int((144*x - log(x)*(4*x^2*exp(2) + 12*x^2) + 4*x^2*exp(2) + 12*x^2)/(log( x)^2*(6*x^2*exp(2) + x^2*exp(4) + 9*x^2) - log(x)*(108*x + 36*x*exp(2)) + 324),x)
Output:
int((144*x - log(x)*(4*x^2*exp(2) + 12*x^2) + 4*x^2*exp(2) + 12*x^2)/(log( x)^2*(6*x^2*exp(2) + x^2*exp(4) + 9*x^2) - log(x)*(108*x + 36*x*exp(2)) + 324), x)
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {144 x+12 x^2+4 e^2 x^2+\left (-12 x^2-4 e^2 x^2\right ) \log (x)}{324+\left (-108 x-36 e^2 x\right ) \log (x)+\left (9 x^2+6 e^2 x^2+e^4 x^2\right ) \log ^2(x)} \, dx=-\frac {4 x^{2}}{\mathrm {log}\left (x \right ) e^{2} x +3 \,\mathrm {log}\left (x \right ) x -18} \] Input:
int(((-4*x^2*exp(2)-12*x^2)*log(x)+4*x^2*exp(2)+12*x^2+144*x)/((x^2*exp(2) ^2+6*x^2*exp(2)+9*x^2)*log(x)^2+(-36*exp(2)*x-108*x)*log(x)+324),x)
Output:
( - 4*x**2)/(log(x)*e**2*x + 3*log(x)*x - 18)