Integrand size = 94, antiderivative size = 29 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=4+\frac {4 \left (\frac {1}{2}+x-\frac {x}{\log ^2(2)-\frac {1}{3} x \log (x)}\right )}{x^2} \] Output:
4*(1/2+x-x/(ln(2)^2-1/3*x*ln(x)))/x^2+4
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=-\frac {2 \left (-1+x \left (-2-\frac {6}{-3 \log ^2(2)+x \log (x)}\right )\right )}{x^2} \] Input:
Integrate[(-12*x^2 + 36*x*Log[2]^2 + (-36 - 36*x)*Log[2]^4 + (-24*x^2 + (2 4*x + 24*x^2)*Log[2]^2)*Log[x] + (-4*x^2 - 4*x^3)*Log[x]^2)/(9*x^3*Log[2]^ 4 - 6*x^4*Log[2]^2*Log[x] + x^5*Log[x]^2),x]
Output:
(-2*(-1 + x*(-2 - 6/(-3*Log[2]^2 + x*Log[x]))))/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.
\(\displaystyle \int \frac {-12 x^2+\left (\left (24 x^2+24 x\right ) \log ^2(2)-24 x^2\right ) \log (x)+\left (-4 x^3-4 x^2\right ) \log ^2(x)+(-36 x-36) \log ^4(2)+36 x \log ^2(2)}{x^5 \log ^2(x)-6 x^4 \log ^2(2) \log (x)+9 x^3 \log ^4(2)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-12 x^2+\left (\left (24 x^2+24 x\right ) \log ^2(2)-24 x^2\right ) \log (x)+\left (-4 x^3-4 x^2\right ) \log ^2(x)+(-36 x-36) \log ^4(2)+36 x \log ^2(2)}{x^3 \left (3 \log ^2(2)-x \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 (x+1)}{x^3}-\frac {24}{x^2 \left (x \log (x)-3 \log ^2(2)\right )}-\frac {12 \left (x+3 \log ^2(2)\right )}{x^2 \left (x \log (x)-3 \log ^2(2)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -36 \log ^2(2) \int \frac {1}{x^2 \left (x \log (x)-3 \log ^2(2)\right )^2}dx-24 \int \frac {1}{x^2 \left (x \log (x)-3 \log ^2(2)\right )}dx-12 \int \frac {1}{x \left (x \log (x)-3 \log ^2(2)\right )^2}dx+\frac {2 (x+1)^2}{x^2}\) |
Input:
Int[(-12*x^2 + 36*x*Log[2]^2 + (-36 - 36*x)*Log[2]^4 + (-24*x^2 + (24*x + 24*x^2)*Log[2]^2)*Log[x] + (-4*x^2 - 4*x^3)*Log[x]^2)/(9*x^3*Log[2]^4 - 6* x^4*Log[2]^2*Log[x] + x^5*Log[x]^2),x]
Output:
$Aborted
Time = 1.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {4 x +2}{x^{2}}-\frac {12}{x \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\) | \(31\) |
default | \(-\frac {4 \left (\left (3-3 \ln \left (2\right )^{2}\right ) x +x^{2} \ln \left (x \right )-\frac {3 \ln \left (2\right )^{2}}{2}+\frac {x \ln \left (x \right )}{2}\right )}{x^{2} \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\) | \(48\) |
norman | \(\frac {\left (12 \ln \left (2\right )^{2}-12\right ) x -4 x^{2} \ln \left (x \right )+6 \ln \left (2\right )^{2}-2 x \ln \left (x \right )}{x^{2} \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\) | \(48\) |
parallelrisch | \(\frac {12 x \ln \left (2\right )^{2}-4 x^{2} \ln \left (x \right )+6 \ln \left (2\right )^{2}-2 x \ln \left (x \right )-12 x}{x^{2} \left (3 \ln \left (2\right )^{2}-x \ln \left (x \right )\right )}\) | \(48\) |
Input:
int(((-4*x^3-4*x^2)*ln(x)^2+((24*x^2+24*x)*ln(2)^2-24*x^2)*ln(x)+(-36*x-36 )*ln(2)^4+36*x*ln(2)^2-12*x^2)/(x^5*ln(x)^2-6*x^4*ln(2)^2*ln(x)+9*x^3*ln(2 )^4),x,method=_RETURNVERBOSE)
Output:
2*(1+2*x)/x^2-12/x/(3*ln(2)^2-x*ln(x))
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {2 \, {\left (3 \, {\left (2 \, x + 1\right )} \log \left (2\right )^{2} - {\left (2 \, x^{2} + x\right )} \log \left (x\right ) - 6 \, x\right )}}{3 \, x^{2} \log \left (2\right )^{2} - x^{3} \log \left (x\right )} \] Input:
integrate(((-4*x^3-4*x^2)*log(x)^2+((24*x^2+24*x)*log(2)^2-24*x^2)*log(x)+ (-36*x-36)*log(2)^4+36*x*log(2)^2-12*x^2)/(x^5*log(x)^2-6*x^4*log(2)^2*log (x)+9*x^3*log(2)^4),x, algorithm="fricas")
Output:
2*(3*(2*x + 1)*log(2)^2 - (2*x^2 + x)*log(x) - 6*x)/(3*x^2*log(2)^2 - x^3* log(x))
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {12}{x^{2} \log {\left (x \right )} - 3 x \log {\left (2 \right )}^{2}} - \frac {- 4 x - 2}{x^{2}} \] Input:
integrate(((-4*x**3-4*x**2)*ln(x)**2+((24*x**2+24*x)*ln(2)**2-24*x**2)*ln( x)+(-36*x-36)*ln(2)**4+36*x*ln(2)**2-12*x**2)/(x**5*ln(x)**2-6*x**4*ln(2)* *2*ln(x)+9*x**3*ln(2)**4),x)
Output:
12/(x**2*log(x) - 3*x*log(2)**2) - (-4*x - 2)/x**2
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {2 \, {\left (6 \, {\left (\log \left (2\right )^{2} - 1\right )} x + 3 \, \log \left (2\right )^{2} - {\left (2 \, x^{2} + x\right )} \log \left (x\right )\right )}}{3 \, x^{2} \log \left (2\right )^{2} - x^{3} \log \left (x\right )} \] Input:
integrate(((-4*x^3-4*x^2)*log(x)^2+((24*x^2+24*x)*log(2)^2-24*x^2)*log(x)+ (-36*x-36)*log(2)^4+36*x*log(2)^2-12*x^2)/(x^5*log(x)^2-6*x^4*log(2)^2*log (x)+9*x^3*log(2)^4),x, algorithm="maxima")
Output:
2*(6*(log(2)^2 - 1)*x + 3*log(2)^2 - (2*x^2 + x)*log(x))/(3*x^2*log(2)^2 - x^3*log(x))
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=-\frac {12}{3 \, x \log \left (2\right )^{2} - x^{2} \log \left (x\right )} + \frac {2 \, {\left (2 \, x + 1\right )}}{x^{2}} \] Input:
integrate(((-4*x^3-4*x^2)*log(x)^2+((24*x^2+24*x)*log(2)^2-24*x^2)*log(x)+ (-36*x-36)*log(2)^4+36*x*log(2)^2-12*x^2)/(x^5*log(x)^2-6*x^4*log(2)^2*log (x)+9*x^3*log(2)^4),x, algorithm="giac")
Output:
-12/(3*x*log(2)^2 - x^2*log(x)) + 2*(2*x + 1)/x^2
Time = 7.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {4\,x^2\,\ln \left (x\right )+x\,\left (2\,\ln \left (x\right )-12\,{\ln \left (2\right )}^2+12\right )-6\,{\ln \left (2\right )}^2}{x^2\,\left (x\,\ln \left (x\right )-3\,{\ln \left (2\right )}^2\right )} \] Input:
int(-(log(2)^4*(36*x + 36) + log(x)^2*(4*x^2 + 4*x^3) - log(x)*(log(2)^2*( 24*x + 24*x^2) - 24*x^2) - 36*x*log(2)^2 + 12*x^2)/(9*x^3*log(2)^4 + x^5*l og(x)^2 - 6*x^4*log(2)^2*log(x)),x)
Output:
(4*x^2*log(x) + x*(2*log(x) - 12*log(2)^2 + 12) - 6*log(2)^2)/(x^2*(x*log( x) - 3*log(2)^2))
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {-12 x^2+36 x \log ^2(2)+(-36-36 x) \log ^4(2)+\left (-24 x^2+\left (24 x+24 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-4 x^2-4 x^3\right ) \log ^2(x)}{9 x^3 \log ^4(2)-6 x^4 \log ^2(2) \log (x)+x^5 \log ^2(x)} \, dx=\frac {4 \,\mathrm {log}\left (x \right ) x^{2}+2 \,\mathrm {log}\left (x \right ) x -12 \mathrm {log}\left (2\right )^{2} x -6 \mathrm {log}\left (2\right )^{2}+12 x}{x^{2} \left (\mathrm {log}\left (x \right ) x -3 \mathrm {log}\left (2\right )^{2}\right )} \] Input:
int(((-4*x^3-4*x^2)*log(x)^2+((24*x^2+24*x)*log(2)^2-24*x^2)*log(x)+(-36*x -36)*log(2)^4+36*x*log(2)^2-12*x^2)/(x^5*log(x)^2-6*x^4*log(2)^2*log(x)+9* x^3*log(2)^4),x)
Output:
(2*(2*log(x)*x**2 + log(x)*x - 6*log(2)**2*x - 3*log(2)**2 + 6*x))/(x**2*( log(x)*x - 3*log(2)**2))