Integrand size = 61, antiderivative size = 25 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=-2+x-2 \left (\frac {1}{3}-2 x\right ) x+\frac {x}{-x+\log \left (\frac {1}{x^2}\right )} \] Output:
x-2*(1/3-2*x)*x+x/(ln(1/x^2)-x)-2
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=\frac {1}{3} \left (x+12 x^2+\frac {3 x}{-x+\log \left (\frac {1}{x^2}\right )}\right ) \] Input:
Integrate[(6 + x^2 + 24*x^3 + (3 - 2*x - 48*x^2)*Log[x^(-2)] + (1 + 24*x)* Log[x^(-2)]^2)/(3*x^2 - 6*x*Log[x^(-2)] + 3*Log[x^(-2)]^2),x]
Output:
(x + 12*x^2 + (3*x)/(-x + Log[x^(-2)]))/3
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.
\(\displaystyle \int \frac {24 x^3+x^2+(24 x+1) \log ^2\left (\frac {1}{x^2}\right )+\left (-48 x^2-2 x+3\right ) \log \left (\frac {1}{x^2}\right )+6}{3 x^2+3 \log ^2\left (\frac {1}{x^2}\right )-6 x \log \left (\frac {1}{x^2}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {24 x^3+x^2+(24 x+1) \log ^2\left (\frac {1}{x^2}\right )+\left (-48 x^2-2 x+3\right ) \log \left (\frac {1}{x^2}\right )+6}{3 \left (x-\log \left (\frac {1}{x^2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {24 x^3+x^2+(24 x+1) \log ^2\left (\frac {1}{x^2}\right )+\left (-48 x^2-2 x+3\right ) \log \left (\frac {1}{x^2}\right )+6}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} \int \left (24 x-\frac {3}{x-\log \left (\frac {1}{x^2}\right )}+\frac {3 (x+2)}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (6 \int \frac {1}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2}dx+3 \int \frac {x}{\left (x-\log \left (\frac {1}{x^2}\right )\right )^2}dx-3 \int \frac {1}{x-\log \left (\frac {1}{x^2}\right )}dx+12 x^2+x\right )\) |
Input:
Int[(6 + x^2 + 24*x^3 + (3 - 2*x - 48*x^2)*Log[x^(-2)] + (1 + 24*x)*Log[x^ (-2)]^2)/(3*x^2 - 6*x*Log[x^(-2)] + 3*Log[x^(-2)]^2),x]
Output:
$Aborted
Time = 0.54 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
method | result | size |
risch | \(4 x^{2}+\frac {x}{3}-\frac {x}{x -\ln \left (\frac {1}{x^{2}}\right )}\) | \(23\) |
parallelrisch | \(\frac {24 x^{3}-24 x^{2} \ln \left (\frac {1}{x^{2}}\right )+2 x^{2}-2 x \ln \left (\frac {1}{x^{2}}\right )-6 x}{6 x -6 \ln \left (\frac {1}{x^{2}}\right )}\) | \(43\) |
norman | \(\frac {-\ln \left (\frac {1}{x^{2}}\right )+\frac {x^{2}}{3}+4 x^{3}-\frac {x \ln \left (\frac {1}{x^{2}}\right )}{3}-4 x^{2} \ln \left (\frac {1}{x^{2}}\right )}{x -\ln \left (\frac {1}{x^{2}}\right )}\) | \(45\) |
Input:
int(((24*x+1)*ln(1/x^2)^2+(-48*x^2-2*x+3)*ln(1/x^2)+24*x^3+x^2+6)/(3*ln(1/ x^2)^2-6*x*ln(1/x^2)+3*x^2),x,method=_RETURNVERBOSE)
Output:
4*x^2+1/3*x-x/(x-ln(1/x^2))
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=\frac {12 \, x^{3} + x^{2} - {\left (12 \, x^{2} + x\right )} \log \left (\frac {1}{x^{2}}\right ) - 3 \, x}{3 \, {\left (x - \log \left (\frac {1}{x^{2}}\right )\right )}} \] Input:
integrate(((24*x+1)*log(1/x^2)^2+(-48*x^2-2*x+3)*log(1/x^2)+24*x^3+x^2+6)/ (3*log(1/x^2)^2-6*x*log(1/x^2)+3*x^2),x, algorithm="fricas")
Output:
1/3*(12*x^3 + x^2 - (12*x^2 + x)*log(x^(-2)) - 3*x)/(x - log(x^(-2)))
Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=4 x^{2} + \frac {x}{3} + \frac {x}{- x + \log {\left (\frac {1}{x^{2}} \right )}} \] Input:
integrate(((24*x+1)*ln(1/x**2)**2+(-48*x**2-2*x+3)*ln(1/x**2)+24*x**3+x**2 +6)/(3*ln(1/x**2)**2-6*x*ln(1/x**2)+3*x**2),x)
Output:
4*x**2 + x/3 + x/(-x + log(x**(-2)))
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=\frac {12 \, x^{3} + x^{2} + 2 \, {\left (12 \, x^{2} + x\right )} \log \left (x\right ) - 3 \, x}{3 \, {\left (x + 2 \, \log \left (x\right )\right )}} \] Input:
integrate(((24*x+1)*log(1/x^2)^2+(-48*x^2-2*x+3)*log(1/x^2)+24*x^3+x^2+6)/ (3*log(1/x^2)^2-6*x*log(1/x^2)+3*x^2),x, algorithm="maxima")
Output:
1/3*(12*x^3 + x^2 + 2*(12*x^2 + x)*log(x) - 3*x)/(x + 2*log(x))
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=4 \, x^{2} + \frac {1}{3} \, x - \frac {x}{x + \log \left (x^{2}\right )} \] Input:
integrate(((24*x+1)*log(1/x^2)^2+(-48*x^2-2*x+3)*log(1/x^2)+24*x^3+x^2+6)/ (3*log(1/x^2)^2-6*x*log(1/x^2)+3*x^2),x, algorithm="giac")
Output:
4*x^2 + 1/3*x - x/(x + log(x^2))
Time = 3.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=\frac {x}{3}-\frac {\ln \left (\frac {1}{x^2}\right )}{x-\ln \left (\frac {1}{x^2}\right )}+4\,x^2 \] Input:
int((log(1/x^2)^2*(24*x + 1) - log(1/x^2)*(2*x + 48*x^2 - 3) + x^2 + 24*x^ 3 + 6)/(3*log(1/x^2)^2 - 6*x*log(1/x^2) + 3*x^2),x)
Output:
x/3 - log(1/x^2)/(x - log(1/x^2)) + 4*x^2
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {6+x^2+24 x^3+\left (3-2 x-48 x^2\right ) \log \left (\frac {1}{x^2}\right )+(1+24 x) \log ^2\left (\frac {1}{x^2}\right )}{3 x^2-6 x \log \left (\frac {1}{x^2}\right )+3 \log ^2\left (\frac {1}{x^2}\right )} \, dx=\frac {-\mathrm {log}\left (x^{2}\right )^{2}+2 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (x \right )+12 \,\mathrm {log}\left (x^{2}\right ) x^{2}+3 \,\mathrm {log}\left (x^{2}\right )+2 \,\mathrm {log}\left (x \right ) x +12 x^{3}+x^{2}}{3 \,\mathrm {log}\left (x^{2}\right )+3 x} \] Input:
int(((24*x+1)*log(1/x^2)^2+(-48*x^2-2*x+3)*log(1/x^2)+24*x^3+x^2+6)/(3*log (1/x^2)^2-6*x*log(1/x^2)+3*x^2),x)
Output:
( - log(x**2)**2 + 2*log(x**2)*log(x) + 12*log(x**2)*x**2 + 3*log(x**2) + 2*log(x)*x + 12*x**3 + x**2)/(3*(log(x**2) + x))