Integrand size = 72, antiderivative size = 21 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{\log (x) \left (1+\frac {2 x^2}{\log \left (x^2\right )}\right )} \] Output:
6/(1+2*x^2/ln(x^2))/ln(x)
Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6 \log \left (x^2\right )}{2 x^2 \log (x)+\log (x) \log \left (x^2\right )} \] Input:
Integrate[(24*x^2*Log[x] + (-12*x^2 - 24*x^2*Log[x])*Log[x^2] - 6*Log[x^2] ^2)/(4*x^5*Log[x]^2 + 4*x^3*Log[x]^2*Log[x^2] + x*Log[x]^2*Log[x^2]^2),x]
Output:
(6*Log[x^2])/(2*x^2*Log[x] + Log[x]*Log[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.
\(\displaystyle \int \frac {-6 \log ^2\left (x^2\right )+24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )}{4 x^5 \log ^2(x)+x \log ^2(x) \log ^2\left (x^2\right )+4 x^3 \log ^2(x) \log \left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {6 \left (-\log ^2\left (x^2\right )+4 x^2 \log (x)-4 x^2 \log (x) \log \left (x^2\right )-2 x^2 \log \left (x^2\right )\right )}{x \log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 6 \int \frac {4 \log (x) x^2-4 \log (x) \log \left (x^2\right ) x^2-2 \log \left (x^2\right ) x^2-\log ^2\left (x^2\right )}{x \log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 6 \int \left (\frac {4 x \left (2 x^2+1\right )}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2}-\frac {2 x (2 \log (x)-1)}{\log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )}-\frac {1}{x \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 \left (2 \int \frac {x}{\log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )}dx+4 \int \frac {x}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2}dx-4 \int \frac {x}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )}dx+8 \int \frac {x^3}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2}dx+\frac {1}{\log (x)}\right )\) |
Input:
Int[(24*x^2*Log[x] + (-12*x^2 - 24*x^2*Log[x])*Log[x^2] - 6*Log[x^2]^2)/(4 *x^5*Log[x]^2 + 4*x^3*Log[x]^2*Log[x^2] + x*Log[x]^2*Log[x^2]^2),x]
Output:
$Aborted
Time = 2.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10
method | result | size |
parallelrisch | \(\frac {6 \ln \left (x^{2}\right )}{\ln \left (x \right ) \left (2 x^{2}+\ln \left (x^{2}\right )\right )}\) | \(23\) |
default | \(-\frac {24 x^{2}}{\left (-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x^{2}+4 \ln \left (x \right )\right ) \ln \left (x \right )}+\frac {6}{\ln \left (x \right )}\) | \(78\) |
parts | \(-\frac {24 x^{2}}{\left (-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x^{2}+4 \ln \left (x \right )\right ) \ln \left (x \right )}+\frac {6}{\ln \left (x \right )}\) | \(78\) |
risch | \(\frac {6 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-12 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+6 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+24 i \ln \left (x \right )}{\left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i x^{2}+4 i \ln \left (x \right )\right ) \ln \left (x \right )}\) | \(115\) |
Input:
int((-6*ln(x^2)^2+(-24*x^2*ln(x)-12*x^2)*ln(x^2)+24*x^2*ln(x))/(x*ln(x)^2* ln(x^2)^2+4*x^3*ln(x)^2*ln(x^2)+4*x^5*ln(x)^2),x,method=_RETURNVERBOSE)
Output:
6*ln(x^2)/ln(x)/(2*x^2+ln(x^2))
Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log \left (x\right )} \] Input:
integrate((-6*log(x^2)^2+(-24*x^2*log(x)-12*x^2)*log(x^2)+24*x^2*log(x))/( x*log(x)^2*log(x^2)^2+4*x^3*log(x)^2*log(x^2)+4*x^5*log(x)^2),x, algorithm ="fricas")
Output:
6/(x^2 + log(x))
Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.33 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log {\left (x \right )}} \] Input:
integrate((-6*ln(x**2)**2+(-24*x**2*ln(x)-12*x**2)*ln(x**2)+24*x**2*ln(x)) /(x*ln(x)**2*ln(x**2)**2+4*x**3*ln(x)**2*ln(x**2)+4*x**5*ln(x)**2),x)
Output:
6/(x**2 + log(x))
Time = 0.09 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log \left (x\right )} \] Input:
integrate((-6*log(x^2)^2+(-24*x^2*log(x)-12*x^2)*log(x^2)+24*x^2*log(x))/( x*log(x)^2*log(x^2)^2+4*x^3*log(x)^2*log(x^2)+4*x^5*log(x)^2),x, algorithm ="maxima")
Output:
6/(x^2 + log(x))
Time = 0.14 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log \left (x\right )} \] Input:
integrate((-6*log(x^2)^2+(-24*x^2*log(x)-12*x^2)*log(x^2)+24*x^2*log(x))/( x*log(x)^2*log(x^2)^2+4*x^3*log(x)^2*log(x^2)+4*x^5*log(x)^2),x, algorithm ="giac")
Output:
6/(x^2 + log(x))
Time = 2.93 (sec) , antiderivative size = 164, normalized size of antiderivative = 7.81 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6\,\ln \left (x^2\right )-12\,\ln \left (x\right )+\frac {12\,\ln \left (x\right )\,\left (x\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+4\,x^3\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+8\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+4\,x^5+8\,x^7\right )}{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )+2\,x^2\right )\,\left (2\,x^3\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+x\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3+4\,x^5\right )}}{2\,{\ln \left (x\right )}^2+\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )+2\,x^2\right )} \] Input:
int(-(log(x^2)*(24*x^2*log(x) + 12*x^2) - 24*x^2*log(x) + 6*log(x^2)^2)/(4 *x^5*log(x)^2 + x*log(x^2)^2*log(x)^2 + 4*x^3*log(x^2)*log(x)^2),x)
Output:
(6*log(x^2) - 12*log(x) + (12*log(x)*(x*(log(x^2) - 2*log(x))^2 + 4*x^3*(l og(x^2) - 2*log(x)) + 8*x^5*(log(x^2) - 2*log(x)) + 2*x^3*(log(x^2) - 2*lo g(x))^2 + 4*x^5 + 8*x^7))/((log(x^2) - 2*log(x) + 2*x^2)*(2*x^3*(log(x^2) - 2*log(x)) + x*(log(x^2) - 2*log(x)) + 2*x^3 + 4*x^5)))/(2*log(x)^2 + log (x)*(log(x^2) - 2*log(x) + 2*x^2))
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6 \,\mathrm {log}\left (x^{2}\right )}{\mathrm {log}\left (x \right ) \left (\mathrm {log}\left (x^{2}\right )+2 x^{2}\right )} \] Input:
int((-6*log(x^2)^2+(-24*x^2*log(x)-12*x^2)*log(x^2)+24*x^2*log(x))/(x*log( x)^2*log(x^2)^2+4*x^3*log(x)^2*log(x^2)+4*x^5*log(x)^2),x)
Output:
(6*log(x**2))/(log(x)*(log(x**2) + 2*x**2))