\(\int \frac {24 x^2 \log (x)+(-12 x^2-24 x^2 \log (x)) \log (x^2)-6 \log ^2(x^2)}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log (x^2)+x \log ^2(x) \log ^2(x^2)} \, dx\) [3460]

Optimal result

Integrand size = 72, antiderivative size = 21 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{\log (x) \left (1+\frac {2 x^2}{\log \left (x^2\right )}\right )} \]

[Out]

6/(1+2*x^2/ln(x^2))/ln(x)

Rubi [F]

\[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx \]

[In]

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]

[Out]

6/Log[x] + 24*Defer[Int][x/(Log[x]*(2*x^2 + Log[x^2])^2), x] + 48*Defer[Int][x^3/(Log[x]*(2*x^2 + Log[x^2])^2)
, x] + 12*Defer[Int][x/(Log[x]^2*(2*x^2 + Log[x^2])), x] - 24*Defer[Int][x/(Log[x]*(2*x^2 + Log[x^2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {6 \left (-4 x^2 \log (x) \left (-1+\log \left (x^2\right )\right )-\log \left (x^2\right ) \left (2 x^2+\log \left (x^2\right )\right )\right )}{x \log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )^2} \, dx \\ & = 6 \int \frac {-4 x^2 \log (x) \left (-1+\log \left (x^2\right )\right )-\log \left (x^2\right ) \left (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 \\ & = 6 \int \left (-\frac {1}{x \log ^2(x)}+\frac {4 x \left (1+2 x^2\right )}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2}-\frac {2 x (-1+2 \log (x))}{\log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )}\right ) \, dx \\ & = -\left (6 \int \frac {1}{x \log ^2(x)} \, dx\right )-12 \int \frac {x (-1+2 \log (x))}{\log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )} \, dx+24 \int \frac {x \left (1+2 x^2\right )}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2} \, dx \\ & = -\left (6 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\right )-12 \int \left (-\frac {x}{\log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )}+\frac {2 x}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )}\right ) \, dx+24 \int \left (\frac {x}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2}+\frac {2 x^3}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2}\right ) \, dx \\ & = \frac {6}{\log (x)}+12 \int \frac {x}{\log ^2(x) \left (2 x^2+\log \left (x^2\right )\right )} \, dx+24 \int \frac {x}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2} \, dx-24 \int \frac {x}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )} \, dx+48 \int \frac {x^3}{\log (x) \left (2 x^2+\log \left (x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6 \log \left (x^2\right )}{2 x^2 \log (x)+\log (x) \log \left (x^2\right )} \]

[In]

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]

[Out]

(6*Log[x^2])/(2*x^2*Log[x] + Log[x]*Log[x^2])

Maple [A] (verified)

Time = 17.88 (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	\(-\frac {24 x^{2}}{\left (-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x^{2}+4 \ln \left (x \right )\right ) \ln \left (x \right )}+\frac {6}{\ln \left (x \right )}\)	\(78\)
parts	\(-\frac {24 x^{2}}{\left (-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x^{2}+4 \ln \left (x \right )\right ) \ln \left (x \right )}+\frac {6}{\ln \left (x \right )}\)	\(78\)
risch	\(\frac {6 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-12 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+6 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+24 i \ln \left (x \right )}{\left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i x^{2}+4 i \ln \left (x \right )\right ) \ln \left (x \right )}\)	\(115\)

[In]

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)

[Out]

6*ln(x^2)/ln(x)/(2*x^2+ln(x^2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log \left (x\right )} \]

[In]

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")

[Out]

6/(x^2 + log(x))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.33 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log {\left (x \right )}} \]

[In]

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)

[Out]

6/(x**2 + log(x))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log \left (x\right )} \]

[In]

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")

[Out]

6/(x^2 + log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6}{x^{2} + \log \left (x\right )} \]

[In]

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")

[Out]

6/(x^2 + log(x))

Mupad [B] (verification not implemented)

Time = 9.37 (sec) , antiderivative size = 164, normalized size of antiderivative = 7.81 \[ \int \frac {24 x^2 \log (x)+\left (-12 x^2-24 x^2 \log (x)\right ) \log \left (x^2\right )-6 \log ^2\left (x^2\right )}{4 x^5 \log ^2(x)+4 x^3 \log ^2(x) \log \left (x^2\right )+x \log ^2(x) \log ^2\left (x^2\right )} \, dx=\frac {6\,\ln \left (x^2\right )-12\,\ln \left (x\right )+\frac {12\,\ln \left (x\right )\,\left (x\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+4\,x^3\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+8\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+4\,x^5+8\,x^7\right )}{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )+2\,x^2\right )\,\left (2\,x^3\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+x\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3+4\,x^5\right )}}{2\,{\ln \left (x\right )}^2+\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )+2\,x^2\right )} \]

[In]

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)

[Out]

(6*log(x^2) - 12*log(x) + (12*log(x)*(x*(log(x^2) - 2*log(x))^2 + 4*x^3*(log(x^2) - 2*log(x)) + 8*x^5*(log(x^2
) - 2*log(x)) + 2*x^3*(log(x^2) - 2*log(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))