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

Optimal. Leaf size=21 \[ \frac {6}{\log (x) \left (1+\frac {2 x^2}{\log \left (x^2\right )}\right )} \]

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Rubi [F]  time = 1.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 23, normalized size = 1.10 \begin {gather*} \frac {6 \log \left (x^2\right )}{2 x^2 \log (x)+\log (x) \log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.57, size = 10, normalized size = 0.48 \begin {gather*} \frac {6}{x^{2} + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.19, size = 10, normalized size = 0.48 \begin {gather*} \frac {6}{x^{2} + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 0.17, size = 115, normalized size = 5.48

method	result	size
risch	\(\frac {6 \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-12 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+6 \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+24 i \ln \relax (x )}{\left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i x^{2}+4 i \ln \relax (x )\right ) \ln \relax (x )}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[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*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+4*I*ln(x))/(Pi*csgn(I*x)^2*csgn(I*
x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+4*I*x^2+4*I*ln(x))/ln(x)

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maxima [A]  time = 0.43, size = 10, normalized size = 0.48 \begin {gather*} \frac {6}{x^{2} + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 2.46, size = 164, normalized size = 7.81 \begin {gather*} \frac {6\,\ln \left (x^2\right )-12\,\ln \relax (x)+\frac {12\,\ln \relax (x)\,\left (x\,{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2+4\,x^3\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )+8\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )+2\,x^3\,{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2+4\,x^5+8\,x^7\right )}{\left (\ln \left (x^2\right )-2\,\ln \relax (x)+2\,x^2\right )\,\left (2\,x^3\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )+x\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )+2\,x^3+4\,x^5\right )}}{2\,{\ln \relax (x)}^2+\ln \relax (x)\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)+2\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.24, size = 7, normalized size = 0.33 \begin {gather*} \frac {6}{x^{2} + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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