3.9.83 \(\int \frac {10 \log (\frac {x^4}{18})-5 \log (x^2) \log (\log (x^2))+(20 \log (x^2)+5 \log (x^2) \log (\frac {x^4}{18})) \log (\log (x^2)) \log (\log (\log (x^2)))}{\log (x^2) \log (\log (x^2))} \, dx\)

Optimal. Leaf size=22 \[ 5 \left (-x+x \log \left (\frac {x^4}{18}\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )\right ) \]

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Rubi [F]  time = 0.26, 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 {10 \log \left (\frac {x^4}{18}\right )-5 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (20 \log \left (x^2\right )+5 \log \left (x^2\right ) \log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*Log[x^4/18] - 5*Log[x^2]*Log[Log[x^2]] + (20*Log[x^2] + 5*Log[x^2]*Log[x^4/18])*Log[Log[x^2]]*Log[Log[
Log[x^2]]])/(Log[x^2]*Log[Log[x^2]]),x]

[Out]

-5*x + 10*Defer[Int][Log[x^4/18]/(Log[x^2]*Log[Log[x^2]]), x] + 20*Defer[Int][Log[Log[Log[x^2]]], x] + 5*Defer
[Int][Log[x^4/18]*Log[Log[Log[x^2]]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 5 \left (-1+\frac {2 \log \left (\frac {x^4}{18}\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}+\left (4+\log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )\right ) \, dx\\ &=5 \int \left (-1+\frac {2 \log \left (\frac {x^4}{18}\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}+\left (4+\log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )\right ) \, dx\\ &=-5 x+5 \int \left (4+\log \left (\frac {x^4}{18}\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right ) \, dx+10 \int \frac {\log \left (\frac {x^4}{18}\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx\\ &=-5 x+5 \int \left (4 \log \left (\log \left (\log \left (x^2\right )\right )\right )+\log \left (\frac {x^4}{18}\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )\right ) \, dx+10 \int \frac {\log \left (\frac {x^4}{18}\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx\\ &=-5 x+5 \int \log \left (\frac {x^4}{18}\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right ) \, dx+10 \int \frac {\log \left (\frac {x^4}{18}\right )}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+20 \int \log \left (\log \left (\log \left (x^2\right )\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 21, normalized size = 0.95 \begin {gather*} -5 x+5 x \log \left (\frac {x^4}{18}\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*Log[x^4/18] - 5*Log[x^2]*Log[Log[x^2]] + (20*Log[x^2] + 5*Log[x^2]*Log[x^4/18])*Log[Log[x^2]]*Lo
g[Log[Log[x^2]]])/(Log[x^2]*Log[Log[x^2]]),x]

[Out]

-5*x + 5*x*Log[x^4/18]*Log[Log[Log[x^2]]]

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fricas [A]  time = 0.70, size = 24, normalized size = 1.09 \begin {gather*} -5 \, {\left (x \log \left (18\right ) - 2 \, x \log \left (x^{2}\right )\right )} \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*log(x^2)*log(1/18*x^4)+20*log(x^2))*log(log(x^2))*log(log(log(x^2)))-5*log(x^2)*log(log(x^2))+10
*log(1/18*x^4))/log(x^2)/log(log(x^2)),x, algorithm="fricas")

[Out]

-5*(x*log(18) - 2*x*log(x^2))*log(log(log(x^2))) - 5*x

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giac [A]  time = 0.90, size = 24, normalized size = 1.09 \begin {gather*} -5 \, {\left (x \log \left (18\right ) - 2 \, x \log \left (x^{2}\right )\right )} \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*log(x^2)*log(1/18*x^4)+20*log(x^2))*log(log(x^2))*log(log(log(x^2)))-5*log(x^2)*log(log(x^2))+10
*log(1/18*x^4))/log(x^2)/log(log(x^2)),x, algorithm="giac")

[Out]

-5*(x*log(18) - 2*x*log(x^2))*log(log(log(x^2))) - 5*x

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maple [C]  time = 0.16, size = 265, normalized size = 12.05

method	result	size
risch	\(\left (20 x \ln \relax (x )+\frac {5 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}}{2}-\frac {5 i \pi x \mathrm {csgn}\left (i x^{4}\right )^{3}}{2}-\frac {5 i \pi x \mathrm {csgn}\left (i x^{3}\right )^{3}}{2}-\frac {5 i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {5 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )}{2}-\frac {5 i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}+5 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\frac {5 i \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}+\frac {5 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}+\frac {5 i \pi x \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}}{2}-\frac {5 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )}{2}-5 x \ln \relax (2)-10 x \ln \relax (3)\right ) \ln \left (\ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )\right )-5 x\)	\(265\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*ln(x^2)*ln(1/18*x^4)+20*ln(x^2))*ln(ln(x^2))*ln(ln(ln(x^2)))-5*ln(x^2)*ln(ln(x^2))+10*ln(1/18*x^4))/ln
(x^2)/ln(ln(x^2)),x,method=_RETURNVERBOSE)

[Out]

(20*x*ln(x)+5/2*I*Pi*x*csgn(I*x)*csgn(I*x^4)^2-5/2*I*Pi*x*csgn(I*x^4)^3-5/2*I*Pi*x*csgn(I*x^3)^3-5/2*I*Pi*x*cs
gn(I*x^2)^3-5/2*I*Pi*x*csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)-5/2*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)+5*I*Pi*x*csgn(I*x)
*csgn(I*x^2)^2+5/2*I*Pi*x*csgn(I*x^2)*csgn(I*x^3)^2+5/2*I*Pi*x*csgn(I*x)*csgn(I*x^3)^2+5/2*I*Pi*x*csgn(I*x^3)*
csgn(I*x^4)^2-5/2*I*Pi*x*csgn(I*x)*csgn(I*x^2)*csgn(I*x^3)-5*x*ln(2)-10*x*ln(3))*ln(ln(2*ln(x)-1/2*I*Pi*csgn(I
*x^2)*(-csgn(I*x^2)+csgn(I*x))^2))-5*x

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maxima [A]  time = 0.93, size = 28, normalized size = 1.27 \begin {gather*} -5 \, {\left (x {\left (2 \, \log \relax (3) + \log \relax (2)\right )} - 4 \, x \log \relax (x)\right )} \log \left (\log \relax (2) + \log \left (\log \relax (x)\right )\right ) - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*log(x^2)*log(1/18*x^4)+20*log(x^2))*log(log(x^2))*log(log(log(x^2)))-5*log(x^2)*log(log(x^2))+10
*log(1/18*x^4))/log(x^2)/log(log(x^2)),x, algorithm="maxima")

[Out]

-5*(x*(2*log(3) + log(2)) - 4*x*log(x))*log(log(2) + log(log(x))) - 5*x

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mupad [B]  time = 0.82, size = 31, normalized size = 1.41 \begin {gather*} -5\,x-\ln \left (\ln \left (\ln \left (x^2\right )\right )\right )\,\left (20\,x+x\,\left (5\,\ln \left (18\right )-20\right )-10\,x\,\ln \left (x^2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*log(x^4/18) - 5*log(x^2)*log(log(x^2)) + log(log(log(x^2)))*log(log(x^2))*(20*log(x^2) + 5*log(x^2)*lo
g(x^4/18)))/(log(x^2)*log(log(x^2))),x)

[Out]

- 5*x - log(log(log(x^2)))*(20*x + x*(5*log(18) - 20) - 10*x*log(x^2))

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sympy [A]  time = 0.87, size = 26, normalized size = 1.18 \begin {gather*} - 5 x + \left (10 x \log {\left (x^{2} \right )} - 5 x \log {\left (18 \right )}\right ) \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*ln(x**2)*ln(1/18*x**4)+20*ln(x**2))*ln(ln(x**2))*ln(ln(ln(x**2)))-5*ln(x**2)*ln(ln(x**2))+10*ln(
1/18*x**4))/ln(x**2)/ln(ln(x**2)),x)

[Out]

-5*x + (10*x*log(x**2) - 5*x*log(18))*log(log(log(x**2)))

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