3.63.34 \(\int \frac {-2 x-x^3+(2 x+3 x^2) \log (x)-3 x \log ^2(x)+\log ^3(x)}{(-x^3-x^4+(x^2+3 x^3) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)) \log (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}) \log (\log (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}))} \, dx\)

Optimal. Leaf size=17 \[ \log \left (\log \left (\log \left (x+\frac {x^2}{(-x+\log (x))^2}\right )\right )\right ) \]

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Rubi [F]  time = 4.03, 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 {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x - x^3 + (2*x + 3*x^2)*Log[x] - 3*x*Log[x]^2 + Log[x]^3)/((-x^3 - x^4 + (x^2 + 3*x^3)*Log[x] - 3*x^2*
Log[x]^2 + x*Log[x]^3)*Log[(x^2 + x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]*Log[Log[(x^2
+ x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]]),x]

[Out]

2*Defer[Int][1/((x^2 + x^3 - x*Log[x] - 3*x^2*Log[x] + 3*x*Log[x]^2 - Log[x]^3)*Log[(x*(x + x^2 - 2*x*Log[x] +
Log[x]^2))/(x - Log[x])^2]*Log[Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2]]), x] + Defer[Int][x
^2/((x^2 + x^3 - x*Log[x] - 3*x^2*Log[x] + 3*x*Log[x]^2 - Log[x]^3)*Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/
(x - Log[x])^2]*Log[Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2]]), x] - 3*Defer[Int][(x*Log[x])/
((x^2 + x^3 - x*Log[x] - 3*x^2*Log[x] + 3*x*Log[x]^2 - Log[x]^3)*Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x
- Log[x])^2]*Log[Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2]]), x] - Defer[Int][Log[x]^3/(x*(x^2
+ x^3 - x*Log[x] - 3*x^2*Log[x] + 3*x*Log[x]^2 - Log[x]^3)*Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log
[x])^2]*Log[Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2]]), x] + 2*Defer[Int][Log[x]/((-x^2 - x^3
+ x*Log[x] + 3*x^2*Log[x] - 3*x*Log[x]^2 + Log[x]^3)*Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2
]*Log[Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2]]), x] - 3*Defer[Int][Log[x]^2/((-x^2 - x^3 + x
*Log[x] + 3*x^2*Log[x] - 3*x*Log[x]^2 + Log[x]^3)*Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2]*Lo
g[Log[(x*(x + x^2 - 2*x*Log[x] + Log[x]^2))/(x - Log[x])^2]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x+x^3-\left (2 x+3 x^2\right ) \log (x)+3 x \log ^2(x)-\log ^3(x)}{\left (x^3+x^4-\left (x^2+3 x^3\right ) \log (x)+3 x^2 \log ^2(x)-x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{(x-\log (x))^2}\right )\right )} \, dx\\ &=\int \left (\frac {2}{\left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}+\frac {x^2}{\left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}-\frac {3 x \log (x)}{\left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}-\frac {\log ^3(x)}{x \left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}+\frac {2 \log (x)}{\left (-x^2-x^3+x \log (x)+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}-\frac {3 \log ^2(x)}{\left (-x^2-x^3+x \log (x)+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )}\right ) \, dx\\ &=2 \int \frac {1}{\left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )} \, dx+2 \int \frac {\log (x)}{\left (-x^2-x^3+x \log (x)+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )} \, dx-3 \int \frac {x \log (x)}{\left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )} \, dx-3 \int \frac {\log ^2(x)}{\left (-x^2-x^3+x \log (x)+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )} \, dx+\int \frac {x^2}{\left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )} \, dx-\int \frac {\log ^3(x)}{x \left (x^2+x^3-x \log (x)-3 x^2 \log (x)+3 x \log ^2(x)-\log ^3(x)\right ) \log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right ) \log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 27, normalized size = 1.59 \begin {gather*} \log \left (\log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - x^3 + (2*x + 3*x^2)*Log[x] - 3*x*Log[x]^2 + Log[x]^3)/((-x^3 - x^4 + (x^2 + 3*x^3)*Log[x] -
3*x^2*Log[x]^2 + x*Log[x]^3)*Log[(x^2 + x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]*Log[Lo
g[(x^2 + x^3 - 2*x^2*Log[x] + x*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2)]]),x]

[Out]

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

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fricas [B]  time = 0.58, size = 39, normalized size = 2.29 \begin {gather*} \log \left (\log \left (\log \left (\frac {x^{3} - 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + x^{2}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^3-3*x*log(x)^2+(3*x^2+2*x)*log(x)-x^3-2*x)/(x*log(x)^3-3*x^2*log(x)^2+(3*x^3+x^2)*log(x)-x^4
-x^3)/log((x*log(x)^2-2*x^2*log(x)+x^3+x^2)/(log(x)^2-2*x*log(x)+x^2))/log(log((x*log(x)^2-2*x^2*log(x)+x^3+x^
2)/(log(x)^2-2*x*log(x)+x^2))),x, algorithm="fricas")

[Out]

log(log(log((x^3 - 2*x^2*log(x) + x*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + log(x)^2))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} + 3 \, x \log \relax (x)^{2} - \log \relax (x)^{3} - {\left (3 \, x^{2} + 2 \, x\right )} \log \relax (x) + 2 \, x}{{\left (x^{4} + 3 \, x^{2} \log \relax (x)^{2} - x \log \relax (x)^{3} + x^{3} - {\left (3 \, x^{3} + x^{2}\right )} \log \relax (x)\right )} \log \left (\frac {x^{3} - 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + x^{2}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}}\right ) \log \left (\log \left (\frac {x^{3} - 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + x^{2}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}}\right )\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^3-3*x*log(x)^2+(3*x^2+2*x)*log(x)-x^3-2*x)/(x*log(x)^3-3*x^2*log(x)^2+(3*x^3+x^2)*log(x)-x^4
-x^3)/log((x*log(x)^2-2*x^2*log(x)+x^3+x^2)/(log(x)^2-2*x*log(x)+x^2))/log(log((x*log(x)^2-2*x^2*log(x)+x^3+x^
2)/(log(x)^2-2*x*log(x)+x^2))),x, algorithm="giac")

[Out]

integrate((x^3 + 3*x*log(x)^2 - log(x)^3 - (3*x^2 + 2*x)*log(x) + 2*x)/((x^4 + 3*x^2*log(x)^2 - x*log(x)^3 + x
^3 - (3*x^3 + x^2)*log(x))*log((x^3 - 2*x^2*log(x) + x*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + log(x)^2))*log(log(
(x^3 - 2*x^2*log(x) + x*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + log(x)^2)))), x)

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maple [C]  time = 0.70, size = 333, normalized size = 19.59

method	result	size
risch	\(\ln \left (\ln \left (\ln \relax (x )-2 \ln \left (x -\ln \relax (x )\right )+\ln \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )+\frac {i \pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )-x \right )^{2}\right ) \left (-\mathrm {csgn}\left (i \left (\ln \relax (x )-x \right )^{2}\right )-\mathrm {csgn}\left (i \left (\ln \relax (x )-x \right )\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )}{\left (\ln \relax (x )-x \right )^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )}{\left (\ln \relax (x )-x \right )^{2}}\right )+\mathrm {csgn}\left (\frac {i}{\left (\ln \relax (x )-x \right )^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )}{\left (\ln \relax (x )-x \right )^{2}}\right )+\mathrm {csgn}\left (i \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )}{\left (\ln \relax (x )-x \right )^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )}{\left (\ln \relax (x )-x \right )^{2}}\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )}{\left (\ln \relax (x )-x \right )^{2}}\right )+\mathrm {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \relax (x )+1\right ) x +\ln \relax (x )^{2}\right )}{\left (\ln \relax (x )-x \right )^{2}}\right )\right )}{2}\right )\right )\)	\(333\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)^3-3*x*ln(x)^2+(3*x^2+2*x)*ln(x)-x^3-2*x)/(x*ln(x)^3-3*x^2*ln(x)^2+(3*x^3+x^2)*ln(x)-x^4-x^3)/ln((x*
ln(x)^2-2*x^2*ln(x)+x^3+x^2)/(ln(x)^2-2*x*ln(x)+x^2))/ln(ln((x*ln(x)^2-2*x^2*ln(x)+x^3+x^2)/(ln(x)^2-2*x*ln(x)
+x^2))),x,method=_RETURNVERBOSE)

[Out]

ln(ln(ln(x)-2*ln(x-ln(x))+ln(x^2+(-2*ln(x)+1)*x+ln(x)^2)+1/2*I*Pi*csgn(I*(ln(x)-x)^2)*(-csgn(I*(ln(x)-x)^2)-cs
gn(I*(ln(x)-x)))^2-1/2*I*Pi*csgn(I/(ln(x)-x)^2*(x^2+(-2*ln(x)+1)*x+ln(x)^2))*(-csgn(I/(ln(x)-x)^2*(x^2+(-2*ln(
x)+1)*x+ln(x)^2))+csgn(I/(ln(x)-x)^2))*(-csgn(I/(ln(x)-x)^2*(x^2+(-2*ln(x)+1)*x+ln(x)^2))+csgn(I*(x^2+(-2*ln(x
)+1)*x+ln(x)^2)))-1/2*I*Pi*csgn(I*x*(x^2+(-2*ln(x)+1)*x+ln(x)^2)/(ln(x)-x)^2)*(-csgn(I*x*(x^2+(-2*ln(x)+1)*x+l
n(x)^2)/(ln(x)-x)^2)+csgn(I*x))*(-csgn(I*x*(x^2+(-2*ln(x)+1)*x+ln(x)^2)/(ln(x)-x)^2)+csgn(I/(ln(x)-x)^2*(x^2+(
-2*ln(x)+1)*x+ln(x)^2)))))

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maxima [A]  time = 0.52, size = 29, normalized size = 1.71 \begin {gather*} \log \left (\log \left (\log \left (x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2} + x\right ) + \log \relax (x) - 2 \, \log \left (-x + \log \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^3-3*x*log(x)^2+(3*x^2+2*x)*log(x)-x^3-2*x)/(x*log(x)^3-3*x^2*log(x)^2+(3*x^3+x^2)*log(x)-x^4
-x^3)/log((x*log(x)^2-2*x^2*log(x)+x^3+x^2)/(log(x)^2-2*x*log(x)+x^2))/log(log((x*log(x)^2-2*x^2*log(x)+x^3+x^
2)/(log(x)^2-2*x*log(x)+x^2))),x, algorithm="maxima")

[Out]

log(log(log(x^2 - 2*x*log(x) + log(x)^2 + x) + log(x) - 2*log(-x + log(x))))

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mupad [B]  time = 5.70, size = 24, normalized size = 1.41 \begin {gather*} \ln \left (\ln \left (\ln \left (x+\frac {x^2}{x^2-2\,x\,\ln \relax (x)+{\ln \relax (x)}^2}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 3*x*log(x)^2 - log(x)^3 - log(x)*(2*x + 3*x^2) + x^3)/(log(log((x*log(x)^2 - 2*x^2*log(x) + x^2 + x
^3)/(log(x)^2 - 2*x*log(x) + x^2)))*log((x*log(x)^2 - 2*x^2*log(x) + x^2 + x^3)/(log(x)^2 - 2*x*log(x) + x^2))
*(3*x^2*log(x)^2 - x*log(x)^3 - log(x)*(x^2 + 3*x^3) + x^3 + x^4)),x)

[Out]

log(log(log(x + x^2/(log(x)^2 - 2*x*log(x) + x^2))))

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sympy [B]  time = 6.17, size = 41, normalized size = 2.41 \begin {gather*} \log {\left (\log {\left (\log {\left (\frac {x^{3} - 2 x^{2} \log {\relax (x )} + x^{2} + x \log {\relax (x )}^{2}}{x^{2} - 2 x \log {\relax (x )} + \log {\relax (x )}^{2}} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)**3-3*x*ln(x)**2+(3*x**2+2*x)*ln(x)-x**3-2*x)/(x*ln(x)**3-3*x**2*ln(x)**2+(3*x**3+x**2)*ln(x)-
x**4-x**3)/ln((x*ln(x)**2-2*x**2*ln(x)+x**3+x**2)/(ln(x)**2-2*x*ln(x)+x**2))/ln(ln((x*ln(x)**2-2*x**2*ln(x)+x*
*3+x**2)/(ln(x)**2-2*x*ln(x)+x**2))),x)

[Out]

log(log(log((x**3 - 2*x**2*log(x) + x**2 + x*log(x)**2)/(x**2 - 2*x*log(x) + log(x)**2))))

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