3.101.51 \(\int \frac {1+\log (x)+(x-x^2-x^3-4 x^4-x^5) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{(x^2-x^3-x^5) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx\)

Optimal. Leaf size=23 \[ x+\log \left (-x+x^2+x^4-\log (\log (\log (-x \log (x))))\right ) \]

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Rubi [A]  time = 1.72, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6688, 6742, 6684} \begin {gather*} \log \left (-x^4-x^2+x+\log (\log (\log (-x \log (x))))\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Log[x] + (x - x^2 - x^3 - 4*x^4 - x^5)*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]] + x*Log[x]*Log[-
(x*Log[x])]*Log[Log[-(x*Log[x])]]*Log[Log[Log[-(x*Log[x])]]])/((x^2 - x^3 - x^5)*Log[x]*Log[-(x*Log[x])]*Log[L
og[-(x*Log[x])]] + x*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]]*Log[Log[Log[-(x*Log[x])]]]),x]

[Out]

x + Log[x - x^2 - x^4 + Log[Log[Log[-(x*Log[x])]]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 23, normalized size = 1.00 \begin {gather*} x+\log \left (-x+x^2+x^4-\log (\log (\log (-x \log (x))))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Log[x] + (x - x^2 - x^3 - 4*x^4 - x^5)*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]] + x*Log[x]
*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]]*Log[Log[Log[-(x*Log[x])]]])/((x^2 - x^3 - x^5)*Log[x]*Log[-(x*Log[x])]
*Log[Log[-(x*Log[x])]] + x*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]]*Log[Log[Log[-(x*Log[x])]]]),x]

[Out]

x + Log[-x + x^2 + x^4 - Log[Log[Log[-(x*Log[x])]]]]

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fricas [A]  time = 0.88, size = 23, normalized size = 1.00 \begin {gather*} x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \left (-x \log \relax (x)\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x))))+(-x^5-4*x^4-x^3-x^2+x)*log(x)*
log(-x*log(x))*log(log(-x*log(x)))+log(x)+1)/(x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x
))))+(-x^5-x^3+x^2)*log(x)*log(-x*log(x))*log(log(-x*log(x)))),x, algorithm="fricas")

[Out]

x + log(-x^4 - x^2 + x + log(log(log(-x*log(x)))))

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giac [A]  time = 8.67, size = 25, normalized size = 1.09 \begin {gather*} x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \relax (x) + \log \left (-\log \relax (x)\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x))))+(-x^5-4*x^4-x^3-x^2+x)*log(x)*
log(-x*log(x))*log(log(-x*log(x)))+log(x)+1)/(x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x
))))+(-x^5-x^3+x^2)*log(x)*log(-x*log(x))*log(log(-x*log(x)))),x, algorithm="giac")

[Out]

x + log(-x^4 - x^2 + x + log(log(log(x) + log(-log(x)))))

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maple [C]  time = 0.16, size = 92, normalized size = 4.00

method	result	size
risch	\(x +\ln \left (-x^{4}-x^{2}+x +\ln \left (\ln \left (i \pi +\ln \relax (x )+\ln \left (\ln \relax (x )\right )-\frac {i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \left (-\mathrm {csgn}\left (i x \ln \relax (x )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \ln \relax (x )\right )+\mathrm {csgn}\left (i \ln \relax (x )\right )\right )}{2}+i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \left (\mathrm {csgn}\left (i x \ln \relax (x )\right )-1\right )\right )\right )\right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x^5-4*x^4-x^3-x^2+x)*ln(x)*ln(-x*ln(x))*ln(l
n(-x*ln(x)))+ln(x)+1)/(x*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x^5-x^3+x^2)*ln(x)*ln(-x*l
n(x))*ln(ln(-x*ln(x)))),x,method=_RETURNVERBOSE)

[Out]

x+ln(-x^4-x^2+x+ln(ln(I*Pi+ln(x)+ln(ln(x))-1/2*I*Pi*csgn(I*x*ln(x))*(-csgn(I*x*ln(x))+csgn(I*x))*(-csgn(I*x*ln
(x))+csgn(I*ln(x)))+I*Pi*csgn(I*x*ln(x))^2*(csgn(I*x*ln(x))-1))))

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maxima [A]  time = 0.44, size = 25, normalized size = 1.09 \begin {gather*} x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \relax (x) + \log \left (-\log \relax (x)\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x))))+(-x^5-4*x^4-x^3-x^2+x)*log(x)*
log(-x*log(x))*log(log(-x*log(x)))+log(x)+1)/(x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x
))))+(-x^5-x^3+x^2)*log(x)*log(-x*log(x))*log(log(-x*log(x)))),x, algorithm="maxima")

[Out]

x + log(-x^4 - x^2 + x + log(log(log(x) + log(-log(x)))))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {\ln \relax (x)-\ln \left (-x\,\ln \relax (x)\right )\,\ln \left (\ln \left (-x\,\ln \relax (x)\right )\right )\,\ln \relax (x)\,\left (x^5+4\,x^4+x^3+x^2-x\right )+x\,\ln \left (\ln \left (\ln \left (-x\,\ln \relax (x)\right )\right )\right )\,\ln \left (-x\,\ln \relax (x)\right )\,\ln \left (\ln \left (-x\,\ln \relax (x)\right )\right )\,\ln \relax (x)+1}{\ln \left (-x\,\ln \relax (x)\right )\,\ln \left (\ln \left (-x\,\ln \relax (x)\right )\right )\,\ln \relax (x)\,\left (x^5+x^3-x^2\right )-x\,\ln \left (\ln \left (\ln \left (-x\,\ln \relax (x)\right )\right )\right )\,\ln \left (-x\,\ln \relax (x)\right )\,\ln \left (\ln \left (-x\,\ln \relax (x)\right )\right )\,\ln \relax (x)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x) - log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^2 - x + x^3 + 4*x^4 + x^5) + x*log(log(log(-x*log(
x))))*log(-x*log(x))*log(log(-x*log(x)))*log(x) + 1)/(log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^3 - x^2 + x
^5) - x*log(log(log(-x*log(x))))*log(-x*log(x))*log(log(-x*log(x)))*log(x)),x)

[Out]

-int((log(x) - log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^2 - x + x^3 + 4*x^4 + x^5) + x*log(log(log(-x*log(
x))))*log(-x*log(x))*log(log(-x*log(x)))*log(x) + 1)/(log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^3 - x^2 + x
^5) - x*log(log(log(-x*log(x))))*log(-x*log(x))*log(log(-x*log(x)))*log(x)), x)

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sympy [A]  time = 1.29, size = 22, normalized size = 0.96 \begin {gather*} x + \log {\left (- x^{4} - x^{2} + x + \log {\left (\log {\left (\log {\left (- x \log {\relax (x )} \right )} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x**5-4*x**4-x**3-x**2+x)*ln(x)*ln(-x*l
n(x))*ln(ln(-x*ln(x)))+ln(x)+1)/(x*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x**5-x**3+x**2)*
ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))),x)

[Out]

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

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