\(\int \frac {1+\log (x)+(x-x^2-x^3-4 x^4-x^5) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{(x^2-x^3-x^5) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx\) [10051]

Optimal result

Integrand size = 126, antiderivative size = 23 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x+\log \left (-x+x^2+x^4-\log (\log (\log (-x \log (x))))\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6820, 6874, 6816} \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=\log \left (-x^4-x^2+x+\log (\log (\log (-x \log (x))))\right )+x \]

[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 6816

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

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x+\log \left (-x+x^2+x^4-\log (\log (\log (-x \log (x))))\right ) \]

[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])]]]]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.01 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00

\[x +\ln \left (-x^{4}-x^{2}+x +\ln \left (\ln \left (i \pi +\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )-\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}{2}+i \pi \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2} \left (\operatorname {csgn}\left (i x \ln \left (x \right )\right )-1\right )\right )\right )\right )\]

[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)

[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))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \left (-x \log \left (x\right )\right )\right )\right )\right ) \]

[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)))))

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log {\left (- x^{4} - x^{2} + x + \log {\left (\log {\left (\log {\left (- x \log {\left (x \right )} \right )} \right )} \right )} \right )} \]

[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)))))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \left (x\right ) + \log \left (-\log \left (x\right )\right )\right )\right )\right ) \]

[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)))))

Giac [A] (verification not implemented)

none

Time = 5.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \left (x\right ) + \log \left (-\log \left (x\right )\right )\right )\right )\right ) \]

[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)))))

Mupad [F(-1)]

Timed out. \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=-\int \frac {\ln \left (x\right )-\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )\,\left (x^5+4\,x^4+x^3+x^2-x\right )+x\,\ln \left (\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\right )\,\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )+1}{\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )\,\left (x^5+x^3-x^2\right )-x\,\ln \left (\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\right )\,\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )} \,d x \]

[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)