\(\int \frac {-6+x^2+(-3-x^2 \log (x)) \log (\frac {-3-x^2 \log (x)}{3 x^2})}{(3 x+x^3 \log (x)) \log (\frac {-3-x^2 \log (x)}{3 x^2})} \, dx\) [3072]

Optimal result

Integrand size = 64, antiderivative size = 21 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=\log \left (\frac {121 \log \left (\frac {1}{3} \left (-\frac {3}{x^2}-\log (x)\right )\right )}{x}\right ) \]

[Out]

ln(121/x*ln(-1/x^2-1/3*ln(x)))

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2641, 6823, 6874, 6816} \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=\log \left (\log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )\right )-\log (x) \]

[In]

Int[(-6 + x^2 + (-3 - x^2*Log[x])*Log[(-3 - x^2*Log[x])/(3*x^2)])/((3*x + x^3*Log[x])*Log[(-3 - x^2*Log[x])/(3
*x^2)]),x]

[Out]

-Log[x] + Log[Log[-x^(-2) - Log[x]/3]]

Rule 2641

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6816

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

Rule 6823

Int[(u_)^(m_.)*((a_.)*(u_)^(n_) + (v_))^(p_.)*(w_), x_Symbol] :> Int[u^(m + n*p)*(a + v/u^n)^p*w, x] /; FreeQ[
{a, m, n}, x] && IntegerQ[p] &&  !GtQ[n, 0] &&  !FreeQ[v, 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 {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{x^3 \left (\frac {3}{x^2}+\log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx \\ & = \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{x \left (3+x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx \\ & = \int \left (-\frac {1}{x}+\frac {-6+x^2}{x \left (3+x^2 \log (x)\right ) \log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )}\right ) \, dx \\ & = -\log (x)+\int \frac {-6+x^2}{x \left (3+x^2 \log (x)\right ) \log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )} \, dx \\ & = -\log (x)+\log \left (\log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log (x)+\log \left (\log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )\right ) \]

[In]

Integrate[(-6 + x^2 + (-3 - x^2*Log[x])*Log[(-3 - x^2*Log[x])/(3*x^2)])/((3*x + x^3*Log[x])*Log[(-3 - x^2*Log[
x])/(3*x^2)]),x]

[Out]

-Log[x] + Log[Log[-x^(-2) - Log[x]/3]]

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\ln \left (\ln \left (-\frac {x^{2} \ln \left (x \right )+3}{3 x^{2}}\right )\right )-\ln \left (x \right )\)	\(21\)
default	\(-\ln \left (x \right )+\ln \left (\ln \left (3\right )-\ln \left (-\frac {x^{2} \ln \left (x \right )+3}{x^{2}}\right )\right )\)	\(26\)
risch	\(-\ln \left (x \right )+\ln \left (\ln \left (x^{2} \ln \left (x \right )+3\right )+\frac {i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 \pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )+3\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )+3\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right )+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{3}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x^{2}}\right )+2 \pi +2 i \ln \left (3\right )+4 i \ln \left (x \right )\right )}{2}\right )\)	\(211\)

[In]

int(((-x^2*ln(x)-3)*ln(1/3*(-x^2*ln(x)-3)/x^2)+x^2-6)/(x^3*ln(x)+3*x)/ln(1/3*(-x^2*ln(x)-3)/x^2),x,method=_RET
URNVERBOSE)

[Out]

ln(ln(-1/3*(x^2*ln(x)+3)/x^2))-ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (-\frac {x^{2} \log \left (x\right ) + 3}{3 \, x^{2}}\right )\right ) \]

[In]

integrate(((-x^2*log(x)-3)*log(1/3*(-x^2*log(x)-3)/x^2)+x^2-6)/(x^3*log(x)+3*x)/log(1/3*(-x^2*log(x)-3)/x^2),x
, algorithm="fricas")

[Out]

-log(x) + log(log(-1/3*(x^2*log(x) + 3)/x^2))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=- \log {\left (x \right )} + \log {\left (\log {\left (\frac {- \frac {x^{2} \log {\left (x \right )}}{3} - 1}{x^{2}} \right )} \right )} \]

[In]

integrate(((-x**2*ln(x)-3)*ln(1/3*(-x**2*ln(x)-3)/x**2)+x**2-6)/(x**3*ln(x)+3*x)/ln(1/3*(-x**2*ln(x)-3)/x**2),
x)

[Out]

-log(x) + log(log((-x**2*log(x)/3 - 1)/x**2))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log \left (x\right ) + \log \left (-\log \left (3\right ) + \log \left (-x^{2} \log \left (x\right ) - 3\right ) - 2 \, \log \left (x\right )\right ) \]

[In]

integrate(((-x^2*log(x)-3)*log(1/3*(-x^2*log(x)-3)/x^2)+x^2-6)/(x^3*log(x)+3*x)/log(1/3*(-x^2*log(x)-3)/x^2),x
, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log \left (x\right ) + \log \left (-\log \left (3\right ) + \log \left (-x^{2} \log \left (x\right ) - 3\right ) - 2 \, \log \left (x\right )\right ) \]

[In]

integrate(((-x^2*log(x)-3)*log(1/3*(-x^2*log(x)-3)/x^2)+x^2-6)/(x^3*log(x)+3*x)/log(1/3*(-x^2*log(x)-3)/x^2),x
, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.44 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=\ln \left (\ln \left (-\frac {\frac {x^2\,\ln \left (x\right )}{3}+1}{x^2}\right )\right )-\ln \left (x\right ) \]

[In]

int(-(log(-((x^2*log(x))/3 + 1)/x^2)*(x^2*log(x) + 3) - x^2 + 6)/(log(-((x^2*log(x))/3 + 1)/x^2)*(3*x + x^3*lo
g(x))),x)

[Out]

log(log(-((x^2*log(x))/3 + 1)/x^2)) - log(x)