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

Optimal. Leaf size=21 \[ \log \left (\frac {121 \log \left (\frac {1}{3} \left (-\frac {3}{x^2}-\log (x)\right )\right )}{x}\right ) \]

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Rubi [A]  time = 0.75, antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2561, 6691, 6742, 6684} \begin {gather*} \log \left (\log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[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 2561

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 6684

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

Rule 6691

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 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 {-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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 19, normalized size = 0.90 \begin {gather*} -\log (x)+\log \left (\log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 0.62, size = 20, normalized size = 0.95 \begin {gather*} -\log \relax (x) + \log \left (\log \left (-\frac {x^{2} \log \relax (x) + 3}{3 \, x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.21, size = 25, normalized size = 1.19 \begin {gather*} -\log \relax (x) + \log \left (-\log \relax (3) + \log \left (-x^{2} \log \relax (x) - 3\right ) - 2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 0.10, size = 211, normalized size = 10.05

method	result	size
risch	\(-\ln \relax (x )+\ln \left (\ln \left (x^{2} \ln \relax (x )+3\right )+\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )+3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )+3\right )}{x^{2}}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )+3\right )}{x^{2}}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 \pi \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )+3\right )}{x^{2}}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )+3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )+3\right )}{x^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )+3\right )}{x^{2}}\right )^{3}+2 \pi +2 i \ln \relax (3)+4 i \ln \relax (x )\right )}{2}\right )\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[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(x)+ln(ln(x^2*ln(x)+3)+1/2*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I/x^2)*csgn(I
*(x^2*ln(x)+3))*csgn(I/x^2*(x^2*ln(x)+3))+Pi*csgn(I/x^2)*csgn(I/x^2*(x^2*ln(x)+3))^2+Pi*csgn(I*x^2)^3-2*Pi*csg
n(I/x^2*(x^2*ln(x)+3))^2+Pi*csgn(I*(x^2*ln(x)+3))*csgn(I/x^2*(x^2*ln(x)+3))^2+Pi*csgn(I/x^2*(x^2*ln(x)+3))^3+2
*Pi+2*I*ln(3)+4*I*ln(x)))

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maxima [A]  time = 0.48, size = 25, normalized size = 1.19 \begin {gather*} -\log \relax (x) + \log \left (-\log \relax (3) + \log \left (-x^{2} \log \relax (x) - 3\right ) - 2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 2.56, size = 21, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (-\frac {\frac {x^2\,\ln \relax (x)}{3}+1}{x^2}\right )\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.36, size = 20, normalized size = 0.95 \begin {gather*} - \log {\relax (x )} + \log {\left (\log {\left (\frac {- \frac {x^{2} \log {\relax (x )}}{3} - 1}{x^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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