Integrand size = 64, antiderivative size = 21 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=\log \left (\frac {121 \log \left (\frac {1}{3} \left (-\frac {3}{x^2}-\log (x)\right )\right )}{x}\right ) \] Output:
ln(121/x*ln(-1/x^2-1/3*ln(x)))
Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log (x)+\log \left (\log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )\right ) \] Input:
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]
Output:
-Log[x] + Log[Log[-x^(-2) - Log[x]/3]]
Time = 0.98 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3041, 7242, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2+\left (x^2 (-\log (x))-3\right ) \log \left (\frac {x^2 (-\log (x))-3}{3 x^2}\right )-6}{\left (x^3 \log (x)+3 x\right ) \log \left (\frac {x^2 (-\log (x))-3}{3 x^2}\right )} \, dx\) |
\(\Big \downarrow \) 3041 |
\(\displaystyle \int \frac {x^2+\left (x^2 (-\log (x))-3\right ) \log \left (\frac {x^2 (-\log (x))-3}{3 x^2}\right )-6}{x^3 \left (\frac {3}{x^2}+\log (x)\right ) \log \left (\frac {x^2 (-\log (x))-3}{3 x^2}\right )}dx\) |
\(\Big \downarrow \) 7242 |
\(\displaystyle \int \frac {x^2+\left (x^2 (-\log (x))-3\right ) \log \left (\frac {x^2 (-\log (x))-3}{3 x^2}\right )-6}{x \left (x^2 \log (x)+3\right ) \log \left (\frac {x^2 (-\log (x))-3}{3 x^2}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2-6}{x \left (x^2 \log (x)+3\right ) \log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )}-\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (\log \left (-\frac {1}{x^2}-\frac {\log (x)}{3}\right )\right )-\log (x)\) |
Input:
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]
Output:
-Log[x] + Log[Log[-x^(-2) - Log[x]/3]]
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]
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]
Time = 0.36 (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 | \(-\ln \left (x \right )+\ln \left (\ln \left (x^{2} \ln \left (x \right )+3\right )-\frac {i \left (2 \pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )+3\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (x \right )+3\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (x \right )+3\right )}{x^{2}}\right )}^{3}-\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 \pi -2 i \ln \left (3\right )-4 i \ln \left (x \right )\right )}{2}\right )\) | \(215\) |
Input:
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=_RETURNVERBOSE)
Output:
ln(ln(-1/3*(x^2*ln(x)+3)/x^2))-ln(x)
Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (-\frac {x^{2} \log \left (x\right ) + 3}{3 \, x^{2}}\right )\right ) \] Input:
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")
Output:
-log(x) + log(log(-1/3*(x^2*log(x) + 3)/x^2))
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=- \log {\left (x \right )} + \log {\left (\log {\left (\frac {- \frac {x^{2} \log {\left (x \right )}}{3} - 1}{x^{2}} \right )} \right )} \] Input:
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)
Output:
-log(x) + log(log((-x**2*log(x)/3 - 1)/x**2))
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log \left (x\right ) + \log \left (-\log \left (3\right ) + \log \left (-x^{2} \log \left (x\right ) - 3\right ) - 2 \, \log \left (x\right )\right ) \] Input:
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")
Output:
-log(x) + log(-log(3) + log(-x^2*log(x) - 3) - 2*log(x))
Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=-\log \left (x\right ) + \log \left (-\log \left (3\right ) + \log \left (-x^{2} \log \left (x\right ) - 3\right ) - 2 \, \log \left (x\right )\right ) \] Input:
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")
Output:
-log(x) + log(-log(3) + log(-x^2*log(x) - 3) - 2*log(x))
Time = 2.88 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=\ln \left (\ln \left (-\frac {\frac {x^2\,\ln \left (x\right )}{3}+1}{x^2}\right )\right )-\ln \left (x\right ) \] Input:
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*log(x))),x)
Output:
log(log(-((x^2*log(x))/3 + 1)/x^2)) - log(x)
Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-6+x^2+\left (-3-x^2 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )}{\left (3 x+x^3 \log (x)\right ) \log \left (\frac {-3-x^2 \log (x)}{3 x^2}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {-\mathrm {log}\left (x \right ) x^{2}-3}{3 x^{2}}\right )\right )-\mathrm {log}\left (x \right ) \] Input:
int(((-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)
Output:
log(log(( - log(x)*x**2 - 3)/(3*x**2))) - log(x)