Integrand size = 205, antiderivative size = 26 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\left (x^2-x \log (4)+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2 \] Output:
(x^2-2*x*ln(2)+ln(x/ln(3)+ln(ln(3/x))))^2
\[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx \] Input:
Integrate[(-2*x^2*Log[3] + 2*x*Log[3]*Log[4] + (2*x^3 + 4*x^5 + (-2*x^2 - 6*x^4)*Log[4] + 2*x^3*Log[4]^2)*Log[3/x] + (4*x^4*Log[3] - 6*x^3*Log[3]*Lo g[4] + 2*x^2*Log[3]*Log[4]^2)*Log[3/x]*Log[Log[3/x]] + (-2*Log[3] + (2*x + 4*x^3 - 2*x^2*Log[4])*Log[3/x] + (4*x^2*Log[3] - 2*x*Log[3]*Log[4])*Log[3 /x]*Log[Log[3/x]])*Log[(x + Log[3]*Log[Log[3/x]])/Log[3]])/(x^2*Log[3/x] + x*Log[3]*Log[3/x]*Log[Log[3/x]]),x]
Output:
Integrate[(-2*x^2*Log[3] + 2*x*Log[3]*Log[4] + (2*x^3 + 4*x^5 + (-2*x^2 - 6*x^4)*Log[4] + 2*x^3*Log[4]^2)*Log[3/x] + (4*x^4*Log[3] - 6*x^3*Log[3]*Lo g[4] + 2*x^2*Log[3]*Log[4]^2)*Log[3/x]*Log[Log[3/x]] + (-2*Log[3] + (2*x + 4*x^3 - 2*x^2*Log[4])*Log[3/x] + (4*x^2*Log[3] - 2*x*Log[3]*Log[4])*Log[3 /x]*Log[Log[3/x]])*Log[(x + Log[3]*Log[Log[3/x]])/Log[3]])/(x^2*Log[3/x] + x*Log[3]*Log[3/x]*Log[Log[3/x]]), x]
Time = 0.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {7239, 27, 25, 7237}
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.
| \(\displaystyle \int \frac {-2 x^2 \log (3)+\left (\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) \log \left (\frac {3}{x}\right )+\left (4 x^3-2 x^2 \log (4)+2 x\right ) \log \left (\frac {3}{x}\right )-2 \log (3)\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (4 x^5+2 x^3+2 x^3 \log ^2(4)+\left (-6 x^4-2 x^2\right ) \log (4)\right ) \log \left (\frac {3}{x}\right )+2 x \log (3) \log (4)}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (\log (3)-x \log \left (\frac {3}{x}\right ) \left (2 x^2-x \log (4)+(x \log (9)-\log (3) \log (4)) \log \left (\log \left (\frac {3}{x}\right )\right )+1\right )\right ) \left (-x (x-\log (4))-\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{x \log \left (\frac {3}{x}\right ) \left (x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\left (\log (3)-x \log \left (\frac {3}{x}\right ) \left (2 x^2-\log (4) x-(\log (3) \log (4)-x \log (9)) \log \left (\log \left (\frac {3}{x}\right )\right )+1\right )\right ) \left (x (x-\log (4))+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{x \log \left (\frac {3}{x}\right ) \left (x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\left (\log (3)-x \log \left (\frac {3}{x}\right ) \left (2 x^2-\log (4) x-(\log (3) \log (4)-x \log (9)) \log \left (\log \left (\frac {3}{x}\right )\right )+1\right )\right ) \left (x (x-\log (4))+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )}{x \log \left (\frac {3}{x}\right ) \left (x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \left (x (x-\log (4))+\log \left (\frac {x}{\log (3)}+\log \left (\log \left (\frac {3}{x}\right )\right )\right )\right )^2\) |
Input:
Int[(-2*x^2*Log[3] + 2*x*Log[3]*Log[4] + (2*x^3 + 4*x^5 + (-2*x^2 - 6*x^4) *Log[4] + 2*x^3*Log[4]^2)*Log[3/x] + (4*x^4*Log[3] - 6*x^3*Log[3]*Log[4] + 2*x^2*Log[3]*Log[4]^2)*Log[3/x]*Log[Log[3/x]] + (-2*Log[3] + (2*x + 4*x^3 - 2*x^2*Log[4])*Log[3/x] + (4*x^2*Log[3] - 2*x*Log[3]*Log[4])*Log[3/x]*Lo g[Log[3/x]])*Log[(x + Log[3]*Log[Log[3/x]])/Log[3]])/(x^2*Log[3/x] + x*Log [3]*Log[3/x]*Log[Log[3/x]]),x]
Output:
(x*(x - Log[4]) + Log[x/Log[3] + Log[Log[3/x]]])^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(26)=52\).
Time = 19.97 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.35
| method | result | size |
| parallelrisch | \(4 x^{2} \ln \left (2\right )^{2}-4 x^{3} \ln \left (2\right )+x^{4}-4 \ln \left (2\right ) x \ln \left (\frac {\ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x}{\ln \left (3\right )}\right )+2 \ln \left (\frac {\ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x}{\ln \left (3\right )}\right ) x^{2}+\ln \left (\frac {\ln \left (3\right ) \ln \left (\ln \left (\frac {3}{x}\right )\right )+x}{\ln \left (3\right )}\right )^{2}\) | \(87\) |
Input:
int((((-4*x*ln(2)*ln(3)+4*x^2*ln(3))*ln(3/x)*ln(ln(3/x))+(-4*x^2*ln(2)+4*x ^3+2*x)*ln(3/x)-2*ln(3))*ln((ln(3)*ln(ln(3/x))+x)/ln(3))+(8*x^2*ln(3)*ln(2 )^2-12*x^3*ln(3)*ln(2)+4*x^4*ln(3))*ln(3/x)*ln(ln(3/x))+(8*x^3*ln(2)^2+2*( -6*x^4-2*x^2)*ln(2)+4*x^5+2*x^3)*ln(3/x)+4*x*ln(2)*ln(3)-2*x^2*ln(3))/(x*l n(3)*ln(3/x)*ln(ln(3/x))+x^2*ln(3/x)),x,method=_RETURNVERBOSE)
Output:
4*x^2*ln(2)^2-4*x^3*ln(2)+x^4-4*ln(2)*x*ln((ln(3)*ln(ln(3/x))+x)/ln(3))+2* ln((ln(3)*ln(ln(3/x))+x)/ln(3))*x^2+ln((ln(3)*ln(ln(3/x))+x)/ln(3))^2
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.65 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{4} - 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2} + 2 \, {\left (x^{2} - 2 \, x \log \left (2\right )\right )} \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \left (3\right )}\right ) + \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \left (3\right )}\right )^{2} \] Input:
integrate((((-4*x*log(2)*log(3)+4*x^2*log(3))*log(3/x)*log(log(3/x))+(-4*x ^2*log(2)+4*x^3+2*x)*log(3/x)-2*log(3))*log((log(3)*log(log(3/x))+x)/log(3 ))+(8*x^2*log(3)*log(2)^2-12*x^3*log(3)*log(2)+4*x^4*log(3))*log(3/x)*log( log(3/x))+(8*x^3*log(2)^2+2*(-6*x^4-2*x^2)*log(2)+4*x^5+2*x^3)*log(3/x)+4* x*log(2)*log(3)-2*x^2*log(3))/(x*log(3)*log(3/x)*log(log(3/x))+x^2*log(3/x )),x, algorithm="fricas")
Output:
x^4 - 4*x^3*log(2) + 4*x^2*log(2)^2 + 2*(x^2 - 2*x*log(2))*log((log(3)*log (log(3/x)) + x)/log(3)) + log((log(3)*log(log(3/x)) + x)/log(3))^2
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{4} - 4 x^{3} \log {\left (2 \right )} + 4 x^{2} \log {\left (2 \right )}^{2} + \left (2 x^{2} - 4 x \log {\left (2 \right )}\right ) \log {\left (\frac {x + \log {\left (3 \right )} \log {\left (\log {\left (\frac {3}{x} \right )} \right )}}{\log {\left (3 \right )}} \right )} + \log {\left (\frac {x + \log {\left (3 \right )} \log {\left (\log {\left (\frac {3}{x} \right )} \right )}}{\log {\left (3 \right )}} \right )}^{2} \] Input:
integrate((((-4*x*ln(2)*ln(3)+4*x**2*ln(3))*ln(3/x)*ln(ln(3/x))+(-4*x**2*l n(2)+4*x**3+2*x)*ln(3/x)-2*ln(3))*ln((ln(3)*ln(ln(3/x))+x)/ln(3))+(8*x**2* ln(3)*ln(2)**2-12*x**3*ln(3)*ln(2)+4*x**4*ln(3))*ln(3/x)*ln(ln(3/x))+(8*x* *3*ln(2)**2+2*(-6*x**4-2*x**2)*ln(2)+4*x**5+2*x**3)*ln(3/x)+4*x*ln(2)*ln(3 )-2*x**2*ln(3))/(x*ln(3)*ln(3/x)*ln(ln(3/x))+x**2*ln(3/x)),x)
Output:
x**4 - 4*x**3*log(2) + 4*x**2*log(2)**2 + (2*x**2 - 4*x*log(2))*log((x + l og(3)*log(log(3/x)))/log(3)) + log((x + log(3)*log(log(3/x)))/log(3))**2
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{4} - 4 \, x^{3} \log \left (2\right ) + 2 \, {\left (2 \, \log \left (2\right )^{2} - \log \left (\log \left (3\right )\right )\right )} x^{2} + 4 \, x \log \left (2\right ) \log \left (\log \left (3\right )\right ) + 2 \, {\left (x^{2} - 2 \, x \log \left (2\right ) - \log \left (\log \left (3\right )\right )\right )} \log \left (\log \left (3\right ) \log \left (\log \left (3\right ) - \log \left (x\right )\right ) + x\right ) + \log \left (\log \left (3\right ) \log \left (\log \left (3\right ) - \log \left (x\right )\right ) + x\right )^{2} \] Input:
integrate((((-4*x*log(2)*log(3)+4*x^2*log(3))*log(3/x)*log(log(3/x))+(-4*x ^2*log(2)+4*x^3+2*x)*log(3/x)-2*log(3))*log((log(3)*log(log(3/x))+x)/log(3 ))+(8*x^2*log(3)*log(2)^2-12*x^3*log(3)*log(2)+4*x^4*log(3))*log(3/x)*log( log(3/x))+(8*x^3*log(2)^2+2*(-6*x^4-2*x^2)*log(2)+4*x^5+2*x^3)*log(3/x)+4* x*log(2)*log(3)-2*x^2*log(3))/(x*log(3)*log(3/x)*log(log(3/x))+x^2*log(3/x )),x, algorithm="maxima")
Output:
x^4 - 4*x^3*log(2) + 2*(2*log(2)^2 - log(log(3)))*x^2 + 4*x*log(2)*log(log (3)) + 2*(x^2 - 2*x*log(2) - log(log(3)))*log(log(3)*log(log(3) - log(x)) + x) + log(log(3)*log(log(3) - log(x)) + x)^2
\[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int { -\frac {2 \, {\left (x^{2} \log \left (3\right ) - 2 \, x \log \left (3\right ) \log \left (2\right ) - 2 \, {\left (x^{4} \log \left (3\right ) - 3 \, x^{3} \log \left (3\right ) \log \left (2\right ) + 2 \, x^{2} \log \left (3\right ) \log \left (2\right )^{2}\right )} \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - {\left (2 \, {\left (x^{2} \log \left (3\right ) - x \log \left (3\right ) \log \left (2\right )\right )} \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + {\left (2 \, x^{3} - 2 \, x^{2} \log \left (2\right ) + x\right )} \log \left (\frac {3}{x}\right ) - \log \left (3\right )\right )} \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x}{\log \left (3\right )}\right ) - {\left (2 \, x^{5} + 4 \, x^{3} \log \left (2\right )^{2} + x^{3} - 2 \, {\left (3 \, x^{4} + x^{2}\right )} \log \left (2\right )\right )} \log \left (\frac {3}{x}\right )\right )}}{x \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x^{2} \log \left (\frac {3}{x}\right )} \,d x } \] Input:
integrate((((-4*x*log(2)*log(3)+4*x^2*log(3))*log(3/x)*log(log(3/x))+(-4*x ^2*log(2)+4*x^3+2*x)*log(3/x)-2*log(3))*log((log(3)*log(log(3/x))+x)/log(3 ))+(8*x^2*log(3)*log(2)^2-12*x^3*log(3)*log(2)+4*x^4*log(3))*log(3/x)*log( log(3/x))+(8*x^3*log(2)^2+2*(-6*x^4-2*x^2)*log(2)+4*x^5+2*x^3)*log(3/x)+4* x*log(2)*log(3)-2*x^2*log(3))/(x*log(3)*log(3/x)*log(log(3/x))+x^2*log(3/x )),x, algorithm="giac")
Output:
integrate(-2*(x^2*log(3) - 2*x*log(3)*log(2) - 2*(x^4*log(3) - 3*x^3*log(3 )*log(2) + 2*x^2*log(3)*log(2)^2)*log(3/x)*log(log(3/x)) - (2*(x^2*log(3) - x*log(3)*log(2))*log(3/x)*log(log(3/x)) + (2*x^3 - 2*x^2*log(2) + x)*log (3/x) - log(3))*log((log(3)*log(log(3/x)) + x)/log(3)) - (2*x^5 + 4*x^3*lo g(2)^2 + x^3 - 2*(3*x^4 + x^2)*log(2))*log(3/x))/(x*log(3)*log(3/x)*log(lo g(3/x)) + x^2*log(3/x)), x)
Timed out. \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int \frac {\ln \left (\frac {3}{x}\right )\,\left (8\,x^3\,{\ln \left (2\right )}^2-2\,\ln \left (2\right )\,\left (6\,x^4+2\,x^2\right )+2\,x^3+4\,x^5\right )-2\,x^2\,\ln \left (3\right )+\ln \left (\frac {x+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (3\right )}{\ln \left (3\right )}\right )\,\left (\ln \left (\frac {3}{x}\right )\,\left (4\,x^3-4\,\ln \left (2\right )\,x^2+2\,x\right )-2\,\ln \left (3\right )+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (\frac {3}{x}\right )\,\left (4\,x^2\,\ln \left (3\right )-4\,x\,\ln \left (2\right )\,\ln \left (3\right )\right )\right )+\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (\frac {3}{x}\right )\,\left (4\,\ln \left (3\right )\,x^4-12\,\ln \left (2\right )\,\ln \left (3\right )\,x^3+8\,{\ln \left (2\right )}^2\,\ln \left (3\right )\,x^2\right )+4\,x\,\ln \left (2\right )\,\ln \left (3\right )}{x^2\,\ln \left (\frac {3}{x}\right )+x\,\ln \left (\ln \left (\frac {3}{x}\right )\right )\,\ln \left (3\right )\,\ln \left (\frac {3}{x}\right )} \,d x \] Input:
int((log(3/x)*(8*x^3*log(2)^2 - 2*log(2)*(2*x^2 + 6*x^4) + 2*x^3 + 4*x^5) - 2*x^2*log(3) + log((x + log(log(3/x))*log(3))/log(3))*(log(3/x)*(2*x - 4 *x^2*log(2) + 4*x^3) - 2*log(3) + log(log(3/x))*log(3/x)*(4*x^2*log(3) - 4 *x*log(2)*log(3))) + log(log(3/x))*log(3/x)*(4*x^4*log(3) - 12*x^3*log(2)* log(3) + 8*x^2*log(2)^2*log(3)) + 4*x*log(2)*log(3))/(x^2*log(3/x) + x*log (log(3/x))*log(3)*log(3/x)),x)
Output:
int((log(3/x)*(8*x^3*log(2)^2 - 2*log(2)*(2*x^2 + 6*x^4) + 2*x^3 + 4*x^5) - 2*x^2*log(3) + log((x + log(log(3/x))*log(3))/log(3))*(log(3/x)*(2*x - 4 *x^2*log(2) + 4*x^3) - 2*log(3) + log(log(3/x))*log(3/x)*(4*x^2*log(3) - 4 *x*log(2)*log(3))) + log(log(3/x))*log(3/x)*(4*x^4*log(3) - 12*x^3*log(2)* log(3) + 8*x^2*log(2)^2*log(3)) + 4*x*log(2)*log(3))/(x^2*log(3/x) + x*log (log(3/x))*log(3)*log(3/x)), x)
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.31 \[ \int \frac {-2 x^2 \log (3)+2 x \log (3) \log (4)+\left (2 x^3+4 x^5+\left (-2 x^2-6 x^4\right ) \log (4)+2 x^3 \log ^2(4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^4 \log (3)-6 x^3 \log (3) \log (4)+2 x^2 \log (3) \log ^2(4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )+\left (-2 \log (3)+\left (2 x+4 x^3-2 x^2 \log (4)\right ) \log \left (\frac {3}{x}\right )+\left (4 x^2 \log (3)-2 x \log (3) \log (4)\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )\right ) \log \left (\frac {x+\log (3) \log \left (\log \left (\frac {3}{x}\right )\right )}{\log (3)}\right )}{x^2 \log \left (\frac {3}{x}\right )+x \log (3) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \, dx=\mathrm {log}\left (\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {3}{x}\right )\right ) \mathrm {log}\left (3\right )+x}{\mathrm {log}\left (3\right )}\right )^{2}-4 \,\mathrm {log}\left (\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {3}{x}\right )\right ) \mathrm {log}\left (3\right )+x}{\mathrm {log}\left (3\right )}\right ) \mathrm {log}\left (2\right ) x +2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {3}{x}\right )\right ) \mathrm {log}\left (3\right )+x}{\mathrm {log}\left (3\right )}\right ) x^{2}+4 \mathrm {log}\left (2\right )^{2} x^{2}-4 \,\mathrm {log}\left (2\right ) x^{3}+x^{4} \] Input:
int((((-4*x*log(2)*log(3)+4*x^2*log(3))*log(3/x)*log(log(3/x))+(-4*x^2*log (2)+4*x^3+2*x)*log(3/x)-2*log(3))*log((log(3)*log(log(3/x))+x)/log(3))+(8* x^2*log(3)*log(2)^2-12*x^3*log(3)*log(2)+4*x^4*log(3))*log(3/x)*log(log(3/ x))+(8*x^3*log(2)^2+2*(-6*x^4-2*x^2)*log(2)+4*x^5+2*x^3)*log(3/x)+4*x*log( 2)*log(3)-2*x^2*log(3))/(x*log(3)*log(3/x)*log(log(3/x))+x^2*log(3/x)),x)
Output:
log((log(log(3/x))*log(3) + x)/log(3))**2 - 4*log((log(log(3/x))*log(3) + x)/log(3))*log(2)*x + 2*log((log(log(3/x))*log(3) + x)/log(3))*x**2 + 4*lo g(2)**2*x**2 - 4*log(2)*x**3 + x**4