Integrand size = 100, antiderivative size = 19 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{16-\log (x)+\log \left (3+\frac {3}{\log (x)}\right )} \] Output:
2/(16-ln(x)+ln(3+3/ln(x)))
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{16-\log (x)+\log \left (3+\frac {3}{\log (x)}\right )} \] Input:
Integrate[(2 + 2*Log[x] + 2*Log[x]^2)/(256*x*Log[x] + 224*x*Log[x]^2 - 31* x*Log[x]^3 + x*Log[x]^4 + (32*x*Log[x] + 30*x*Log[x]^2 - 2*x*Log[x]^3)*Log [(3 + 3*Log[x])/Log[x]] + (x*Log[x] + x*Log[x]^2)*Log[(3 + 3*Log[x])/Log[x ]]^2),x]
Output:
2/(16 - Log[x] + Log[3 + 3/Log[x]])
Time = 0.44 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {3039, 27, 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.
\(\displaystyle \int \frac {2 \log ^2(x)+2 \log (x)+2}{x \log ^4(x)-31 x \log ^3(x)+224 x \log ^2(x)+\left (x \log ^2(x)+x \log (x)\right ) \log ^2\left (\frac {3 \log (x)+3}{\log (x)}\right )+\left (-2 x \log ^3(x)+30 x \log ^2(x)+32 x \log (x)\right ) \log \left (\frac {3 \log (x)+3}{\log (x)}\right )+256 x \log (x)} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \int \frac {2 \left (\log ^2(x)+\log (x)+1\right )}{\log (x) (\log (x)+1) \left (-\log (x)+\log \left (\frac {3 (\log (x)+1)}{\log (x)}\right )+16\right )^2}d\log (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\log ^2(x)+\log (x)+1}{\log (x) (\log (x)+1) \left (-\log (x)+\log \left (\frac {3 (\log (x)+1)}{\log (x)}\right )+16\right )^2}d\log (x)\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {2}{-\log (x)+\log \left (\frac {3 (\log (x)+1)}{\log (x)}\right )+16}\) |
Input:
Int[(2 + 2*Log[x] + 2*Log[x]^2)/(256*x*Log[x] + 224*x*Log[x]^2 - 31*x*Log[ x]^3 + x*Log[x]^4 + (32*x*Log[x] + 30*x*Log[x]^2 - 2*x*Log[x]^3)*Log[(3 + 3*Log[x])/Log[x]] + (x*Log[x] + x*Log[x]^2)*Log[(3 + 3*Log[x])/Log[x]]^2), x]
Output:
2/(16 - Log[x] + Log[(3*(1 + Log[x]))/Log[x]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
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]
Time = 2.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
parallelrisch | \(-\frac {2}{-16+\ln \left (x \right )-\ln \left (\frac {3 \ln \left (x \right )+3}{\ln \left (x \right )}\right )}\) | \(22\) |
default | \(\frac {2}{\ln \left (x \right ) \left (\ln \left (3\right ) \left (1+\frac {1}{\ln \left (x \right )}\right )+\ln \left (1+\frac {1}{\ln \left (x \right )}\right ) \left (1+\frac {1}{\ln \left (x \right )}\right )-\ln \left (3\right )-1+\frac {16}{\ln \left (x \right )}-\ln \left (1+\frac {1}{\ln \left (x \right )}\right )\right )}\) | \(53\) |
risch | \(\frac {4}{32-i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{3}+2 \ln \left (3\right )-2 \ln \left (x \right )-2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (\ln \left (x \right )+1\right )}\) | \(129\) |
Input:
int((2*ln(x)^2+2*ln(x)+2)/((x*ln(x)^2+x*ln(x))*ln((3*ln(x)+3)/ln(x))^2+(-2 *x*ln(x)^3+30*x*ln(x)^2+32*x*ln(x))*ln((3*ln(x)+3)/ln(x))+x*ln(x)^4-31*x*l n(x)^3+224*x*ln(x)^2+256*x*ln(x)),x,method=_RETURNVERBOSE)
Output:
-2/(ln(x)-ln(3*(ln(x)+1)/ln(x))-16)
Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=-\frac {2}{\log \left (x\right ) - \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right ) - 16} \] Input:
integrate((2*log(x)^2+2*log(x)+2)/((x*log(x)^2+x*log(x))*log((3*log(x)+3)/ log(x))^2+(-2*x*log(x)^3+30*x*log(x)^2+32*x*log(x))*log((3*log(x)+3)/log(x ))+x*log(x)^4-31*x*log(x)^3+224*x*log(x)^2+256*x*log(x)),x, algorithm="fri cas")
Output:
-2/(log(x) - log(3*(log(x) + 1)/log(x)) - 16)
Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{- \log {\left (x \right )} + \log {\left (\frac {3 \log {\left (x \right )} + 3}{\log {\left (x \right )}} \right )} + 16} \] Input:
integrate((2*ln(x)**2+2*ln(x)+2)/((x*ln(x)**2+x*ln(x))*ln((3*ln(x)+3)/ln(x ))**2+(-2*x*ln(x)**3+30*x*ln(x)**2+32*x*ln(x))*ln((3*ln(x)+3)/ln(x))+x*ln( x)**4-31*x*ln(x)**3+224*x*ln(x)**2+256*x*ln(x)),x)
Output:
2/(-log(x) + log((3*log(x) + 3)/log(x)) + 16)
Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{\log \left (3\right ) - \log \left (x\right ) + \log \left (\log \left (x\right ) + 1\right ) - \log \left (\log \left (x\right )\right ) + 16} \] Input:
integrate((2*log(x)^2+2*log(x)+2)/((x*log(x)^2+x*log(x))*log((3*log(x)+3)/ log(x))^2+(-2*x*log(x)^3+30*x*log(x)^2+32*x*log(x))*log((3*log(x)+3)/log(x ))+x*log(x)^4-31*x*log(x)^3+224*x*log(x)^2+256*x*log(x)),x, algorithm="max ima")
Output:
2/(log(3) - log(x) + log(log(x) + 1) - log(log(x)) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2 \, {\left (\frac {\log \left (x\right ) + 1}{\log \left (x\right )} - 1\right )}}{\frac {{\left (\log \left (x\right ) + 1\right )} \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right )}{\log \left (x\right )} + \frac {16 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )} - \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right ) - 17} \] Input:
integrate((2*log(x)^2+2*log(x)+2)/((x*log(x)^2+x*log(x))*log((3*log(x)+3)/ log(x))^2+(-2*x*log(x)^3+30*x*log(x)^2+32*x*log(x))*log((3*log(x)+3)/log(x ))+x*log(x)^4-31*x*log(x)^3+224*x*log(x)^2+256*x*log(x)),x, algorithm="gia c")
Output:
2*((log(x) + 1)/log(x) - 1)/((log(x) + 1)*log(3*(log(x) + 1)/log(x))/log(x ) + 16*(log(x) + 1)/log(x) - log(3*(log(x) + 1)/log(x)) - 17)
Time = 2.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{\ln \left (\frac {3\,\ln \left (x\right )+3}{\ln \left (x\right )}\right )-\ln \left (x\right )+16} \] Input:
int((2*log(x) + 2*log(x)^2 + 2)/(224*x*log(x)^2 - 31*x*log(x)^3 + x*log(x) ^4 + log((3*log(x) + 3)/log(x))*(30*x*log(x)^2 - 2*x*log(x)^3 + 32*x*log(x )) + log((3*log(x) + 3)/log(x))^2*(x*log(x)^2 + x*log(x)) + 256*x*log(x)), x)
Output:
2/(log((3*log(x) + 3)/log(x)) - log(x) + 16)
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{\mathrm {log}\left (\frac {3 \,\mathrm {log}\left (x \right )+3}{\mathrm {log}\left (x \right )}\right )-\mathrm {log}\left (x \right )+16} \] Input:
int((2*log(x)^2+2*log(x)+2)/((x*log(x)^2+x*log(x))*log((3*log(x)+3)/log(x) )^2+(-2*x*log(x)^3+30*x*log(x)^2+32*x*log(x))*log((3*log(x)+3)/log(x))+x*l og(x)^4-31*x*log(x)^3+224*x*log(x)^2+256*x*log(x)),x)
Output:
2/(log((3*log(x) + 3)/log(x)) - log(x) + 16)