\(\int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+(32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)) \log (\frac {3+3 \log (x)}{\log (x)})+(x \log (x)+x \log ^2(x)) \log ^2(\frac {3+3 \log (x)}{\log (x)})} \, dx\) [1842]

Optimal result

Integrand size = 100, antiderivative size = 19 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{16-\log (x)+\log \left (3+\frac {3}{\log (x)}\right )} \]

[Out]

2/(16-ln(x)+ln(3+3/ln(x)))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {12, 6818} \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{-\log (x)+\log \left (\frac {3 (\log (x)+1)}{\log (x)}\right )+16} \]

[In]

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] + 3
0*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]

[Out]

2/(16 - Log[x] + Log[(3*(1 + Log[x]))/Log[x]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {2 \left (1+x+x^2\right )}{x (1+x) \left (-16+x-\log \left (\frac {3+3 x}{x}\right )\right )^2} \, dx,x,\log (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {1+x+x^2}{x (1+x) \left (-16+x-\log \left (\frac {3+3 x}{x}\right )\right )^2} \, dx,x,\log (x)\right ) \\ & = \frac {2}{16-\log (x)+\log \left (\frac {3 (1+\log (x))}{\log (x)}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{16-\log (x)+\log \left (3+\frac {3}{\log (x)}\right )} \]

[In]

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]

[Out]

2/(16 - Log[x] + Log[3 + 3/Log[x]])

Maple [A] (verified)

Time = 1.47 (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	\(\frac {4 i}{\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+1\right )}{\ln \left (x \right )}\right )^{3}+2 i \ln \left (3\right )-2 i \ln \left (x \right )-2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (\ln \left (x \right )+1\right )+32 i}\)	\(129\)

[In]

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*ln(x)^3+224*x*ln(x)^2+256*x*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-2/(ln(x)-ln(3*(ln(x)+1)/ln(x))-16)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=-\frac {2}{\log \left (x\right ) - \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right ) - 16} \]

[In]

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="f
ricas")

[Out]

-2/(log(x) - log(3*(log(x) + 1)/log(x)) - 16)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{- \log {\left (x \right )} + \log {\left (\frac {3 \log {\left (x \right )} + 3}{\log {\left (x \right )}} \right )} + 16} \]

[In]

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+3
2*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)

[Out]

2/(-log(x) + log((3*log(x) + 3)/log(x)) + 16)

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{\log \left (3\right ) - \log \left (x\right ) + \log \left (\log \left (x\right ) + 1\right ) - \log \left (\log \left (x\right )\right ) + 16} \]

[In]

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="m
axima")

[Out]

2/(log(3) - log(x) + log(log(x) + 1) - log(log(x)) + 16)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2 \, {\left (\frac {\log \left (x\right ) + 1}{\log \left (x\right )} - 1\right )}}{\frac {{\left (\log \left (x\right ) + 1\right )} \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right )}{\log \left (x\right )} + \frac {16 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )} - \log \left (\frac {3 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )}\right ) - 17} \]

[In]

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="g
iac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 9.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {2+2 \log (x)+2 \log ^2(x)}{256 x \log (x)+224 x \log ^2(x)-31 x \log ^3(x)+x \log ^4(x)+\left (32 x \log (x)+30 x \log ^2(x)-2 x \log ^3(x)\right ) \log \left (\frac {3+3 \log (x)}{\log (x)}\right )+\left (x \log (x)+x \log ^2(x)\right ) \log ^2\left (\frac {3+3 \log (x)}{\log (x)}\right )} \, dx=\frac {2}{\ln \left (\frac {3\,\ln \left (x\right )+3}{\ln \left (x\right )}\right )-\ln \left (x\right )+16} \]

[In]

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)

[Out]

2/(log((3*log(x) + 3)/log(x)) - log(x) + 16)