3.68.81 \(\int \frac {(-2916 x-1458 x^2) \log (4+4 x+x^2)-108 x \log (\log (4+4 x+x^2))+(-54-27 x) \log (4+4 x+x^2) \log ^2(\log (4+4 x+x^2))}{(1458 x^4+729 x^5) \log (4+4 x+x^2)+(108 x^3+54 x^4) \log (4+4 x+x^2) \log ^2(\log (4+4 x+x^2))+(2 x^2+x^3) \log (4+4 x+x^2) \log ^4(\log (4+4 x+x^2))} \, dx\)

Optimal. Leaf size=24 \[ \frac {3}{x \left (3 x+\frac {1}{9} \log ^2\left (\log \left ((2+x)^2\right )\right )\right )} \]

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Rubi [A]  time = 0.37, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 3, integrand size = 149, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6688, 12, 6687} \begin {gather*} \frac {27}{x \left (27 x+\log ^2\left (\log \left ((x+2)^2\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-2916*x - 1458*x^2)*Log[4 + 4*x + x^2] - 108*x*Log[Log[4 + 4*x + x^2]] + (-54 - 27*x)*Log[4 + 4*x + x^2]
*Log[Log[4 + 4*x + x^2]]^2)/((1458*x^4 + 729*x^5)*Log[4 + 4*x + x^2] + (108*x^3 + 54*x^4)*Log[4 + 4*x + x^2]*L
og[Log[4 + 4*x + x^2]]^2 + (2*x^2 + x^3)*Log[4 + 4*x + x^2]*Log[Log[4 + 4*x + x^2]]^4),x]

[Out]

27/(x*(27*x + Log[Log[(2 + x)^2]]^2))

Rule 12

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

Rule 6687

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27 \left (-4 x \log \left (\log \left ((2+x)^2\right )\right )-(2+x) \log \left ((2+x)^2\right ) \left (54 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )\right )}{x^2 (2+x) \log \left ((2+x)^2\right ) \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )^2} \, dx\\ &=27 \int \frac {-4 x \log \left (\log \left ((2+x)^2\right )\right )-(2+x) \log \left ((2+x)^2\right ) \left (54 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )}{x^2 (2+x) \log \left ((2+x)^2\right ) \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )^2} \, dx\\ &=\frac {27}{x \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 20, normalized size = 0.83 \begin {gather*} \frac {27}{x \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2916*x - 1458*x^2)*Log[4 + 4*x + x^2] - 108*x*Log[Log[4 + 4*x + x^2]] + (-54 - 27*x)*Log[4 + 4*x
+ x^2]*Log[Log[4 + 4*x + x^2]]^2)/((1458*x^4 + 729*x^5)*Log[4 + 4*x + x^2] + (108*x^3 + 54*x^4)*Log[4 + 4*x +
x^2]*Log[Log[4 + 4*x + x^2]]^2 + (2*x^2 + x^3)*Log[4 + 4*x + x^2]*Log[Log[4 + 4*x + x^2]]^4),x]

[Out]

27/(x*(27*x + Log[Log[(2 + x)^2]]^2))

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fricas [A]  time = 0.61, size = 24, normalized size = 1.00 \begin {gather*} \frac {27}{x \log \left (\log \left (x^{2} + 4 \, x + 4\right )\right )^{2} + 27 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x-54)*log(x^2+4*x+4)*log(log(x^2+4*x+4))^2-108*x*log(log(x^2+4*x+4))+(-1458*x^2-2916*x)*log(x^
2+4*x+4))/((x^3+2*x^2)*log(x^2+4*x+4)*log(log(x^2+4*x+4))^4+(54*x^4+108*x^3)*log(x^2+4*x+4)*log(log(x^2+4*x+4)
)^2+(729*x^5+1458*x^4)*log(x^2+4*x+4)),x, algorithm="fricas")

[Out]

27/(x*log(log(x^2 + 4*x + 4))^2 + 27*x^2)

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giac [A]  time = 1.95, size = 24, normalized size = 1.00 \begin {gather*} \frac {27}{x \log \left (\log \left (x^{2} + 4 \, x + 4\right )\right )^{2} + 27 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x-54)*log(x^2+4*x+4)*log(log(x^2+4*x+4))^2-108*x*log(log(x^2+4*x+4))+(-1458*x^2-2916*x)*log(x^
2+4*x+4))/((x^3+2*x^2)*log(x^2+4*x+4)*log(log(x^2+4*x+4))^4+(54*x^4+108*x^3)*log(x^2+4*x+4)*log(log(x^2+4*x+4)
)^2+(729*x^5+1458*x^4)*log(x^2+4*x+4)),x, algorithm="giac")

[Out]

27/(x*log(log(x^2 + 4*x + 4))^2 + 27*x^2)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-27 x -54\right ) \ln \left (x^{2}+4 x +4\right ) \ln \left (\ln \left (x^{2}+4 x +4\right )\right )^{2}-108 x \ln \left (\ln \left (x^{2}+4 x +4\right )\right )+\left (-1458 x^{2}-2916 x \right ) \ln \left (x^{2}+4 x +4\right )}{\left (x^{3}+2 x^{2}\right ) \ln \left (x^{2}+4 x +4\right ) \ln \left (\ln \left (x^{2}+4 x +4\right )\right )^{4}+\left (54 x^{4}+108 x^{3}\right ) \ln \left (x^{2}+4 x +4\right ) \ln \left (\ln \left (x^{2}+4 x +4\right )\right )^{2}+\left (729 x^{5}+1458 x^{4}\right ) \ln \left (x^{2}+4 x +4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-27*x-54)*ln(x^2+4*x+4)*ln(ln(x^2+4*x+4))^2-108*x*ln(ln(x^2+4*x+4))+(-1458*x^2-2916*x)*ln(x^2+4*x+4))/((
x^3+2*x^2)*ln(x^2+4*x+4)*ln(ln(x^2+4*x+4))^4+(54*x^4+108*x^3)*ln(x^2+4*x+4)*ln(ln(x^2+4*x+4))^2+(729*x^5+1458*
x^4)*ln(x^2+4*x+4)),x)

[Out]

int(((-27*x-54)*ln(x^2+4*x+4)*ln(ln(x^2+4*x+4))^2-108*x*ln(ln(x^2+4*x+4))+(-1458*x^2-2916*x)*ln(x^2+4*x+4))/((
x^3+2*x^2)*ln(x^2+4*x+4)*ln(ln(x^2+4*x+4))^4+(54*x^4+108*x^3)*ln(x^2+4*x+4)*ln(ln(x^2+4*x+4))^2+(729*x^5+1458*
x^4)*ln(x^2+4*x+4)),x)

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maxima [A]  time = 4.00, size = 35, normalized size = 1.46 \begin {gather*} \frac {27}{x \log \relax (2)^{2} + 2 \, x \log \relax (2) \log \left (\log \left (x + 2\right )\right ) + x \log \left (\log \left (x + 2\right )\right )^{2} + 27 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x-54)*log(x^2+4*x+4)*log(log(x^2+4*x+4))^2-108*x*log(log(x^2+4*x+4))+(-1458*x^2-2916*x)*log(x^
2+4*x+4))/((x^3+2*x^2)*log(x^2+4*x+4)*log(log(x^2+4*x+4))^4+(54*x^4+108*x^3)*log(x^2+4*x+4)*log(log(x^2+4*x+4)
)^2+(729*x^5+1458*x^4)*log(x^2+4*x+4)),x, algorithm="maxima")

[Out]

27/(x*log(2)^2 + 2*x*log(2)*log(log(x + 2)) + x*log(log(x + 2))^2 + 27*x^2)

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mupad [B]  time = 4.75, size = 199, normalized size = 8.29 \begin {gather*} \frac {27\,{\left (2\,\ln \left (x^2+4\,x+4\right )+x\,\ln \left (x^2+4\,x+4\right )\right )}^2\,\left (27\,x^2\,{\ln \left (x^2+4\,x+4\right )}^2+108\,x\,{\ln \left (x^2+4\,x+4\right )}^2+16\,x+108\,{\ln \left (x^2+4\,x+4\right )}^2\right )}{x\,\ln \left (x^2+4\,x+4\right )\,\left ({\ln \left (\ln \left (x^2+4\,x+4\right )\right )}^2+27\,x\right )\,\left (x+2\right )\,\left (27\,x^3\,{\ln \left (x^2+4\,x+4\right )}^3+162\,x^2\,{\ln \left (x^2+4\,x+4\right )}^3+16\,x^2\,\ln \left (x^2+4\,x+4\right )+324\,x\,{\ln \left (x^2+4\,x+4\right )}^3+32\,x\,\ln \left (x^2+4\,x+4\right )+216\,{\ln \left (x^2+4\,x+4\right )}^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(4*x + x^2 + 4)*(2916*x + 1458*x^2) + 108*x*log(log(4*x + x^2 + 4)) + log(4*x + x^2 + 4)*log(log(4*x
+ x^2 + 4))^2*(27*x + 54))/(log(4*x + x^2 + 4)*(1458*x^4 + 729*x^5) + log(4*x + x^2 + 4)*log(log(4*x + x^2 + 4
))^4*(2*x^2 + x^3) + log(4*x + x^2 + 4)*log(log(4*x + x^2 + 4))^2*(108*x^3 + 54*x^4)),x)

[Out]

(27*(2*log(4*x + x^2 + 4) + x*log(4*x + x^2 + 4))^2*(16*x + 27*x^2*log(4*x + x^2 + 4)^2 + 108*x*log(4*x + x^2
+ 4)^2 + 108*log(4*x + x^2 + 4)^2))/(x*log(4*x + x^2 + 4)*(27*x + log(log(4*x + x^2 + 4))^2)*(x + 2)*(162*x^2*
log(4*x + x^2 + 4)^3 + 27*x^3*log(4*x + x^2 + 4)^3 + 32*x*log(4*x + x^2 + 4) + 16*x^2*log(4*x + x^2 + 4) + 324
*x*log(4*x + x^2 + 4)^3 + 216*log(4*x + x^2 + 4)^3))

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sympy [A]  time = 0.36, size = 20, normalized size = 0.83 \begin {gather*} \frac {27}{27 x^{2} + x \log {\left (\log {\left (x^{2} + 4 x + 4 \right )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x-54)*ln(x**2+4*x+4)*ln(ln(x**2+4*x+4))**2-108*x*ln(ln(x**2+4*x+4))+(-1458*x**2-2916*x)*ln(x**
2+4*x+4))/((x**3+2*x**2)*ln(x**2+4*x+4)*ln(ln(x**2+4*x+4))**4+(54*x**4+108*x**3)*ln(x**2+4*x+4)*ln(ln(x**2+4*x
+4))**2+(729*x**5+1458*x**4)*ln(x**2+4*x+4)),x)

[Out]

27/(27*x**2 + x*log(log(x**2 + 4*x + 4))**2)

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