3.57.24 \(\int \frac {-24 x^2+16 x^5-8 x^6+(-144 x^2+96 x^3+96 x^5-112 x^6+32 x^7) \log (x) \log (\log (x))+(8 x^5+(-72 x^2+96 x^5-56 x^6) \log (x) \log (\log (x))) \log (\log (\log (x)))+24 x^5 \log (x) \log (\log (x)) \log ^2(\log (\log (x)))}{\log (x) \log (\log (x))} \, dx\)

Optimal. Leaf size=18 \[ \left (6-2 x^3 (2-x+\log (\log (\log (x))))\right )^2 \]

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Rubi [A]  time = 0.27, antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 3, integrand size = 107, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6688, 12, 6686} \begin {gather*} 4 \left (x^4-2 x^3-x^3 \log (\log (\log (x)))+3\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-24*x^2 + 16*x^5 - 8*x^6 + (-144*x^2 + 96*x^3 + 96*x^5 - 112*x^6 + 32*x^7)*Log[x]*Log[Log[x]] + (8*x^5 +
(-72*x^2 + 96*x^5 - 56*x^6)*Log[x]*Log[Log[x]])*Log[Log[Log[x]]] + 24*x^5*Log[x]*Log[Log[x]]*Log[Log[Log[x]]]^
2)/(Log[x]*Log[Log[x]]),x]

[Out]

4*(3 - 2*x^3 + x^4 - x^3*Log[Log[Log[x]]])^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 6686

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]

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 {8 x^2 (1-\log (x) \log (\log (x)) (-6+4 x-3 \log (\log (\log (x))))) \left (-3+2 x^3-x^4+x^3 \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx\\ &=8 \int \frac {x^2 (1-\log (x) \log (\log (x)) (-6+4 x-3 \log (\log (\log (x))))) \left (-3+2 x^3-x^4+x^3 \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx\\ &=4 \left (3-2 x^3+x^4-x^3 \log (\log (\log (x)))\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 23, normalized size = 1.28 \begin {gather*} 4 \left (3-2 x^3+x^4-x^3 \log (\log (\log (x)))\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24*x^2 + 16*x^5 - 8*x^6 + (-144*x^2 + 96*x^3 + 96*x^5 - 112*x^6 + 32*x^7)*Log[x]*Log[Log[x]] + (8*
x^5 + (-72*x^2 + 96*x^5 - 56*x^6)*Log[x]*Log[Log[x]])*Log[Log[Log[x]]] + 24*x^5*Log[x]*Log[Log[x]]*Log[Log[Log
[x]]]^2)/(Log[x]*Log[Log[x]]),x]

[Out]

4*(3 - 2*x^3 + x^4 - x^3*Log[Log[Log[x]]])^2

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fricas [B]  time = 0.67, size = 57, normalized size = 3.17 \begin {gather*} 4 \, x^{8} + 4 \, x^{6} \log \left (\log \left (\log \relax (x)\right )\right )^{2} - 16 \, x^{7} + 16 \, x^{6} + 24 \, x^{4} - 48 \, x^{3} - 8 \, {\left (x^{7} - 2 \, x^{6} + 3 \, x^{3}\right )} \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^5*log(x)*log(log(x))*log(log(log(x)))^2+((-56*x^6+96*x^5-72*x^2)*log(x)*log(log(x))+8*x^5)*log
(log(log(x)))+(32*x^7-112*x^6+96*x^5+96*x^3-144*x^2)*log(x)*log(log(x))-8*x^6+16*x^5-24*x^2)/log(x)/log(log(x)
),x, algorithm="fricas")

[Out]

4*x^8 + 4*x^6*log(log(log(x)))^2 - 16*x^7 + 16*x^6 + 24*x^4 - 48*x^3 - 8*(x^7 - 2*x^6 + 3*x^3)*log(log(log(x))
)

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giac [B]  time = 0.31, size = 122, normalized size = 6.78 \begin {gather*} 2 \, x^{6} \log \relax (x)^{2} \log \left (\log \relax (x)\right )^{2} - 8 \, x^{7} \log \relax (x) \log \left (\log \relax (x)\right ) + 4 \, x^{6} \log \relax (x) \log \left (\log \relax (x)\right ) \log \left (\log \left (\log \relax (x)\right )\right ) + 4 \, x^{8} + 8 \, x^{6} \log \relax (x) \log \left (\log \relax (x)\right ) - 8 \, x^{7} \log \left (\log \left (\log \relax (x)\right )\right ) + 4 \, x^{6} \log \left (\log \left (\log \relax (x)\right )\right )^{2} - 16 \, x^{7} + 16 \, x^{6} \log \left (\log \left (\log \relax (x)\right )\right ) + \frac {46}{3} \, x^{6} - 24 \, x^{3} \log \relax (x) \log \left (\log \relax (x)\right ) + 24 \, x^{4} - 24 \, x^{3} \log \left (\log \left (\log \relax (x)\right )\right ) - 48 \, x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^5*log(x)*log(log(x))*log(log(log(x)))^2+((-56*x^6+96*x^5-72*x^2)*log(x)*log(log(x))+8*x^5)*log
(log(log(x)))+(32*x^7-112*x^6+96*x^5+96*x^3-144*x^2)*log(x)*log(log(x))-8*x^6+16*x^5-24*x^2)/log(x)/log(log(x)
),x, algorithm="giac")

[Out]

2*x^6*log(x)^2*log(log(x))^2 - 8*x^7*log(x)*log(log(x)) + 4*x^6*log(x)*log(log(x))*log(log(log(x))) + 4*x^8 +
8*x^6*log(x)*log(log(x)) - 8*x^7*log(log(log(x))) + 4*x^6*log(log(log(x)))^2 - 16*x^7 + 16*x^6*log(log(log(x))
) + 46/3*x^6 - 24*x^3*log(x)*log(log(x)) + 24*x^4 - 24*x^3*log(log(log(x))) - 48*x^3

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maple [B]  time = 0.04, size = 59, normalized size = 3.28

method	result	size
risch	\(4 x^{6} \ln \left (\ln \left (\ln \relax (x )\right )\right )^{2}+\left (-8 x^{7}+16 x^{6}-24 x^{3}\right ) \ln \left (\ln \left (\ln \relax (x )\right )\right )+4 x^{8}-16 x^{7}+16 x^{6}+24 x^{4}-48 x^{3}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^5*ln(x)*ln(ln(x))*ln(ln(ln(x)))^2+((-56*x^6+96*x^5-72*x^2)*ln(x)*ln(ln(x))+8*x^5)*ln(ln(ln(x)))+(32*
x^7-112*x^6+96*x^5+96*x^3-144*x^2)*ln(x)*ln(ln(x))-8*x^6+16*x^5-24*x^2)/ln(x)/ln(ln(x)),x,method=_RETURNVERBOS
E)

[Out]

4*x^6*ln(ln(ln(x)))^2+(-8*x^7+16*x^6-24*x^3)*ln(ln(ln(x)))+4*x^8-16*x^7+16*x^6+24*x^4-48*x^3

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maxima [B]  time = 0.44, size = 57, normalized size = 3.17 \begin {gather*} 4 \, x^{8} + 4 \, x^{6} \log \left (\log \left (\log \relax (x)\right )\right )^{2} - 16 \, x^{7} + 16 \, x^{6} + 24 \, x^{4} - 48 \, x^{3} - 8 \, {\left (x^{7} - 2 \, x^{6} + 3 \, x^{3}\right )} \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^5*log(x)*log(log(x))*log(log(log(x)))^2+((-56*x^6+96*x^5-72*x^2)*log(x)*log(log(x))+8*x^5)*log
(log(log(x)))+(32*x^7-112*x^6+96*x^5+96*x^3-144*x^2)*log(x)*log(log(x))-8*x^6+16*x^5-24*x^2)/log(x)/log(log(x)
),x, algorithm="maxima")

[Out]

4*x^8 + 4*x^6*log(log(log(x)))^2 - 16*x^7 + 16*x^6 + 24*x^4 - 48*x^3 - 8*(x^7 - 2*x^6 + 3*x^3)*log(log(log(x))
)

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mupad [B]  time = 3.86, size = 62, normalized size = 3.44 \begin {gather*} 4\,x^6\,{\ln \left (\ln \left (\ln \relax (x)\right )\right )}^2-48\,x^3+24\,x^4+16\,x^6-16\,x^7+4\,x^8-\frac {\ln \left (\ln \left (\ln \relax (x)\right )\right )\,\left (8\,x^8-16\,x^7+24\,x^4\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(log(x)))*(8*x^5 - log(log(x))*log(x)*(72*x^2 - 96*x^5 + 56*x^6)) - 24*x^2 + 16*x^5 - 8*x^6 + log(
log(x))*log(x)*(96*x^3 - 144*x^2 + 96*x^5 - 112*x^6 + 32*x^7) + 24*x^5*log(log(x))*log(log(log(x)))^2*log(x))/
(log(log(x))*log(x)),x)

[Out]

4*x^6*log(log(log(x)))^2 - 48*x^3 + 24*x^4 + 16*x^6 - 16*x^7 + 4*x^8 - (log(log(log(x)))*(24*x^4 - 16*x^7 + 8*
x^8))/x

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sympy [B]  time = 0.93, size = 60, normalized size = 3.33 \begin {gather*} 4 x^{8} - 16 x^{7} + 4 x^{6} \log {\left (\log {\left (\log {\relax (x )} \right )} \right )}^{2} + 16 x^{6} + 24 x^{4} - 48 x^{3} + \left (- 8 x^{7} + 16 x^{6} - 24 x^{3}\right ) \log {\left (\log {\left (\log {\relax (x )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x**5*ln(x)*ln(ln(x))*ln(ln(ln(x)))**2+((-56*x**6+96*x**5-72*x**2)*ln(x)*ln(ln(x))+8*x**5)*ln(ln(
ln(x)))+(32*x**7-112*x**6+96*x**5+96*x**3-144*x**2)*ln(x)*ln(ln(x))-8*x**6+16*x**5-24*x**2)/ln(x)/ln(ln(x)),x)

[Out]

4*x**8 - 16*x**7 + 4*x**6*log(log(log(x)))**2 + 16*x**6 + 24*x**4 - 48*x**3 + (-8*x**7 + 16*x**6 - 24*x**3)*lo
g(log(log(x)))

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