3.11.84 \(\int \frac {-3300 x+(26400 x-6600 x \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(-576+144 \log (x)) \log (4-\log (x))+(-26400 x^2+6600 x^2 \log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))+(-302500 x^4+75625 x^4 \log (x)) \log (4-\log (x)) \log ^2(\log (4-\log (x)))} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{1+\frac {275}{12} x^2 \log (\log (4-\log (x)))} \]

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Rubi [A]  time = 0.22, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*} \frac {12}{275 x^2 \log (\log (4-\log (x)))+12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3300*x + (26400*x - 6600*x*Log[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]])/((-576 + 144*Log[x])*Log[4 - Lo
g[x]] + (-26400*x^2 + 6600*x^2*Log[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]] + (-302500*x^4 + 75625*x^4*Log[x])
*Log[4 - Log[x]]*Log[Log[4 - Log[x]]]^2),x]

[Out]

12/(12 + 275*x^2*Log[Log[4 - 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 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 {3300 (x+2 x (-4+\log (x)) \log (4-\log (x)) \log (\log (4-\log (x))))}{(4-\log (x)) \log (4-\log (x)) \left (12+275 x^2 \log (\log (4-\log (x)))\right )^2} \, dx\\ &=3300 \int \frac {x+2 x (-4+\log (x)) \log (4-\log (x)) \log (\log (4-\log (x)))}{(4-\log (x)) \log (4-\log (x)) \left (12+275 x^2 \log (\log (4-\log (x)))\right )^2} \, dx\\ &=\frac {12}{12+275 x^2 \log (\log (4-\log (x)))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} \frac {12}{12+275 x^2 \log (\log (4-\log (x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3300*x + (26400*x - 6600*x*Log[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]])/((-576 + 144*Log[x])*Log[
4 - Log[x]] + (-26400*x^2 + 6600*x^2*Log[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]] + (-302500*x^4 + 75625*x^4*L
og[x])*Log[4 - Log[x]]*Log[Log[4 - Log[x]]]^2),x]

[Out]

12/(12 + 275*x^2*Log[Log[4 - Log[x]]])

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fricas [A]  time = 0.69, size = 19, normalized size = 1.00 \begin {gather*} \frac {12}{275 \, x^{2} \log \left (\log \left (-\log \relax (x) + 4\right )\right ) + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6600*x*log(x)+26400*x)*log(-log(x)+4)*log(log(-log(x)+4))-3300*x)/((75625*x^4*log(x)-302500*x^4)*
log(-log(x)+4)*log(log(-log(x)+4))^2+(6600*x^2*log(x)-26400*x^2)*log(-log(x)+4)*log(log(-log(x)+4))+(144*log(x
)-576)*log(-log(x)+4)),x, algorithm="fricas")

[Out]

12/(275*x^2*log(log(-log(x) + 4)) + 12)

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giac [A]  time = 0.73, size = 19, normalized size = 1.00 \begin {gather*} \frac {12}{275 \, x^{2} \log \left (\log \left (-\log \relax (x) + 4\right )\right ) + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6600*x*log(x)+26400*x)*log(-log(x)+4)*log(log(-log(x)+4))-3300*x)/((75625*x^4*log(x)-302500*x^4)*
log(-log(x)+4)*log(log(-log(x)+4))^2+(6600*x^2*log(x)-26400*x^2)*log(-log(x)+4)*log(log(-log(x)+4))+(144*log(x
)-576)*log(-log(x)+4)),x, algorithm="giac")

[Out]

12/(275*x^2*log(log(-log(x) + 4)) + 12)

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maple [A]  time = 0.04, size = 20, normalized size = 1.05

method	result	size
risch	\(\frac {12}{275 x^{2} \ln \left (\ln \left (-\ln \relax (x )+4\right )\right )+12}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6600*x*ln(x)+26400*x)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))-3300*x)/((75625*x^4*ln(x)-302500*x^4)*ln(-ln(x)+4)*
ln(ln(-ln(x)+4))^2+(6600*x^2*ln(x)-26400*x^2)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))+(144*ln(x)-576)*ln(-ln(x)+4)),x,me
thod=_RETURNVERBOSE)

[Out]

12/(275*x^2*ln(ln(-ln(x)+4))+12)

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maxima [A]  time = 0.56, size = 19, normalized size = 1.00 \begin {gather*} \frac {12}{275 \, x^{2} \log \left (\log \left (-\log \relax (x) + 4\right )\right ) + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6600*x*log(x)+26400*x)*log(-log(x)+4)*log(log(-log(x)+4))-3300*x)/((75625*x^4*log(x)-302500*x^4)*
log(-log(x)+4)*log(log(-log(x)+4))^2+(6600*x^2*log(x)-26400*x^2)*log(-log(x)+4)*log(log(-log(x)+4))+(144*log(x
)-576)*log(-log(x)+4)),x, algorithm="maxima")

[Out]

12/(275*x^2*log(log(-log(x) + 4)) + 12)

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mupad [B]  time = 1.81, size = 19, normalized size = 1.00 \begin {gather*} \frac {12}{275\,x^2\,\ln \left (\ln \left (4-\ln \relax (x)\right )\right )+12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3300*x - log(4 - log(x))*log(log(4 - log(x)))*(26400*x - 6600*x*log(x)))/(log(4 - log(x))*(144*log(x) -
576) + log(4 - log(x))*log(log(4 - log(x)))*(6600*x^2*log(x) - 26400*x^2) + log(4 - log(x))*log(log(4 - log(x)
))^2*(75625*x^4*log(x) - 302500*x^4)),x)

[Out]

12/(275*x^2*log(log(4 - log(x))) + 12)

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sympy [A]  time = 0.39, size = 15, normalized size = 0.79 \begin {gather*} \frac {12}{275 x^{2} \log {\left (\log {\left (4 - \log {\relax (x )} \right )} \right )} + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6600*x*ln(x)+26400*x)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))-3300*x)/((75625*x**4*ln(x)-302500*x**4)*ln(-l
n(x)+4)*ln(ln(-ln(x)+4))**2+(6600*x**2*ln(x)-26400*x**2)*ln(-ln(x)+4)*ln(ln(-ln(x)+4))+(144*ln(x)-576)*ln(-ln(
x)+4)),x)

[Out]

12/(275*x**2*log(log(4 - log(x))) + 12)

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