3.53.83 \(\int \frac {4}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx\)

Optimal. Leaf size=12 \[ \log (\log (4 (-5+\log (5-20 x)))) \]

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Rubi [A]  time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 6684} \begin {gather*} \log (\log (4 \log (5-20 x)-20)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4/((5 - 20*x + (-1 + 4*x)*Log[5 - 20*x])*Log[-20 + 4*Log[5 - 20*x]]),x]

[Out]

Log[Log[-20 + 4*Log[5 - 20*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 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 \int \frac {1}{(5-20 x+(-1+4 x) \log (5-20 x)) \log (-20+4 \log (5-20 x))} \, dx\\ &=\log (\log (-20+4 \log (5-20 x)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 12, normalized size = 1.00 \begin {gather*} \log (\log (4 (-5+\log (5-20 x)))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4/((5 - 20*x + (-1 + 4*x)*Log[5 - 20*x])*Log[-20 + 4*Log[5 - 20*x]]),x]

[Out]

Log[Log[4*(-5 + Log[5 - 20*x])]]

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fricas [A]  time = 0.64, size = 12, normalized size = 1.00 \begin {gather*} \log \left (\log \left (4 \, \log \left (-20 \, x + 5\right ) - 20\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/((4*x-1)*log(-20*x+5)-20*x+5)/log(4*log(-20*x+5)-20),x, algorithm="fricas")

[Out]

log(log(4*log(-20*x + 5) - 20))

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giac [A]  time = 0.15, size = 12, normalized size = 1.00 \begin {gather*} \log \left (\log \left (4 \, \log \left (-20 \, x + 5\right ) - 20\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/((4*x-1)*log(-20*x+5)-20*x+5)/log(4*log(-20*x+5)-20),x, algorithm="giac")

[Out]

log(log(4*log(-20*x + 5) - 20))

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maple [A]  time = 0.04, size = 13, normalized size = 1.08

method	result	size
norman	\(\ln \left (\ln \left (4 \ln \left (-20 x +5\right )-20\right )\right )\)	\(13\)
risch	\(\ln \left (\ln \left (4 \ln \left (-20 x +5\right )-20\right )\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4/((4*x-1)*ln(-20*x+5)-20*x+5)/ln(4*ln(-20*x+5)-20),x,method=_RETURNVERBOSE)

[Out]

ln(ln(4*ln(-20*x+5)-20))

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maxima [C]  time = 0.46, size = 20, normalized size = 1.67 \begin {gather*} \log \left (2 \, \log \relax (2) + \log \left (i \, \pi + \log \relax (5) + \log \left (4 \, x - 1\right ) - 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/((4*x-1)*log(-20*x+5)-20*x+5)/log(4*log(-20*x+5)-20),x, algorithm="maxima")

[Out]

log(2*log(2) + log(I*pi + log(5) + log(4*x - 1) - 5))

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mupad [B]  time = 4.04, size = 12, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (4\,\ln \left (5-20\,x\right )-20\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4/(log(4*log(5 - 20*x) - 20)*(log(5 - 20*x)*(4*x - 1) - 20*x + 5)),x)

[Out]

log(log(4*log(5 - 20*x) - 20))

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sympy [A]  time = 0.31, size = 12, normalized size = 1.00 \begin {gather*} \log {\left (\log {\left (4 \log {\left (5 - 20 x \right )} - 20 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/((4*x-1)*ln(-20*x+5)-20*x+5)/ln(4*ln(-20*x+5)-20),x)

[Out]

log(log(4*log(5 - 20*x) - 20))

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