3.6.9 \(\int \frac {(-36 x+x^2) \log (\log (2))+(72-x+(-72 x+x^2) \log (\log (2))) \log (-1+x \log (\log (2)))}{(36 x-x^2+(-36 x^2+x^3) \log (\log (2))) \log (-1+x \log (\log (2)))} \, dx\) [509]

Optimal. Leaf size=25 \[ \log \left (\frac {x^2 \log (-1+x \log (\log (2)))}{3 \left (-9+\frac {x}{4}\right )}\right ) \]

[Out]

ln(1/3*x^2/(1/4*x-9)*ln(x*ln(ln(2))-1))

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Rubi [A]
time = 0.12, antiderivative size = 22, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 5, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6820, 78, 2437, 2339, 29} \begin {gather*} -\log (36-x)+2 \log (x)+\log (\log (x \log (\log (2))-1)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-36*x + x^2)*Log[Log[2]] + (72 - x + (-72*x + x^2)*Log[Log[2]])*Log[-1 + x*Log[Log[2]]])/((36*x - x^2 +
(-36*x^2 + x^3)*Log[Log[2]])*Log[-1 + x*Log[Log[2]]]),x]

[Out]

-Log[36 - x] + 2*Log[x] + Log[Log[-1 + x*Log[Log[2]]]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 6820

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 \left (\frac {-72+x}{(-36+x) x}+\frac {\log (\log (2))}{(-1+x \log (\log (2))) \log (-1+x \log (\log (2)))}\right ) \, dx\\ &=\log (\log (2)) \int \frac {1}{(-1+x \log (\log (2))) \log (-1+x \log (\log (2)))} \, dx+\int \frac {-72+x}{(-36+x) x} \, dx\\ &=\int \left (\frac {1}{36-x}+\frac {2}{x}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,-1+x \log (\log (2))\right )\\ &=-\log (36-x)+2 \log (x)+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (-1+x \log (\log (2)))\right )\\ &=-\log (36-x)+2 \log (x)+\log (\log (-1+x \log (\log (2))))\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.88 \begin {gather*} -\log (36-x)+2 \log (x)+\log (\log (-1+x \log (\log (2)))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-36*x + x^2)*Log[Log[2]] + (72 - x + (-72*x + x^2)*Log[Log[2]])*Log[-1 + x*Log[Log[2]]])/((36*x -
x^2 + (-36*x^2 + x^3)*Log[Log[2]])*Log[-1 + x*Log[Log[2]]]),x]

[Out]

-Log[36 - x] + 2*Log[x] + Log[Log[-1 + x*Log[Log[2]]]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(21)=42\).
time = 0.27, size = 49, normalized size = 1.96

method	result	size
norman	\(2 \ln \left (x \right )-\ln \left (x -36\right )+\ln \left (\ln \left (x \ln \left (\ln \left (2\right )\right )-1\right )\right )\)	\(21\)
risch	\(2 \ln \left (x \right )-\ln \left (x -36\right )+\ln \left (\ln \left (x \ln \left (\ln \left (2\right )\right )-1\right )\right )\)	\(21\)
derivativedivides	\(\frac {\ln \left (\ln \left (2\right )\right ) \ln \left (\ln \left (x \ln \left (\ln \left (2\right )\right )-1\right )\right )+2 \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )-\ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )-36 \ln \left (\ln \left (2\right )\right )\right )}{\ln \left (\ln \left (2\right )\right )}\)	\(49\)
default	\(\frac {\ln \left (\ln \left (2\right )\right ) \ln \left (\ln \left (x \ln \left (\ln \left (2\right )\right )-1\right )\right )+2 \ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )\right )-\ln \left (\ln \left (2\right )\right ) \ln \left (x \ln \left (\ln \left (2\right )\right )-36 \ln \left (\ln \left (2\right )\right )\right )}{\ln \left (\ln \left (2\right )\right )}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-72*x)*ln(ln(2))-x+72)*ln(x*ln(ln(2))-1)+(x^2-36*x)*ln(ln(2)))/((x^3-36*x^2)*ln(ln(2))-x^2+36*x)/ln(
x*ln(ln(2))-1),x,method=_RETURNVERBOSE)

[Out]

1/ln(ln(2))*(ln(ln(2))*ln(ln(x*ln(ln(2))-1))+2*ln(ln(2))*ln(x*ln(ln(2)))-ln(ln(2))*ln(x*ln(ln(2))-36*ln(ln(2))
))

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Maxima [A]
time = 0.51, size = 20, normalized size = 0.80 \begin {gather*} -\log \left (x - 36\right ) + 2 \, \log \left (x\right ) + \log \left (\log \left (x \log \left (\log \left (2\right )\right ) - 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-72*x)*log(log(2))-x+72)*log(x*log(log(2))-1)+(x^2-36*x)*log(log(2)))/((x^3-36*x^2)*log(log(2)
)-x^2+36*x)/log(x*log(log(2))-1),x, algorithm="maxima")

[Out]

-log(x - 36) + 2*log(x) + log(log(x*log(log(2)) - 1))

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Fricas [A]
time = 0.31, size = 20, normalized size = 0.80 \begin {gather*} -\log \left (x - 36\right ) + 2 \, \log \left (x\right ) + \log \left (\log \left (x \log \left (\log \left (2\right )\right ) - 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-72*x)*log(log(2))-x+72)*log(x*log(log(2))-1)+(x^2-36*x)*log(log(2)))/((x^3-36*x^2)*log(log(2)
)-x^2+36*x)/log(x*log(log(2))-1),x, algorithm="fricas")

[Out]

-log(x - 36) + 2*log(x) + log(log(x*log(log(2)) - 1))

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Sympy [A]
time = 0.08, size = 20, normalized size = 0.80 \begin {gather*} 2 \log {\left (x \right )} - \log {\left (x - 36 \right )} + \log {\left (\log {\left (x \log {\left (\log {\left (2 \right )} \right )} - 1 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-72*x)*ln(ln(2))-x+72)*ln(x*ln(ln(2))-1)+(x**2-36*x)*ln(ln(2)))/((x**3-36*x**2)*ln(ln(2))-x**
2+36*x)/ln(x*ln(ln(2))-1),x)

[Out]

2*log(x) - log(x - 36) + log(log(x*log(log(2)) - 1))

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Giac [A]
time = 0.44, size = 20, normalized size = 0.80 \begin {gather*} -\log \left (x - 36\right ) + 2 \, \log \left (x\right ) + \log \left (\log \left (x \log \left (\log \left (2\right )\right ) - 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-72*x)*log(log(2))-x+72)*log(x*log(log(2))-1)+(x^2-36*x)*log(log(2)))/((x^3-36*x^2)*log(log(2)
)-x^2+36*x)/log(x*log(log(2))-1),x, algorithm="giac")

[Out]

-log(x - 36) + 2*log(x) + log(log(x*log(log(2)) - 1))

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Mupad [B]
time = 0.87, size = 20, normalized size = 0.80 \begin {gather*} 2\,\ln \left (x\right )-\ln \left (x-36\right )+\ln \left (\ln \left (x\,\ln \left (\ln \left (2\right )\right )-1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x*log(log(2)) - 1)*(x + log(log(2))*(72*x - x^2) - 72) + log(log(2))*(36*x - x^2))/(log(x*log(log(2))
- 1)*(log(log(2))*(36*x^2 - x^3) - 36*x + x^2)),x)

[Out]

2*log(x) - log(x - 36) + log(log(x*log(log(2)) - 1))

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