3.25.61 \(\int \frac {-x+8 \log (16)-x \log (x)}{(-x^2+8 x \log (16)) \log (x)} \, dx\)

Optimal. Leaf size=11 \[ \log (5 (x-8 \log (16)) \log (x)) \]

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Rubi [A]  time = 0.26, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1593, 6742, 2302, 29} \begin {gather*} \log (x-8 \log (16))+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x + 8*Log[16] - x*Log[x])/((-x^2 + 8*x*Log[16])*Log[x]),x]

[Out]

Log[x - 8*Log[16]] + Log[Log[x]]

Rule 29

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2302

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+8 \log (16)-x \log (x)}{x (-x+8 \log (16)) \log (x)} \, dx\\ &=\int \left (\frac {1}{x-8 \log (16)}+\frac {1}{x \log (x)}\right ) \, dx\\ &=\log (x-8 \log (16))+\int \frac {1}{x \log (x)} \, dx\\ &=\log (x-8 \log (16))+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\log (x-8 \log (16))+\log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 11, normalized size = 1.00 \begin {gather*} \log (x-8 \log (16))+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + 8*Log[16] - x*Log[x])/((-x^2 + 8*x*Log[16])*Log[x]),x]

[Out]

Log[x - 8*Log[16]] + Log[Log[x]]

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fricas [A]  time = 0.67, size = 11, normalized size = 1.00 \begin {gather*} \log \left (x - 32 \, \log \relax (2)\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)+32*log(2)-x)/(32*x*log(2)-x^2)/log(x),x, algorithm="fricas")

[Out]

log(x - 32*log(2)) + log(log(x))

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giac [A]  time = 0.34, size = 11, normalized size = 1.00 \begin {gather*} \log \left (x - 32 \, \log \relax (2)\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)+32*log(2)-x)/(32*x*log(2)-x^2)/log(x),x, algorithm="giac")

[Out]

log(x - 32*log(2)) + log(log(x))

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maple [A]  time = 0.40, size = 12, normalized size = 1.09

method	result	size
risch	\(\ln \left (-32 \ln \relax (2)+x \right )+\ln \left (\ln \relax (x )\right )\)	\(12\)
default	\(\ln \left (\ln \relax (x )\right )+\ln \left (32 \ln \relax (2)-x \right )\)	\(14\)
norman	\(\ln \left (\ln \relax (x )\right )+\ln \left (32 \ln \relax (2)-x \right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*ln(x)+32*ln(2)-x)/(32*x*ln(2)-x^2)/ln(x),x,method=_RETURNVERBOSE)

[Out]

ln(-32*ln(2)+x)+ln(ln(x))

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maxima [A]  time = 0.62, size = 11, normalized size = 1.00 \begin {gather*} \log \left (x - 32 \, \log \relax (2)\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)+32*log(2)-x)/(32*x*log(2)-x^2)/log(x),x, algorithm="maxima")

[Out]

log(x - 32*log(2)) + log(log(x))

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mupad [B]  time = 1.47, size = 11, normalized size = 1.00 \begin {gather*} \ln \left (x-32\,\ln \relax (2)\right )+\ln \left (\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 32*log(2) + x*log(x))/(log(x)*(32*x*log(2) - x^2)),x)

[Out]

log(x - 32*log(2)) + log(log(x))

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sympy [A]  time = 0.11, size = 12, normalized size = 1.09 \begin {gather*} \log {\left (x - 32 \log {\relax (2 )} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*ln(x)+32*ln(2)-x)/(32*x*ln(2)-x**2)/ln(x),x)

[Out]

log(x - 32*log(2)) + log(log(x))

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