3.43.14 \(\int \frac {-28 x+14 x \log (x)+2 \log (-2+\log (x))}{-28 x^2+14 x^2 \log (x)+(-2 x+x \log (x)) \log ^2(-2+\log (x))} \, dx\)

Optimal. Leaf size=12 \[ \log \left (14 x+\log ^2(-2+\log (x))\right ) \]

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Rubi [A]  time = 0.19, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6741, 12, 6684} \begin {gather*} \log \left (14 x+\log ^2(\log (x)-2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-28*x + 14*x*Log[x] + 2*Log[-2 + Log[x]])/(-28*x^2 + 14*x^2*Log[x] + (-2*x + x*Log[x])*Log[-2 + Log[x]]^2
),x]

[Out]

Log[14*x + Log[-2 + 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 6684

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

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 (14 x-7 x \log (x)-\log (-2+\log (x)))}{x (2-\log (x)) \left (14 x+\log ^2(-2+\log (x))\right )} \, dx\\ &=2 \int \frac {14 x-7 x \log (x)-\log (-2+\log (x))}{x (2-\log (x)) \left (14 x+\log ^2(-2+\log (x))\right )} \, dx\\ &=\log \left (14 x+\log ^2(-2+\log (x))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 12, normalized size = 1.00 \begin {gather*} \log \left (14 x+\log ^2(-2+\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-28*x + 14*x*Log[x] + 2*Log[-2 + Log[x]])/(-28*x^2 + 14*x^2*Log[x] + (-2*x + x*Log[x])*Log[-2 + Log
[x]]^2),x]

[Out]

Log[14*x + Log[-2 + Log[x]]^2]

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fricas [A]  time = 0.68, size = 12, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\log \relax (x) - 2\right )^{2} + 14 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(log(x)-2)+14*x*log(x)-28*x)/((x*log(x)-2*x)*log(log(x)-2)^2+14*x^2*log(x)-28*x^2),x, algorith
m="fricas")

[Out]

log(log(log(x) - 2)^2 + 14*x)

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giac [A]  time = 0.29, size = 12, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\log \relax (x) - 2\right )^{2} + 14 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(log(x)-2)+14*x*log(x)-28*x)/((x*log(x)-2*x)*log(log(x)-2)^2+14*x^2*log(x)-28*x^2),x, algorith
m="giac")

[Out]

log(log(log(x) - 2)^2 + 14*x)

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maple [A]  time = 0.02, size = 13, normalized size = 1.08

method	result	size
risch	\(\ln \left (14 x +\ln \left (\ln \relax (x )-2\right )^{2}\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(ln(x)-2)+14*x*ln(x)-28*x)/((x*ln(x)-2*x)*ln(ln(x)-2)^2+14*x^2*ln(x)-28*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(14*x+ln(ln(x)-2)^2)

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maxima [A]  time = 0.38, size = 12, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\log \relax (x) - 2\right )^{2} + 14 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(log(x)-2)+14*x*log(x)-28*x)/((x*log(x)-2*x)*log(log(x)-2)^2+14*x^2*log(x)-28*x^2),x, algorith
m="maxima")

[Out]

log(log(log(x) - 2)^2 + 14*x)

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mupad [B]  time = 3.41, size = 12, normalized size = 1.00 \begin {gather*} \ln \left ({\ln \left (\ln \relax (x)-2\right )}^2+14\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(log(x) - 2) - 28*x + 14*x*log(x))/(log(log(x) - 2)^2*(2*x - x*log(x)) - 14*x^2*log(x) + 28*x^2),x)

[Out]

log(14*x + log(log(x) - 2)^2)

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sympy [A]  time = 0.39, size = 12, normalized size = 1.00 \begin {gather*} \log {\left (14 x + \log {\left (\log {\relax (x )} - 2 \right )}^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(ln(x)-2)+14*x*ln(x)-28*x)/((x*ln(x)-2*x)*ln(ln(x)-2)**2+14*x**2*ln(x)-28*x**2),x)

[Out]

log(14*x + log(log(x) - 2)**2)

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