3.37.51 \(\int -\frac {20}{x \log (\frac {2 x^2}{1+\log (\log (5))})} \, dx\)

Optimal. Leaf size=26 \[ 5 \left (1+\log \left (\frac {e^4}{\log ^2\left (\frac {2 x^3}{x+x \log (\log (5))}\right )}\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2302, 29} \begin {gather*} -10 \log \left (\log \left (\frac {2 x^2}{1+\log (\log (5))}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-20/(x*Log[(2*x^2)/(1 + Log[Log[5]])]),x]

[Out]

-10*Log[Log[(2*x^2)/(1 + Log[Log[5]])]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

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

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (20 \int \frac {1}{x \log \left (\frac {2 x^2}{1+\log (\log (5))}\right )} \, dx\right )\\ &=-\left (10 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {2 x^2}{1+\log (\log (5))}\right )\right )\right )\\ &=-10 \log \left (\log \left (\frac {2 x^2}{1+\log (\log (5))}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.62 \begin {gather*} -10 \log \left (\log \left (\frac {2 x^2}{1+\log (\log (5))}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-20/(x*Log[(2*x^2)/(1 + Log[Log[5]])]),x]

[Out]

-10*Log[Log[(2*x^2)/(1 + Log[Log[5]])]]

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fricas [A]  time = 0.64, size = 16, normalized size = 0.62 \begin {gather*} -10 \, \log \left (\log \left (\frac {2 \, x^{2}}{\log \left (\log \relax (5)\right ) + 1}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-20/x/log(2*x^2/(log(log(5))+1)),x, algorithm="fricas")

[Out]

-10*log(log(2*x^2/(log(log(5)) + 1)))

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giac [A]  time = 2.68, size = 17, normalized size = 0.65 \begin {gather*} -10 \, \log \left ({\left | \log \left (\frac {2 \, x^{2}}{\log \left (\log \relax (5)\right ) + 1}\right ) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-20/x/log(2*x^2/(log(log(5))+1)),x, algorithm="giac")

[Out]

-10*log(abs(log(2*x^2/(log(log(5)) + 1))))

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maple [A]  time = 0.02, size = 17, normalized size = 0.65

method	result	size
derivativedivides	\(-10 \ln \left (\ln \left (\frac {2 x^{2}}{\ln \left (\ln \relax (5)\right )+1}\right )\right )\)	\(17\)
default	\(-10 \ln \left (\ln \left (\frac {2 x^{2}}{\ln \left (\ln \relax (5)\right )+1}\right )\right )\)	\(17\)
norman	\(-10 \ln \left (\ln \left (\frac {2 x^{2}}{\ln \left (\ln \relax (5)\right )+1}\right )\right )\)	\(17\)
risch	\(-10 \ln \left (\ln \left (\frac {2 x^{2}}{\ln \left (\ln \relax (5)\right )+1}\right )\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-20/x/ln(2*x^2/(ln(ln(5))+1)),x,method=_RETURNVERBOSE)

[Out]

-10*ln(ln(2*x^2/(ln(ln(5))+1)))

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maxima [A]  time = 0.49, size = 16, normalized size = 0.62 \begin {gather*} -10 \, \log \left (\log \left (\frac {2 \, x^{2}}{\log \left (\log \relax (5)\right ) + 1}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-20/x/log(2*x^2/(log(log(5))+1)),x, algorithm="maxima")

[Out]

-10*log(log(2*x^2/(log(log(5)) + 1)))

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mupad [B]  time = 2.25, size = 18, normalized size = 0.69 \begin {gather*} -10\,\ln \left (\ln \left (x^2\right )+\ln \relax (2)-\ln \left (\ln \left (\ln \relax (5)\right )+1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-20/(x*log((2*x^2)/(log(log(5)) + 1))),x)

[Out]

-10*log(log(x^2) + log(2) - log(log(log(5)) + 1))

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sympy [A]  time = 0.11, size = 17, normalized size = 0.65 \begin {gather*} - 10 \log {\left (\log {\left (\frac {2 x^{2}}{\log {\left (\log {\relax (5 )} \right )} + 1} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-20/x/ln(2*x**2/(ln(ln(5))+1)),x)

[Out]

-10*log(log(2*x**2/(log(log(5)) + 1)))

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