3.71.67 \(\int \frac {-36 \log ^2(4)-72 x^2 \log (4) \log ^2(\log (2))+108 x^4 \log ^4(\log (2))}{x^2 \log ^4(\log (2))} \, dx\)

Optimal. Leaf size=20 \[ 36 x^3 \left (-1+\frac {\log (4)}{x^2 \log ^2(\log (2))}\right )^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 14} \begin {gather*} 36 x^3-\frac {72 x \log (4)}{\log ^2(\log (2))}+\frac {36 \log ^2(4)}{x \log ^4(\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36*Log[4]^2 - 72*x^2*Log[4]*Log[Log[2]]^2 + 108*x^4*Log[Log[2]]^4)/(x^2*Log[Log[2]]^4),x]

[Out]

36*x^3 + (36*Log[4]^2)/(x*Log[Log[2]]^4) - (72*x*Log[4])/Log[Log[2]]^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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-36 \log ^2(4)-72 x^2 \log (4) \log ^2(\log (2))+108 x^4 \log ^4(\log (2))}{x^2} \, dx}{\log ^4(\log (2))}\\ &=\frac {\int \left (-\frac {36 \log ^2(4)}{x^2}-72 \log (4) \log ^2(\log (2))+108 x^2 \log ^4(\log (2))\right ) \, dx}{\log ^4(\log (2))}\\ &=36 x^3+\frac {36 \log ^2(4)}{x \log ^4(\log (2))}-\frac {72 x \log (4)}{\log ^2(\log (2))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.45 \begin {gather*} 36 \left (x^3+\frac {\log ^2(4)}{x \log ^4(\log (2))}-\frac {x \log (16)}{\log ^2(\log (2))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36*Log[4]^2 - 72*x^2*Log[4]*Log[Log[2]]^2 + 108*x^4*Log[Log[2]]^4)/(x^2*Log[Log[2]]^4),x]

[Out]

36*(x^3 + Log[4]^2/(x*Log[Log[2]]^4) - (x*Log[16])/Log[Log[2]]^2)

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fricas [A]  time = 0.55, size = 38, normalized size = 1.90 \begin {gather*} \frac {36 \, {\left (x^{4} \log \left (\log \relax (2)\right )^{4} - 4 \, x^{2} \log \relax (2) \log \left (\log \relax (2)\right )^{2} + 4 \, \log \relax (2)^{2}\right )}}{x \log \left (\log \relax (2)\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((108*x^4*log(log(2))^4-144*x^2*log(2)*log(log(2))^2-144*log(2)^2)/x^2/log(log(2))^4,x, algorithm="fr
icas")

[Out]

36*(x^4*log(log(2))^4 - 4*x^2*log(2)*log(log(2))^2 + 4*log(2)^2)/(x*log(log(2))^4)

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giac [A]  time = 0.12, size = 36, normalized size = 1.80 \begin {gather*} \frac {36 \, {\left (x^{3} \log \left (\log \relax (2)\right )^{4} - 4 \, x \log \relax (2) \log \left (\log \relax (2)\right )^{2} + \frac {4 \, \log \relax (2)^{2}}{x}\right )}}{\log \left (\log \relax (2)\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((108*x^4*log(log(2))^4-144*x^2*log(2)*log(log(2))^2-144*log(2)^2)/x^2/log(log(2))^4,x, algorithm="gi
ac")

[Out]

36*(x^3*log(log(2))^4 - 4*x*log(2)*log(log(2))^2 + 4*log(2)^2/x)/log(log(2))^4

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maple [A]  time = 0.05, size = 31, normalized size = 1.55

method	result	size
risch	\(36 x^{3}-\frac {144 x \ln \relax (2)}{\ln \left (\ln \relax (2)\right )^{2}}+\frac {144 \ln \relax (2)^{2}}{\ln \left (\ln \relax (2)\right )^{4} x}\)	\(31\)
default	\(\frac {36 x^{3} \ln \left (\ln \relax (2)\right )^{4}-144 x \ln \relax (2) \ln \left (\ln \relax (2)\right )^{2}+\frac {144 \ln \relax (2)^{2}}{x}}{\ln \left (\ln \relax (2)\right )^{4}}\)	\(37\)
gosper	\(\frac {36 x^{4} \ln \left (\ln \relax (2)\right )^{4}-144 x^{2} \ln \relax (2) \ln \left (\ln \relax (2)\right )^{2}+144 \ln \relax (2)^{2}}{\ln \left (\ln \relax (2)\right )^{4} x}\)	\(39\)
norman	\(\frac {\frac {144 \ln \relax (2)^{2}}{\ln \left (\ln \relax (2)\right )}+36 \ln \left (\ln \relax (2)\right )^{3} x^{4}-144 \ln \relax (2) \ln \left (\ln \relax (2)\right ) x^{2}}{x \ln \left (\ln \relax (2)\right )^{3}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((108*x^4*ln(ln(2))^4-144*x^2*ln(2)*ln(ln(2))^2-144*ln(2)^2)/x^2/ln(ln(2))^4,x,method=_RETURNVERBOSE)

[Out]

36*x^3-144/ln(ln(2))^2*x*ln(2)+144/ln(ln(2))^4*ln(2)^2/x

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maxima [A]  time = 0.36, size = 36, normalized size = 1.80 \begin {gather*} \frac {36 \, {\left (x^{3} \log \left (\log \relax (2)\right )^{4} - 4 \, x \log \relax (2) \log \left (\log \relax (2)\right )^{2} + \frac {4 \, \log \relax (2)^{2}}{x}\right )}}{\log \left (\log \relax (2)\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((108*x^4*log(log(2))^4-144*x^2*log(2)*log(log(2))^2-144*log(2)^2)/x^2/log(log(2))^4,x, algorithm="ma
xima")

[Out]

36*(x^3*log(log(2))^4 - 4*x*log(2)*log(log(2))^2 + 4*log(2)^2/x)/log(log(2))^4

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mupad [B]  time = 0.07, size = 27, normalized size = 1.35 \begin {gather*} \frac {36\,{\left (2\,\ln \relax (2)-x^2\,{\ln \left (\ln \relax (2)\right )}^2\right )}^2}{x\,{\ln \left (\ln \relax (2)\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(144*log(2)^2 - 108*x^4*log(log(2))^4 + 144*x^2*log(2)*log(log(2))^2)/(x^2*log(log(2))^4),x)

[Out]

(36*(2*log(2) - x^2*log(log(2))^2)^2)/(x*log(log(2))^4)

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sympy [A]  time = 0.09, size = 39, normalized size = 1.95 \begin {gather*} \frac {36 x^{3} \log {\left (\log {\relax (2 )} \right )}^{4} - 144 x \log {\relax (2 )} \log {\left (\log {\relax (2 )} \right )}^{2} + \frac {144 \log {\relax (2 )}^{2}}{x}}{\log {\left (\log {\relax (2 )} \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((108*x**4*ln(ln(2))**4-144*x**2*ln(2)*ln(ln(2))**2-144*ln(2)**2)/x**2/ln(ln(2))**4,x)

[Out]

(36*x**3*log(log(2))**4 - 144*x*log(2)*log(log(2))**2 + 144*log(2)**2/x)/log(log(2))**4

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