3.58.19 \(\int \frac {5+3 x^2 \log (\frac {\log (4)}{2})}{3 x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{3} \left (25-\frac {5}{x}\right )+x \log \left (\frac {\log (4)}{2}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 14} \begin {gather*} x \log (\log (2))-\frac {5}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 3*x^2*Log[Log[4]/2])/(3*x^2),x]

[Out]

-5/(3*x) + x*Log[Log[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 {1}{3} \int \frac {5+3 x^2 \log \left (\frac {\log (4)}{2}\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {5}{x^2}+3 \log (\log (2))\right ) \, dx\\ &=-\frac {5}{3 x}+x \log (\log (2))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.62 \begin {gather*} -\frac {5}{3 x}+x \log (\log (2)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 3*x^2*Log[Log[4]/2])/(3*x^2),x]

[Out]

-5/(3*x) + x*Log[Log[2]]

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fricas [A]  time = 0.79, size = 15, normalized size = 0.71 \begin {gather*} \frac {3 \, x^{2} \log \left (\log \relax (2)\right ) - 5}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x^2*log(log(2))+5)/x^2,x, algorithm="fricas")

[Out]

1/3*(3*x^2*log(log(2)) - 5)/x

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giac [A]  time = 0.14, size = 11, normalized size = 0.52 \begin {gather*} x \log \left (\log \relax (2)\right ) - \frac {5}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x^2*log(log(2))+5)/x^2,x, algorithm="giac")

[Out]

x*log(log(2)) - 5/3/x

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maple [A]  time = 0.04, size = 12, normalized size = 0.57

method	result	size
default	\(x \ln \left (\ln \relax (2)\right )-\frac {5}{3 x}\)	\(12\)
risch	\(x \ln \left (\ln \relax (2)\right )-\frac {5}{3 x}\)	\(12\)
norman	\(\frac {-\frac {5}{3}+x^{2} \ln \left (\ln \relax (2)\right )}{x}\)	\(14\)
gosper	\(\frac {3 x^{2} \ln \left (\ln \relax (2)\right )-5}{3 x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(3*x^2*ln(ln(2))+5)/x^2,x,method=_RETURNVERBOSE)

[Out]

x*ln(ln(2))-5/3/x

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maxima [A]  time = 0.35, size = 11, normalized size = 0.52 \begin {gather*} x \log \left (\log \relax (2)\right ) - \frac {5}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x^2*log(log(2))+5)/x^2,x, algorithm="maxima")

[Out]

x*log(log(2)) - 5/3/x

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mupad [B]  time = 0.03, size = 11, normalized size = 0.52 \begin {gather*} x\,\ln \left (\ln \relax (2)\right )-\frac {5}{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*log(log(2)) + 5/3)/x^2,x)

[Out]

x*log(log(2)) - 5/(3*x)

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sympy [A]  time = 0.07, size = 10, normalized size = 0.48 \begin {gather*} x \log {\left (\log {\relax (2 )} \right )} - \frac {5}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*x**2*ln(ln(2))+5)/x**2,x)

[Out]

x*log(log(2)) - 5/(3*x)

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