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

Optimal. Leaf size=24 \[ \frac {3}{2 x}+2 \left (\frac {2}{\log ^2(2)}-\frac {\log (4)}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.54, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 30} \begin {gather*} \frac {3-4 \log (4)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 4*Log[4])/(2*x^2),x]

[Out]

(3 - 4*Log[4])/(2*x)

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} (-3+\log (256)) \int \frac {1}{x^2} \, dx\\ &=\frac {3-\log (256)}{2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.46 \begin {gather*} -\frac {-3+\log (256)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 4*Log[4])/(2*x^2),x]

[Out]

-1/2*(-3 + Log[256])/x

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fricas [A]  time = 0.75, size = 11, normalized size = 0.46 \begin {gather*} -\frac {8 \, \log \relax (2) - 3}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(8*log(2)-3)/x^2,x, algorithm="fricas")

[Out]

-1/2*(8*log(2) - 3)/x

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giac [A]  time = 0.18, size = 11, normalized size = 0.46 \begin {gather*} -\frac {8 \, \log \relax (2) - 3}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(8*log(2)-3)/x^2,x, algorithm="giac")

[Out]

-1/2*(8*log(2) - 3)/x

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maple [A]  time = 0.02, size = 11, normalized size = 0.46

method	result	size
norman	\(\frac {-4 \ln \relax (2)+\frac {3}{2}}{x}\)	\(11\)
gosper	\(-\frac {8 \ln \relax (2)-3}{2 x}\)	\(12\)
default	\(-\frac {4 \ln \relax (2)-\frac {3}{2}}{x}\)	\(12\)
risch	\(-\frac {4 \ln \relax (2)}{x}+\frac {3}{2 x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(8*ln(2)-3)/x^2,x,method=_RETURNVERBOSE)

[Out]

(-4*ln(2)+3/2)/x

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maxima [A]  time = 0.52, size = 11, normalized size = 0.46 \begin {gather*} -\frac {8 \, \log \relax (2) - 3}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(8*log(2)-3)/x^2,x, algorithm="maxima")

[Out]

-1/2*(8*log(2) - 3)/x

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mupad [B]  time = 0.02, size = 9, normalized size = 0.38 \begin {gather*} -\frac {\ln \left (16\right )-\frac {3}{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(2) - 3/2)/x^2,x)

[Out]

-(log(16) - 3/2)/x

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sympy [A]  time = 0.06, size = 10, normalized size = 0.42 \begin {gather*} - \frac {- \frac {3}{2} + 4 \log {\relax (2 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(8*ln(2)-3)/x**2,x)

[Out]

-(-3/2 + 4*log(2))/x

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