3.77.11 \(\int -\frac {2 \log ^2(3)}{x^3} \, dx\) [7611]

Optimal. Leaf size=12 \[ \frac {2}{25}+\frac {\log ^2(3)}{x^2} \]

[Out]

2/25+ln(3)^2/x^2

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 30} \begin {gather*} \frac {\log ^2(3)}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*Log[3]^2)/x^3,x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log ^2(3)}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*Log[3]^2)/x^3,x]

[Out]

Log[3]^2/x^2

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Maple [A]
time = 0.28, size = 9, normalized size = 0.75

method	result	size
gosper	\(\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(9\)
default	\(\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(9\)
norman	\(\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(9\)
risch	\(\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(3)^2/x^2

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Maxima [A]
time = 0.28, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log \left (3\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3)^2/x^2

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Fricas [A]
time = 0.37, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log \left (3\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3)^2/x^2

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Sympy [A]
time = 0.01, size = 7, normalized size = 0.58 \begin {gather*} \frac {\log {\left (3 \right )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3)**2/x**2

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Giac [A]
time = 0.41, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log \left (3\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3)^2/x^2

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Mupad [B]
time = 0.04, size = 8, normalized size = 0.67 \begin {gather*} \frac {{\ln \left (3\right )}^2}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(3)^2)/x^3,x)

[Out]

log(3)^2/x^2

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