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

Optimal result

Integrand size = 9, antiderivative size = 12 \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {2}{25}+\frac {\log ^2(3)}{x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 30} \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {\log ^2(3)}{x^2} \]

[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{align*} \text {integral}& = -\left (\left (2 \log ^2(3)\right ) \int \frac {1}{x^3} \, dx\right ) \\ & = \frac {\log ^2(3)}{x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {\log ^2(3)}{x^2} \]

[In]

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

[Out]

Log[3]^2/x^2

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 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	\(\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(9\)
parallelrisch	\(\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(9\)

[In]

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

[Out]

ln(3)^2/x^2

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {\log \left (3\right )^{2}}{x^{2}} \]

[In]

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

[Out]

log(3)^2/x^2

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {\log {\left (3 \right )}^{2}}{x^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {\log \left (3\right )^{2}}{x^{2}} \]

[In]

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

[Out]

log(3)^2/x^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {\log \left (3\right )^{2}}{x^{2}} \]

[In]

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

[Out]

log(3)^2/x^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int -\frac {2 \log ^2(3)}{x^3} \, dx=\frac {{\ln \left (3\right )}^2}{x^2} \]

[In]

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

[Out]

log(3)^2/x^2