3.44.61 \(\int \frac {2 \log (-5 x)}{x} \, dx\)

Optimal. Leaf size=6 \[ \log ^2(-5 x) \]

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Rubi [A]  time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, 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, 2301} \begin {gather*} \log ^2(-5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*Log[-5*x])/x,x]

[Out]

Log[-5*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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 \int \frac {\log (-5 x)}{x} \, dx\\ &=\log ^2(-5 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} \log ^2(-5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*Log[-5*x])/x,x]

[Out]

Log[-5*x]^2

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fricas [A]  time = 0.63, size = 6, normalized size = 1.00 \begin {gather*} \log \left (-5 \, x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(-5*x)/x,x, algorithm="fricas")

[Out]

log(-5*x)^2

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giac [A]  time = 0.13, size = 6, normalized size = 1.00 \begin {gather*} \log \left (-5 \, x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(-5*x)/x,x, algorithm="giac")

[Out]

log(-5*x)^2

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maple [A]  time = 0.02, size = 7, normalized size = 1.17

method	result	size
derivativedivides	\(\ln \left (-5 x \right )^{2}\)	\(7\)
default	\(\ln \left (-5 x \right )^{2}\)	\(7\)
norman	\(\ln \left (-5 x \right )^{2}\)	\(7\)
risch	\(\ln \left (-5 x \right )^{2}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*ln(-5*x)/x,x,method=_RETURNVERBOSE)

[Out]

ln(-5*x)^2

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maxima [A]  time = 0.35, size = 6, normalized size = 1.00 \begin {gather*} \log \left (-5 \, x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(-5*x)/x,x, algorithm="maxima")

[Out]

log(-5*x)^2

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mupad [B]  time = 3.41, size = 6, normalized size = 1.00 \begin {gather*} {\ln \left (-5\,x\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(-5*x))/x,x)

[Out]

log(-5*x)^2

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sympy [A]  time = 0.08, size = 7, normalized size = 1.17 \begin {gather*} \log {\left (- 5 x \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*ln(-5*x)/x,x)

[Out]

log(-5*x)**2

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