3.30.77 \(\int -\frac {2}{x} \, dx\)

Optimal. Leaf size=18 \[ 1-18 \log \left (\frac {3}{5 e^3}\right )-\log \left (x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 4, normalized size of antiderivative = 0.22, number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {12, 29} \begin {gather*} -2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2/x,x]

[Out]

-2*Log[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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (2 \int \frac {1}{x} \, dx\right )\\ &=-2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 4, normalized size = 0.22 \begin {gather*} -2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2/x,x]

[Out]

-2*Log[x]

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fricas [A]  time = 0.59, size = 4, normalized size = 0.22 \begin {gather*} -2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/x,x, algorithm="fricas")

[Out]

-2*log(x)

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giac [A]  time = 0.21, size = 5, normalized size = 0.28 \begin {gather*} -2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/x,x, algorithm="giac")

[Out]

-2*log(abs(x))

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maple [A]  time = 0.02, size = 5, normalized size = 0.28

method	result	size
default	\(-2 \ln \relax (x )\)	\(5\)
norman	\(-2 \ln \relax (x )\)	\(5\)
risch	\(-2 \ln \relax (x )\)	\(5\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/x,x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)

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maxima [A]  time = 0.43, size = 4, normalized size = 0.22 \begin {gather*} -2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/x,x, algorithm="maxima")

[Out]

-2*log(x)

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mupad [B]  time = 0.01, size = 4, normalized size = 0.22 \begin {gather*} -2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/x,x)

[Out]

-2*log(x)

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sympy [A]  time = 0.05, size = 5, normalized size = 0.28 \begin {gather*} - 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/x,x)

[Out]

-2*log(x)

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