3.65.67 \(\int -\frac {1}{x} \, dx\)

Optimal. Leaf size=14 \[ 4-\frac {1}{4} \log \left (\frac {x^4}{16}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 4, normalized size of antiderivative = 0.29, number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {29} \begin {gather*} -\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-x^(-1),x]

[Out]

-Log[x]

Rule 29

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 4, normalized size = 0.29 \begin {gather*} -\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-x^(-1),x]

[Out]

-Log[x]

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fricas [A]  time = 0.64, size = 4, normalized size = 0.29 \begin {gather*} -\log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/x,x, algorithm="fricas")

[Out]

-log(x)

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giac [A]  time = 0.18, size = 5, normalized size = 0.36 \begin {gather*} -\log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/x,x, algorithm="giac")

[Out]

-log(abs(x))

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maple [A]  time = 0.01, size = 5, normalized size = 0.36

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/x,x,method=_RETURNVERBOSE)

[Out]

-ln(x)

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maxima [A]  time = 0.37, size = 4, normalized size = 0.29 \begin {gather*} -\log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/x,x, algorithm="maxima")

[Out]

-log(x)

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mupad [B]  time = 0.01, size = 4, normalized size = 0.29 \begin {gather*} -\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/x,x)

[Out]

-log(x)

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sympy [A]  time = 0.05, size = 3, normalized size = 0.21 \begin {gather*} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/x,x)

[Out]

-log(x)

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