3.64.51 \(\int \frac {-1-2 x}{x} \, dx\)

Optimal. Leaf size=11 \[ -4-2 x+\log \left (\frac {4}{x}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.73, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} -2 x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 2*x)/x,x]

[Out]

-2*x - Log[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.73 \begin {gather*} -2 x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 2*x)/x,x]

[Out]

-2*x - Log[x]

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fricas [A]  time = 0.55, size = 8, normalized size = 0.73 \begin {gather*} -2 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-1)/x,x, algorithm="fricas")

[Out]

-2*x - log(x)

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giac [A]  time = 0.14, size = 9, normalized size = 0.82 \begin {gather*} -2 \, x - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-1)/x,x, algorithm="giac")

[Out]

-2*x - log(abs(x))

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maple [A]  time = 0.02, size = 9, normalized size = 0.82

method	result	size
default	\(-2 x -\ln \relax (x )\)	\(9\)
norman	\(-2 x -\ln \relax (x )\)	\(9\)
risch	\(-2 x -\ln \relax (x )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-1)/x,x,method=_RETURNVERBOSE)

[Out]

-2*x-ln(x)

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maxima [A]  time = 0.37, size = 8, normalized size = 0.73 \begin {gather*} -2 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-1)/x,x, algorithm="maxima")

[Out]

-2*x - log(x)

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mupad [B]  time = 0.02, size = 8, normalized size = 0.73 \begin {gather*} -2\,x-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 1)/x,x)

[Out]

- 2*x - log(x)

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sympy [A]  time = 0.06, size = 7, normalized size = 0.64 \begin {gather*} - 2 x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-1)/x,x)

[Out]

-2*x - log(x)

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