3.15.37 \(\int \frac {-4-4 x}{x^2} \, dx\)

Optimal. Leaf size=19 \[ 4 \left (-1+\frac {1-x}{x}+\log (4)-\log (x)\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.53, 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*} \frac {4}{x}-4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 4*x)/x^2,x]

[Out]

4/x - 4*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 (-\frac {4}{x^2}-\frac {4}{x}\right ) \, dx\\ &=\frac {4}{x}-4 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.53 \begin {gather*} -4 \left (-\frac {1}{x}+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 4*x)/x^2,x]

[Out]

-4*(-x^(-1) + Log[x])

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fricas [A]  time = 0.61, size = 11, normalized size = 0.58 \begin {gather*} -\frac {4 \, {\left (x \log \relax (x) - 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)/x^2,x, algorithm="fricas")

[Out]

-4*(x*log(x) - 1)/x

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giac [A]  time = 0.29, size = 11, normalized size = 0.58 \begin {gather*} \frac {4}{x} - 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)/x^2,x, algorithm="giac")

[Out]

4/x - 4*log(abs(x))

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maple [A]  time = 0.02, size = 11, normalized size = 0.58

method	result	size
default	\(-4 \ln \relax (x )+\frac {4}{x}\)	\(11\)
norman	\(-4 \ln \relax (x )+\frac {4}{x}\)	\(11\)
risch	\(-4 \ln \relax (x )+\frac {4}{x}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x-4)/x^2,x,method=_RETURNVERBOSE)

[Out]

-4*ln(x)+4/x

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maxima [A]  time = 0.43, size = 10, normalized size = 0.53 \begin {gather*} \frac {4}{x} - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)/x^2,x, algorithm="maxima")

[Out]

4/x - 4*log(x)

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mupad [B]  time = 0.02, size = 10, normalized size = 0.53 \begin {gather*} \frac {4}{x}-4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 4)/x^2,x)

[Out]

4/x - 4*log(x)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.37 \begin {gather*} - 4 \log {\relax (x )} + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-4)/x**2,x)

[Out]

-4*log(x) + 4/x

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