3.87.37 \(\int \frac {2-2 x}{x} \, dx\) [8637]

Optimal. Leaf size=9 \[ -6-2 x+\log \left (x^2\right ) \]

[Out]

-2*x+ln(x^2)-6

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.89, 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 = {45} \begin {gather*} 2 \log (x)-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 2*x)/x,x]

[Out]

-2*x + 2*Log[x]

Rule 45

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 {2}{x}\right ) \, dx\\ &=-2 x+2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 0.89 \begin {gather*} -2 x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 2*x)/x,x]

[Out]

-2*x + 2*Log[x]

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Maple [A]
time = 0.12, size = 9, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*ln(x)-2*x

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Maxima [A]
time = 0.27, size = 8, normalized size = 0.89 \begin {gather*} -2 \, x + 2 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + 2*log(x)

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Fricas [A]
time = 0.39, size = 8, normalized size = 0.89 \begin {gather*} -2 \, x + 2 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + 2*log(x)

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Sympy [A]
time = 0.02, size = 7, normalized size = 0.78 \begin {gather*} - 2 x + 2 \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + 2*log(x)

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Giac [A]
time = 0.41, size = 9, normalized size = 1.00 \begin {gather*} -2 \, x + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + 2*log(abs(x))

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Mupad [B]
time = 0.02, size = 8, normalized size = 0.89 \begin {gather*} 2\,\ln \left (x\right )-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 2)/x,x)

[Out]

2*log(x) - 2*x

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