3.41.35 \(\int \frac {1-x}{x} \, dx\) [4035]

Optimal. Leaf size=9 \[ 1-x+\log (4)+\log (x) \]

[Out]

-x+ln(x)+2*ln(2)+1

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Rubi [A]
time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.67, 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*} \log (x)-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)/x,x]

[Out]

-x + 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 (-1+\frac {1}{x}\right ) \, dx\\ &=-x+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 6, normalized size = 0.67 \begin {gather*} -x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)/x,x]

[Out]

-x + Log[x]

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Maple [A]
time = 0.00, size = 7, normalized size = 0.78

method	result	size
default	\(\ln \left (x \right )-x\)	\(7\)
norman	\(\ln \left (x \right )-x\)	\(7\)
risch	\(\ln \left (x \right )-x\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)-x

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Maxima [A]
time = 0.28, size = 6, normalized size = 0.67 \begin {gather*} -x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(x)

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Fricas [A]
time = 0.37, size = 6, normalized size = 0.67 \begin {gather*} -x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(x)

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Sympy [A]
time = 0.02, size = 3, normalized size = 0.33 \begin {gather*} - x + \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(x)

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Giac [A]
time = 0.42, size = 7, normalized size = 0.78 \begin {gather*} -x + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(abs(x))

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Mupad [B]
time = 0.02, size = 6, normalized size = 0.67 \begin {gather*} \ln \left (x\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 1)/x,x)

[Out]

log(x) - x

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