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

Optimal. Leaf size=34 \[ -1+x+\log \left (\frac {1}{2} \left (\frac {4}{x}+\frac {5 e^{\frac {5}{4} \left (-2+e^{e^3}\right )^2}}{x}\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.18, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {43} \begin {gather*} x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/x,x]

[Out]

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 (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 = 0.18 \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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fricas [A]  time = 0.66, size = 6, normalized size = 0.18 \begin {gather*} x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - log(x)

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giac [A]  time = 0.16, size = 7, normalized size = 0.21 \begin {gather*} x - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - log(abs(x))

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maple [A]  time = 0.02, size = 7, normalized size = 0.21

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-ln(x)

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maxima [A]  time = 0.47, size = 6, normalized size = 0.18 \begin {gather*} x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - log(x)

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mupad [B]  time = 0.01, size = 6, normalized size = 0.18 \begin {gather*} x-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/x,x)

[Out]

x - log(x)

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sympy [A]  time = 0.06, size = 3, normalized size = 0.09 \begin {gather*} x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - log(x)

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