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

Optimal. Leaf size=12 \[ \log (1-b x)-\log (x) \]

[Out]

-Log[x] + Log[1 - b*x]

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Rubi [A]  time = 0.0023875, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {36, 29, 31} \[ \log (1-b x)-\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(-1 + b*x)),x]

[Out]

-Log[x] + Log[1 - b*x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{x (-1+b x)} \, dx &=b \int \frac{1}{-1+b x} \, dx-\int \frac{1}{x} \, dx\\ &=-\log (x)+\log (1-b x)\\ \end{align*}

Mathematica [A]  time = 0.0028893, size = 12, normalized size = 1. \[ \log (1-b x)-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-1 + b*x)),x]

[Out]

-Log[x] + Log[1 - b*x]

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Maple [A]  time = 0.006, size = 12, normalized size = 1. \begin{align*} \ln \left ( bx-1 \right ) -\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x-1),x)

[Out]

ln(b*x-1)-ln(x)

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Maxima [A]  time = 1.07908, size = 15, normalized size = 1.25 \begin{align*} \log \left (b x - 1\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x - 1) - log(x)

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Fricas [A]  time = 1.4917, size = 31, normalized size = 2.58 \begin{align*} \log \left (b x - 1\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x - 1) - log(x)

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Sympy [A]  time = 0.131059, size = 8, normalized size = 0.67 \begin{align*} - \log{\left (x \right )} + \log{\left (x - \frac{1}{b} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(x - 1/b)

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Giac [A]  time = 1.13137, size = 18, normalized size = 1.5 \begin{align*} \log \left ({\left | b x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*x - 1)) - log(abs(x))