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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.00, size = 12, normalized size = 1.00 \[ \log (1-b x)-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-1 + b*x)),x]

[Out]

-Log[x] + Log[1 - b*x]

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 11, normalized size = 0.92 \[ \log \left (b x - 1\right ) - \log \relax (x) \]

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)

________________________________________________________________________________________

giac [A]  time = 1.02, size = 13, normalized size = 1.08 \[ \log \left ({\left | b x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 12, normalized size = 1.00 \[ -\ln \relax (x )+\ln \left (b x -1\right ) \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 11, normalized size = 0.92 \[ \log \left (b x - 1\right ) - \log \relax (x) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 9, normalized size = 0.75 \[ -2\,\mathrm {atanh}\left (2\,b\,x-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x - 1)),x)

[Out]

-2*atanh(2*b*x - 1)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 8, normalized size = 0.67 \[ - \log {\relax (x )} + \log {\left (x - \frac {1}{b} \right )} \]

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)

________________________________________________________________________________________