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 = 0.0113802, 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 \[ \log (1-b x)-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-1 + b*x)),x]

[Out]

-Log[x] + Log[1 - b*x]

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Rubi in Sympy [A]  time = 2.48967, size = 8, normalized size = 0.67 \[ - \log{\left (x \right )} + \log{\left (- b x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x-1),x)

[Out]

-log(x) + log(-b*x + 1)

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Mathematica [A]  time = 0.00361037, 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 = 0.009, size = 12, normalized size = 1. \[ -\ln \left ( x \right ) +\ln \left ( bx-1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x-1),x)

[Out]

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

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Maxima [A]  time = 1.34284, size = 15, normalized size = 1.25 \[ \log \left (b x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x - 1) - log(x)

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Fricas [A]  time = 0.212492, size = 15, normalized size = 1.25 \[ \log \left (b x - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x - 1) - log(x)

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Sympy [A]  time = 0.227036, size = 8, normalized size = 0.67 \[ - \log{\left (x \right )} + \log{\left (x - \frac{1}{b} \right )} \]

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)

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GIAC/XCAS [A]  time = 0.208412, size = 18, normalized size = 1.5 \[{\rm ln}\left ({\left | b x - 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*x - 1)) - ln(abs(x))