3.1544 \(\int \frac{1}{2-b x} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\log (2-b x)}{b} \]

[Out]

-(Log[2 - b*x]/b)

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Rubi [A]  time = 0.0014485, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {31} \[ -\frac{\log (2-b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(2 - b*x)^(-1),x]

[Out]

-(Log[2 - b*x]/b)

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}{2-b x} \, dx &=-\frac{\log (2-b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0011097, size = 12, normalized size = 1. \[ -\frac{\log (2-b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - b*x)^(-1),x]

[Out]

-(Log[2 - b*x]/b)

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Maple [A]  time = 0.001, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( -bx+2 \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x+2),x)

[Out]

-ln(-b*x+2)/b

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Maxima [A]  time = 0.954928, size = 15, normalized size = 1.25 \begin{align*} -\frac{\log \left (b x - 2\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+2),x, algorithm="maxima")

[Out]

-log(b*x - 2)/b

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Fricas [A]  time = 1.93477, size = 23, normalized size = 1.92 \begin{align*} -\frac{\log \left (b x - 2\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+2),x, algorithm="fricas")

[Out]

-log(b*x - 2)/b

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Sympy [A]  time = 0.060902, size = 8, normalized size = 0.67 \begin{align*} - \frac{\log{\left (b x - 2 \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+2),x)

[Out]

-log(b*x - 2)/b

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Giac [A]  time = 1.05433, size = 16, normalized size = 1.33 \begin{align*} -\frac{\log \left ({\left | b x - 2 \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x+2),x, algorithm="giac")

[Out]

-log(abs(b*x - 2))/b