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

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

[Out]

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

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ -\frac {\log (2-b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - b*x)^(-1),x]

[Out]

-(Log[2 - b*x]/b)

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fricas [A]  time = 0.44, size = 11, normalized size = 0.92 \[ -\frac {\log \left (b x - 2\right )}{b} \]

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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giac [A]  time = 1.10, size = 12, normalized size = 1.00 \[ -\frac {\log \left ({\left | b x - 2 \right |}\right )}{b} \]

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

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maple [A]  time = 0.00, size = 13, normalized size = 1.08 \[ -\frac {\ln \left (-b x +2\right )}{b} \]

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 = 1.42, size = 11, normalized size = 0.92 \[ -\frac {\log \left (b x - 2\right )}{b} \]

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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mupad [B]  time = 0.03, size = 11, normalized size = 0.92 \[ -\frac {\ln \left (b\,x-2\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(b*x - 2),x)

[Out]

-log(b*x - 2)/b

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sympy [A]  time = 0.07, size = 8, normalized size = 0.67 \[ - \frac {\log {\left (b x - 2 \right )}}{b} \]

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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