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

Optimal. Leaf size=11 \[ \log (x)-\log (b x+1) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, 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} \begin {gather*} \log (x)-\log (b x+1) \end {gather*}

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \log (x)-\log (b x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + b*x)),x]

[Out]

Log[x] - Log[1 + b*x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (1+b x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x*(1 + b*x)),x]

[Out]

IntegrateAlgebraic[1/(x*(1 + b*x)), x]

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fricas [A]  time = 0.77, size = 11, normalized size = 1.00 \begin {gather*} -\log \left (b x + 1\right ) + \log \relax (x) \end {gather*}

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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giac [A]  time = 0.99, size = 13, normalized size = 1.18 \begin {gather*} -\log \left ({\left | b x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

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))

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maple [A]  time = 0.00, size = 12, normalized size = 1.09 \begin {gather*} \ln \relax (x )-\ln \left (b x +1\right ) \end {gather*}

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 = 1.31, size = 11, normalized size = 1.00 \begin {gather*} -\log \left (b x + 1\right ) + \log \relax (x) \end {gather*}

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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mupad [B]  time = 0.10, size = 9, normalized size = 0.82 \begin {gather*} -2\,\mathrm {atanh}\left (2\,b\,x+1\right ) \end {gather*}

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)

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sympy [A]  time = 0.14, size = 8, normalized size = 0.73 \begin {gather*} \log {\relax (x )} - \log {\left (x + \frac {1}{b} \right )} \end {gather*}

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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