3.242 \(\int \log (\frac{1}{13+x}) \, dx\)

Optimal. Leaf size=12 \[ x+(x+13) \log \left (\frac{1}{x+13}\right ) \]

[Out]

x + (13 + x)*Log[(13 + x)^(-1)]

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Rubi [A]  time = 0.0026972, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2389, 2295} \[ x+(x+13) \log \left (\frac{1}{x+13}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[(13 + x)^(-1)],x]

[Out]

x + (13 + x)*Log[(13 + x)^(-1)]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int \log \left (\frac{1}{13+x}\right ) \, dx &=\operatorname{Subst}\left (\int \log \left (\frac{1}{x}\right ) \, dx,x,13+x\right )\\ &=x+(13+x) \log \left (\frac{1}{13+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0019943, size = 12, normalized size = 1. \[ x+(x+13) \log \left (\frac{1}{x+13}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(13 + x)^(-1)],x]

[Out]

x + (13 + x)*Log[(13 + x)^(-1)]

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Maple [A]  time = 0.007, size = 14, normalized size = 1.2 \begin{align*} \left ( 13+x \right ) \ln \left ( \left ( 13+x \right ) ^{-1} \right ) +13+x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(1/(13+x)),x)

[Out]

(13+x)*ln(1/(13+x))+13+x

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Maxima [A]  time = 1.0395, size = 16, normalized size = 1.33 \begin{align*} -{\left (x + 13\right )} \log \left (x + 13\right ) + x + 13 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/(13+x)),x, algorithm="maxima")

[Out]

-(x + 13)*log(x + 13) + x + 13

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Fricas [A]  time = 1.84408, size = 41, normalized size = 3.42 \begin{align*}{\left (x + 13\right )} \log \left (\frac{1}{x + 13}\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/(13+x)),x, algorithm="fricas")

[Out]

(x + 13)*log(1/(x + 13)) + x

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Sympy [A]  time = 0.105836, size = 15, normalized size = 1.25 \begin{align*} x \log{\left (\frac{1}{x + 13} \right )} + x - 13 \log{\left (x + 13 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(1/(13+x)),x)

[Out]

x*log(1/(x + 13)) + x - 13*log(x + 13)

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Giac [A]  time = 1.24363, size = 23, normalized size = 1.92 \begin{align*} x \log \left (\frac{1}{x + 13}\right ) + x - 13 \, \log \left ({\left | x + 13 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/(13+x)),x, algorithm="giac")

[Out]

x*log(1/(x + 13)) + x - 13*log(abs(x + 13))