3.3.42 \(\int \log (\frac {1}{13+x}) \, dx\) [242]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {2436, 2332} \begin {gather*} x+(x+13) \log \left (\frac {1}{x+13}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[(13 + x)^(-1)],x]

[Out]

x + (13 + x)*Log[(13 + x)^(-1)]

Rule 2332

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

Rule 2436

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} x+(13+x) \log \left (\frac {1}{13+x}\right ) \end {gather*}

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 = 0.01, size = 14, normalized size = 1.17

method	result	size
derivativedivides	\(\left (13+x \right ) \ln \left (\frac {1}{13+x}\right )+13+x\)	\(14\)
default	\(\left (13+x \right ) \ln \left (\frac {1}{13+x}\right )+13+x\)	\(14\)
risch	\(x \ln \left (\frac {1}{13+x}\right )+x -13 \ln \left (13+x \right )\)	\(17\)
norman	\(x +x \ln \left (\frac {1}{13+x}\right )+13 \ln \left (\frac {1}{13+x}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(1/(13+x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 1.00 \begin {gather*} -{\left (x + 13\right )} \log \left (x + 13\right ) + x + 13 \end {gather*}

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 = 0.50, size = 12, normalized size = 1.00 \begin {gather*} {\left (x + 13\right )} \log \left (\frac {1}{x + 13}\right ) + x \end {gather*}

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 = 0.03, size = 15, normalized size = 1.25 \begin {gather*} x \log {\left (\frac {1}{x + 13} \right )} + x - 13 \log {\left (x + 13 \right )} \end {gather*}

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 = 3.08, size = 12, normalized size = 1.00 \begin {gather*} -{\left (x + 13\right )} \log \left (x + 13\right ) + x + 13 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 13)*log(x + 13) + x + 13

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Mupad [B]
time = 0.05, size = 12, normalized size = 1.00 \begin {gather*} \left (\ln \left (\frac {1}{x+13}\right )+1\right )\,\left (x+13\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(1/(x + 13)),x)

[Out]

(log(1/(x + 13)) + 1)*(x + 13)

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