3.65 \(\int \frac{\log (1-t)}{1-t} \, dt\)

Optimal. Leaf size=12 \[ -\frac{1}{2} \log ^2(1-t) \]

[Out]

-Log[1 - t]^2/2

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Rubi [A]  time = 0.0154553, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2390, 2301} \[ -\frac{1}{2} \log ^2(1-t) \]

Antiderivative was successfully verified.

[In]

Int[Log[1 - t]/(1 - t),t]

[Out]

-Log[1 - t]^2/2

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{\log (1-t)}{1-t} \, dt &=-\operatorname{Subst}\left (\int \frac{\log (t)}{t} \, dt,t,1-t\right )\\ &=-\frac{1}{2} \log ^2(1-t)\\ \end{align*}

Mathematica [A]  time = 0.0017971, size = 12, normalized size = 1. \[ -\frac{1}{2} \log ^2(1-t) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 - t]/(1 - t),t]

[Out]

-Log[1 - t]^2/2

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \ln \left ( 1-t \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(1-t)/(1-t),t)

[Out]

-1/2*ln(1-t)^2

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Maxima [A]  time = 0.961706, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{2} \, \log \left (-t + 1\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1-t)/(1-t),t, algorithm="maxima")

[Out]

-1/2*log(-t + 1)^2

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Fricas [A]  time = 0.516226, size = 27, normalized size = 2.25 \begin{align*} -\frac{1}{2} \, \log \left (-t + 1\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1-t)/(1-t),t, algorithm="fricas")

[Out]

-1/2*log(-t + 1)^2

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Sympy [A]  time = 0.084218, size = 8, normalized size = 0.67 \begin{align*} - \frac{\log{\left (1 - t \right )}^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(1-t)/(1-t),t)

[Out]

-log(1 - t)**2/2

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Giac [A]  time = 1.06302, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{2} \, \log \left (-t + 1\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1-t)/(1-t),t, algorithm="giac")

[Out]

-1/2*log(-t + 1)^2