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 = 0.0255394, 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 \[ -\frac{1}{2} \log ^2(1-t) \]

Antiderivative was successfully verified.

[In]  Int[Log[1 - t]/(1 - t),t]

[Out]

-Log[1 - t]^2/2

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Rubi in Sympy [A]  time = 1.46301, size = 8, normalized size = 0.67 \[ - \frac{\log{\left (- t + 1 \right )}^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(1-t)/(1-t),t)

[Out]

-log(-t + 1)**2/2

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Mathematica [A]  time = 0.00309904, 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 \[ -{\frac{ \left ( \ln \left ( 1-t \right ) \right ) ^{2}}{2}} \]

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 = 1.44768, size = 14, normalized size = 1.17 \[ -\frac{1}{2} \, \log \left (-t + 1\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-log(-t + 1)/(t - 1),t, algorithm="maxima")

[Out]

-1/2*log(-t + 1)^2

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Fricas [A]  time = 0.206628, size = 14, normalized size = 1.17 \[ -\frac{1}{2} \, \log \left (-t + 1\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-log(-t + 1)/(t - 1),t, algorithm="fricas")

[Out]

-1/2*log(-t + 1)^2

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Sympy [A]  time = 0.079763, size = 8, normalized size = 0.67 \[ - \frac{\log{\left (- t + 1 \right )}^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(1-t)/(1-t),t)

[Out]

-log(-t + 1)**2/2

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GIAC/XCAS [A]  time = 0.240403, size = 14, normalized size = 1.17 \[ -\frac{1}{2} \,{\rm ln}\left (-t + 1\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-log(-t + 1)/(t - 1),t, algorithm="giac")

[Out]

-1/2*ln(-t + 1)^2