3.172 \(\int \log ^{-1-n}(t) \, dt\)

Optimal. Leaf size=22 \[ (-\log (t))^n \log ^{-n}(t) (-\text{Gamma}(-n,-\log (t))) \]

[Out]

-((Gamma[-n, -Log[t]]*(-Log[t])^n)/Log[t]^n)

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Rubi [A]  time = 0.0290269, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ (-\log (t))^n \log ^{-n}(t) (-\text{Gamma}(-n,-\log (t))) \]

Antiderivative was successfully verified.

[In]  Int[Log[t]^(-1 - n),t]

[Out]

-((Gamma[-n, -Log[t]]*(-Log[t])^n)/Log[t]^n)

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Rubi in Sympy [A]  time = 0.584314, size = 24, normalized size = 1.09 \[ \left (- \log{\left (t \right )}\right )^{n + 1} \Gamma{\left (- n,- \log{\left (t \right )} \right )} \log{\left (t \right )}^{- n - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(t)**(-1-n),t)

[Out]

(-log(t))**(n + 1)*Gamma(-n, -log(t))*log(t)**(-n - 1)

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Mathematica [A]  time = 0.0331394, size = 22, normalized size = 1. \[ (-\log (t))^n \log ^{-n}(t) (-\text{Gamma}(-n,-\log (t))) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[t]^(-1 - n),t]

[Out]

-((Gamma[-n, -Log[t]]*(-Log[t])^n)/Log[t]^n)

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \[ \int \left ( \ln \left ( t \right ) \right ) ^{-1-n}\, dt \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(t)^(-1-n),t)

[Out]

int(ln(t)^(-1-n),t)

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Maxima [A]  time = 1.50065, size = 30, normalized size = 1.36 \[ -\left (-\log \left (t\right )\right )^{n} \log \left (t\right )^{-n} \Gamma \left (-n, -\log \left (t\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(t)^(-n - 1),t, algorithm="maxima")

[Out]

-(-log(t))^n*log(t)^(-n)*gamma(-n, -log(t))

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Fricas [A]  time = 0.226325, size = 20, normalized size = 0.91 \[ \cos \left (\pi + \pi n\right ) \Gamma \left (-n, -\log \left (t\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(t)^(-n - 1),t, algorithm="fricas")

[Out]

cos(pi + pi*n)*gamma(-n, -log(t))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \log{\left (t \right )}^{- n - 1}\, dt \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(t)**(-1-n),t)

[Out]

Integral(log(t)**(-n - 1), t)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \log \left (t\right )^{-n - 1}\,{d t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(t)^(-n - 1),t, algorithm="giac")

[Out]

integrate(log(t)^(-n - 1), t)