Optimal. Leaf size=22 \[ (-\log (t))^n \log ^{-n}(t) (-\text{Gamma}(-n,-\log (t))) \]
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Rubi [A] time = 0.0174485, 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, Rules used = {2299, 2181} \[ (-\log (t))^n \log ^{-n}(t) (-\text{Gamma}(-n,-\log (t))) \]
Antiderivative was successfully verified.
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Rule 2299
Rule 2181
Rubi steps
\begin{align*} \int \log ^{-1-n}(t) \, dt &=\operatorname{Subst}\left (\int e^t t^{-1-n} \, dt,t,\log (t)\right )\\ &=-\Gamma (-n,-\log (t)) (-\log (t))^n \log ^{-n}(t)\\ \end{align*}
Mathematica [A] time = 0.0106361, size = 22, normalized size = 1. \[ (-\log (t))^n \log ^{-n}(t) (-\text{Gamma}(-n,-\log (t))) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( t \right ) \right ) ^{-1-n}\, dt \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03843, size = 30, normalized size = 1.36 \begin{align*} -\left (-\log \left (t\right )\right )^{n} \log \left (t\right )^{-n} \Gamma \left (-n, -\log \left (t\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15611, size = 47, normalized size = 2.14 \begin{align*} \cos \left (\pi + \pi n\right ) \Gamma \left (-n, -\log \left (t\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.90796, size = 24, normalized size = 1.09 \begin{align*} \left (- \log{\left (t \right )}\right )^{n + 1} \log{\left (t \right )}^{- n - 1} \Gamma \left (- n, - \log{\left (t \right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (t\right )^{-n - 1}\,{d t} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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