Optimal. Leaf size=22 \[ (-\log (t))^n \log ^{-n}(t) (-\operatorname {Gamma}(-n,-\log (t))) \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2299, 2181} \[ (-\log (t))^n \log ^{-n}(t) (-\text {Gamma}(-n,-\log (t))) \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2299
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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ (-\log (t))^n \log ^{-n}(t) (-\operatorname {Gamma}(-n,-\log (t))) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 15, normalized size = 0.68 \[ \cos \left (\pi + \pi n\right ) \Gamma \left (-n, -\log \relax (t)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \ln \relax (t )^{-n -1}\, dt \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 22, normalized size = 1.00 \[ -\left (-\log \relax (t)\right )^{n} \log \relax (t)^{-n} \Gamma \left (-n, -\log \relax (t)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 22, normalized size = 1.00 \[ -\frac {{\left (-\ln \relax (t)\right )}^n\,\Gamma \left (-n,-\ln \relax (t)\right )}{{\ln \relax (t)}^n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.74, size = 24, normalized size = 1.09 \[ \left (- \log {\relax (t )}\right )^{n + 1} \log {\relax (t )}^{- n - 1} \Gamma \left (- n, - \log {\relax (t )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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