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 = 0.0174485, 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.

[In]

Int[Log[t]^(-1 - n),t]

[Out]

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

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

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.

[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 = 0.02, 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.

[In]

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

[Out]

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

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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.

[In]

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

[Out]

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

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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.

[In]

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

[Out]

cos(pi + pi*n)*gamma(-n, -log(t))

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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.

[In]

integrate(ln(t)**(-1-n),t)

[Out]

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

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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.

[In]

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

[Out]

integrate(log(t)^(-n - 1), t)