3.2.72 \(\int \log ^{-1-n}(t) \, dt\) [172]

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

[Out]

-GAMMA(-n,-ln(t))*(-ln(t))^n/(ln(t)^n)

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Rubi [A]
time = 0.01, 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 = {2336, 2212} \begin {gather*} (-\log (t))^n \log ^{-n}(t) (-\Gamma (-n,-\log (t))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[t]^(-1 - n),t]

[Out]

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

Rule 2212

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

Rule 2336

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]

Rubi steps

\begin {align*} \int \log ^{-1-n}(t) \, dt &=\text {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 = 0.02, size = 22, normalized size = 1.00 \begin {gather*} -\Gamma (-n,-\log (t)) (-\log (t))^n \log ^{-n}(t) \end {gather*}

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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Mathics [A]
time = 3.14, size = 22, normalized size = 1.00 \begin {gather*} -\text {Gamma}\left [-n,-\text {Log}\left [t\right ]\right ] \text {Log}\left [t\right ]^{-n} \left (-\text {Log}\left [t\right ]\right )^n \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/Log[t]^(n + 1),t]')

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \ln \left (t \right )^{-1-n}\, dt\]

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 = 0.31, size = 22, normalized size = 1.00 \begin {gather*} -\left (-\log \left (t\right )\right )^{n} \log \left (t\right )^{-n} \Gamma \left (-n, -\log \left (t\right )\right ) \end {gather*}

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 = 0.35, size = 15, normalized size = 0.68 \begin {gather*} \cos \left (\pi + \pi n\right ) \Gamma \left (-n, -\log \left (t\right )\right ) \end {gather*}

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 = 1.52, size = 24, normalized size = 1.09 \begin {gather*} \left (- \log {\left (t \right )}\right )^{n + 1} \log {\left (t \right )}^{- n - 1} \Gamma \left (- n, - \log {\left (t \right )}\right ) \end {gather*}

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] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(t)^(-1-n),t)

[Out]

Could not integrate

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Mupad [B]
time = 0.06, size = 22, normalized size = 1.00 \begin {gather*} -\frac {{\left (-\ln \left (t\right )\right )}^n\,\Gamma \left (-n,-\ln \left (t\right )\right )}{{\ln \left (t\right )}^n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(t)^(n + 1),t)

[Out]

-((-log(t))^n*igamma(-n, -log(t)))/log(t)^n

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