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

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

[Out]

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

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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.

[In]

Int[Log[t]^(-1 - n),t]

[Out]

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

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]

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.

[In]

Integrate[Log[t]^(-1 - n),t]

[Out]

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

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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.

[In]

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

[Out]

cos(pi + pi*n)*gamma(-n, -log(t))

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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.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: integrate(ln(t)^(-n-1)*t/t,t)

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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.

[In]

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

[Out]

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

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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.

[In]

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

[Out]

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

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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.

[In]

int(1/log(t)^(n + 1),t)

[Out]

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

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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.

[In]

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

[Out]

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

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