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

Optimal result

Integrand size = 8, antiderivative size = 22 \[ \int \log ^{-1-n}(t) \, dt=-\Gamma (-n,-\log (t)) (-\log (t))^n \log ^{-n}(t) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2336, 2212} \[ \int \log ^{-1-n}(t) \, dt=(-\log (t))^n \log ^{-n}(t) (-\Gamma (-n,-\log (t))) \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \log ^{-1-n}(t) \, dt=-\Gamma (-n,-\log (t)) (-\log (t))^n \log ^{-n}(t) \]

[In]

Integrate[Log[t]^(-1 - n),t]

[Out]

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

Maple [F]

\[\int \ln \left (t \right )^{-1-n}d t\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \log ^{-1-n}(t) \, dt=e^{\left (i \, \pi + i \, \pi n\right )} \Gamma \left (-n, -\log \left (t\right )\right ) \]

[In]

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

[Out]

e^(I*pi + I*pi*n)*gamma(-n, -log(t))

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \log ^{-1-n}(t) \, dt=\left (- \log {\left (t \right )}\right )^{n + 1} \log {\left (t \right )}^{- n - 1} \Gamma \left (- n, - \log {\left (t \right )}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \log ^{-1-n}(t) \, dt=-\left (-\log \left (t\right )\right )^{n} \log \left (t\right )^{-n} \Gamma \left (-n, -\log \left (t\right )\right ) \]

[In]

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

[Out]

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

Giac [F]

\[ \int \log ^{-1-n}(t) \, dt=\int { \log \left (t\right )^{-n - 1} \,d t } \]

[In]

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

[Out]

integrate(log(t)^(-n - 1), t)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \log ^{-1-n}(t) \, dt=-\frac {{\left (-\ln \left (t\right )\right )}^n\,\Gamma \left (-n,-\ln \left (t\right )\right )}{{\ln \left (t\right )}^n} \]

[In]

int(1/log(t)^(n + 1),t)

[Out]

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