3.350 \(\int \frac {\log (a x^{1-n})}{a x-x^n} \, dx\)

Optimal. Leaf size=26 \[ -\frac {\text {Li}_2\left (1-a x^{1-n}\right )}{a (1-n)} \]

[Out]

-polylog(2,1-a*x^(1-n))/a/(1-n)

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Rubi [A]  time = 0.09, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1593, 2336, 2315} \[ -\frac {\text {PolyLog}\left (2,1-a x^{1-n}\right )}{a (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[Log[a*x^(1 - n)]/(a*x - x^n),x]

[Out]

-(PolyLog[2, 1 - a*x^(1 - n)]/(a*(1 - n)))

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :>
Dist[f^m/n, Subst[Int[(d + e*x)^q*(a + b*Log[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}
, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && EqQ[r, n]

Rubi steps

\begin {align*} \int \frac {\log \left (a x^{1-n}\right )}{a x-x^n} \, dx &=\int \frac {x^{-n} \log \left (a x^{1-n}\right )}{-1+a x^{1-n}} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log (a x)}{-1+a x} \, dx,x,x^{1-n}\right )}{1-n}\\ &=-\frac {\text {Li}_2\left (1-a x^{1-n}\right )}{a (1-n)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.88 \[ \frac {\text {Li}_2\left (1-a x^{1-n}\right )}{a (n-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a*x^(1 - n)]/(a*x - x^n),x]

[Out]

PolyLog[2, 1 - a*x^(1 - n)]/(a*(-1 + n))

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fricas [B]  time = 0.42, size = 89, normalized size = 3.42 \[ \frac {2 \, {\left (n - 1\right )} \log \relax (a) \log \relax (x) - {\left (n^{2} - 2 \, n + 1\right )} \log \relax (x)^{2} + 2 \, {\left (n - 1\right )} \log \relax (x) \log \left (\frac {a - x^{n - 1}}{a}\right ) - 2 \, \log \relax (a) \log \left (-a + x^{n - 1}\right ) + 2 \, {\rm Li}_2\left (-\frac {a - x^{n - 1}}{a} + 1\right )}{2 \, {\left (a n - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*x^(1-n))/(a*x-x^n),x, algorithm="fricas")

[Out]

1/2*(2*(n - 1)*log(a)*log(x) - (n^2 - 2*n + 1)*log(x)^2 + 2*(n - 1)*log(x)*log((a - x^(n - 1))/a) - 2*log(a)*l
og(-a + x^(n - 1)) + 2*dilog(-(a - x^(n - 1))/a + 1))/(a*n - a)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (a x^{-n + 1}\right )}{a x - x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*x^(1-n))/(a*x-x^n),x, algorithm="giac")

[Out]

integrate(log(a*x^(-n + 1))/(a*x - x^n), x)

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maple [F]  time = 0.93, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (a \,x^{-n +1}\right )}{a x -x^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*x^(-n+1))/(a*x-x^n),x)

[Out]

int(ln(a*x^(-n+1))/(a*x-x^n),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (a x^{-n + 1}\right )}{a x - x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*x^(1-n))/(a*x-x^n),x, algorithm="maxima")

[Out]

integrate(log(a*x^(-n + 1))/(a*x - x^n), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (a\,x^{1-n}\right )}{a\,x-x^n} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a*x^(1 - n))/(a*x - x^n),x)

[Out]

int(log(a*x^(1 - n))/(a*x - x^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (a x x^{- n} \right )}}{a x - x^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*x**(1-n))/(a*x-x**n),x)

[Out]

Integral(log(a*x*x**(-n))/(a*x - x**n), x)

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