3.97 \(\int \frac{\cot ^{-1}(a x^n)}{x} \, dx\)

Optimal. Leaf size=47 \[ \frac{i \text{PolyLog}\left (2,\frac{i x^{-n}}{a}\right )}{2 n}-\frac{i \text{PolyLog}\left (2,-\frac{i x^{-n}}{a}\right )}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.0351754, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5032, 4849, 2391} \[ \frac{i \text{PolyLog}\left (2,\frac{i x^{-n}}{a}\right )}{2 n}-\frac{i \text{PolyLog}\left (2,-\frac{i x^{-n}}{a}\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x^n]/x,x]

[Out]

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

Rule 5032

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcCot[c*x])^p
/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

Rule 4849

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I/(c*
x)]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I/(c*x)]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 2391

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

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}\left (a x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(a x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i}{a x}\right )}{x} \, dx,x,x^n\right )}{2 n}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i}{a x}\right )}{x} \, dx,x,x^n\right )}{2 n}\\ &=-\frac{i \text{Li}_2\left (-\frac{i x^{-n}}{a}\right )}{2 n}+\frac{i \text{Li}_2\left (\frac{i x^{-n}}{a}\right )}{2 n}\\ \end{align*}

Mathematica [A]  time = 0.0150394, size = 40, normalized size = 0.85 \[ -\frac{i \left (\text{PolyLog}\left (2,-\frac{i x^{-n}}{a}\right )-\text{PolyLog}\left (2,\frac{i x^{-n}}{a}\right )\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x^n]/x,x]

[Out]

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

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Maple [B]  time = 0.052, size = 94, normalized size = 2. \begin{align*}{\frac{\ln \left ( a{x}^{n} \right ){\rm arccot} \left (a{x}^{n}\right )}{n}}-{\frac{{\frac{i}{2}}\ln \left ( a{x}^{n} \right ) \ln \left ( 1+ia{x}^{n} \right ) }{n}}+{\frac{{\frac{i}{2}}\ln \left ( a{x}^{n} \right ) \ln \left ( 1-ia{x}^{n} \right ) }{n}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+ia{x}^{n} \right ) }{n}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1-ia{x}^{n} \right ) }{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x^n)/x,x)

[Out]

1/n*ln(a*x^n)*arccot(a*x^n)-1/2*I/n*ln(a*x^n)*ln(1+I*a*x^n)+1/2*I/n*ln(a*x^n)*ln(1-I*a*x^n)-1/2*I/n*dilog(1+I*
a*x^n)+1/2*I/n*dilog(1-I*a*x^n)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a n \int \frac{x^{n} \log \left (x\right )}{a^{2} x x^{2 \, n} + x}\,{d x} + \arctan \left (1, a x^{n}\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^n)/x,x, algorithm="maxima")

[Out]

a*n*integrate(x^n*log(x)/(a^2*x*x^(2*n) + x), x) + arctan2(1, a*x^n)*log(x)

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Fricas [A]  time = 2.32139, size = 181, normalized size = 3.85 \begin{align*} \frac{2 \, n \operatorname{arccot}\left (a x^{n}\right ) \log \left (x\right ) - i \, n \log \left (i \, a x^{n} + 1\right ) \log \left (x\right ) + i \, n \log \left (-i \, a x^{n} + 1\right ) \log \left (x\right ) + i \,{\rm Li}_2\left (i \, a x^{n}\right ) - i \,{\rm Li}_2\left (-i \, a x^{n}\right )}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^n)/x,x, algorithm="fricas")

[Out]

1/2*(2*n*arccot(a*x^n)*log(x) - I*n*log(I*a*x^n + 1)*log(x) + I*n*log(-I*a*x^n + 1)*log(x) + I*dilog(I*a*x^n)
- I*dilog(-I*a*x^n))/n

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (a x^{n} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x**n)/x,x)

[Out]

Integral(acot(a*x**n)/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x^{n}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^n)/x,x, algorithm="giac")

[Out]

integrate(arccot(a*x^n)/x, x)