3.95 \(\int \frac{\coth ^{-1}(a x^n)}{x} \, dx\)

Optimal. Leaf size=38 \[ \frac{\text{PolyLog}\left (2,-\frac{x^{-n}}{a}\right )}{2 n}-\frac{\text{PolyLog}\left (2,\frac{x^{-n}}{a}\right )}{2 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.023247, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6096, 5913} \[ \frac{\text{PolyLog}\left (2,-\frac{x^{-n}}{a}\right )}{2 n}-\frac{\text{PolyLog}\left (2,\frac{x^{-n}}{a}\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6096

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

Rule 5913

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

Rubi steps

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

Mathematica [B]  time = 0.0454459, size = 97, normalized size = 2.55 \[ \frac{-\text{PolyLog}\left (2,1-a x^n\right )+\text{PolyLog}\left (2,a x^n+1\right )+n \log (x) \log \left (a x^n-1\right )-n \log (x) \log \left (a x^n+1\right )-\log \left (a x^n\right ) \log \left (a x^n-1\right )+\log \left (-a x^n\right ) \log \left (a x^n+1\right )+2 n \log (x) \coth ^{-1}\left (a x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.09, size = 61, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( a{x}^{n} \right ){\rm arccoth} \left (a{x}^{n}\right )}{n}}-{\frac{{\it dilog} \left ( a{x}^{n} \right ) }{2\,n}}-{\frac{{\it dilog} \left ( a{x}^{n}+1 \right ) }{2\,n}}-{\frac{\ln \left ( a{x}^{n} \right ) \ln \left ( a{x}^{n}+1 \right ) }{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.26882, size = 198, normalized size = 5.21 \begin{align*} -\frac{1}{2} \, a n{\left (\frac{\log \left (\frac{a x^{n} + 1}{a}\right )}{a n} - \frac{\log \left (\frac{a x^{n} - 1}{a}\right )}{a n}\right )} \log \left (x\right ) + \frac{1}{2} \, a n{\left (\frac{\log \left (a x^{n} + 1\right ) \log \left (x\right ) - \log \left (a x^{n} - 1\right ) \log \left (x\right )}{a n} - \frac{n \log \left (a x^{n} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x^{n}\right )}{a n^{2}} + \frac{n \log \left (-a x^{n} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x^{n}\right )}{a n^{2}}\right )} + \operatorname{arcoth}\left (a x^{n}\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.00394, size = 421, normalized size = 11.08 \begin{align*} -\frac{n \log \left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (\frac{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1}{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) - 1}\right ) -{\rm Li}_2\left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right )\right ) +{\rm Li}_2\left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(n*log(a*cosh(n*log(x)) + a*sinh(n*log(x)) + 1)*log(x) - n*log(-a*cosh(n*log(x)) - a*sinh(n*log(x)) + 1)*
log(x) - n*log(x)*log((a*cosh(n*log(x)) + a*sinh(n*log(x)) + 1)/(a*cosh(n*log(x)) + a*sinh(n*log(x)) - 1)) - d
ilog(a*cosh(n*log(x)) + a*sinh(n*log(x))) + dilog(-a*cosh(n*log(x)) - a*sinh(n*log(x))))/n

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(a*x**n)/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(a*x^n)/x, x)