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

Optimal. Leaf size=38 \[ \frac {\text {Li}_2\left (-\frac {x^{-n}}{a}\right )}{2 n}-\frac {\text {Li}_2\left (\frac {x^{-n}}{a}\right )}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]

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*}

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Mathematica [B]  time = 0.05, size = 97, normalized size = 2.55 \[ \frac {-\text {Li}_2\left (1-a x^n\right )+\text {Li}_2\left (a x^n+1\right )+n \log (x) \log \left (a x^n-1\right )-n \log (x) \log \left (a x^n+1\right )-\log \left (a x^n\right ) \log \left (a x^n-1\right )+\log \left (-a x^n\right ) \log \left (a x^n+1\right )+2 n \log (x) \coth ^{-1}\left (a x^n\right )}{2 n} \]

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)

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fricas [B]  time = 0.58, size = 128, normalized size = 3.37 \[ -\frac {n \log \left (a \cosh \left (n \log \relax (x)\right ) + a \sinh \left (n \log \relax (x)\right ) + 1\right ) \log \relax (x) - n \log \left (-a \cosh \left (n \log \relax (x)\right ) - a \sinh \left (n \log \relax (x)\right ) + 1\right ) \log \relax (x) - n \log \relax (x) \log \left (\frac {a \cosh \left (n \log \relax (x)\right ) + a \sinh \left (n \log \relax (x)\right ) + 1}{a \cosh \left (n \log \relax (x)\right ) + a \sinh \left (n \log \relax (x)\right ) - 1}\right ) - {\rm Li}_2\left (a \cosh \left (n \log \relax (x)\right ) + a \sinh \left (n \log \relax (x)\right )\right ) + {\rm Li}_2\left (-a \cosh \left (n \log \relax (x)\right ) - a \sinh \left (n \log \relax (x)\right )\right )}{2 \, n} \]

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

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

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)

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maple [A]  time = 0.09, size = 61, normalized size = 1.61 \[ \frac {\ln \left (a \,x^{n}\right ) \mathrm {arccoth}\left (a \,x^{n}\right )}{n}-\frac {\dilog \left (a \,x^{n}\right )}{2 n}-\frac {\dilog \left (a \,x^{n}+1\right )}{2 n}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2 n} \]

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)

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maxima [B]  time = 0.42, size = 147, normalized size = 3.87 \[ -\frac {1}{2} \, a n {\left (\frac {\log \left (\frac {a x^{n} + 1}{a}\right )}{a n} - \frac {\log \left (\frac {a x^{n} - 1}{a}\right )}{a n}\right )} \log \relax (x) + \frac {1}{2} \, a n {\left (\frac {\log \left (a x^{n} + 1\right ) \log \relax (x) - \log \left (a x^{n} - 1\right ) \log \relax (x)}{a n} - \frac {n \log \left (a x^{n} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x^{n}\right )}{a n^{2}} + \frac {n \log \left (-a x^{n} + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x^{n}\right )}{a n^{2}}\right )} + \operatorname {arcoth}\left (a x^{n}\right ) \log \relax (x) \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acoth}\left (a\,x^n\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acoth(a*x^n)/x, x)

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

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)

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