\(\int \frac {\coth ^{-1}(a x^n)}{x} \, dx\) [95]

Optimal result

Integrand size = 10, antiderivative size = 38 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {x^{-n}}{a}\right )}{2 n}-\frac {\operatorname {PolyLog}\left (2,\frac {x^{-n}}{a}\right )}{2 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6036, 6032} \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {x^{-n}}{a}\right )}{2 n}-\frac {\operatorname {PolyLog}\left (2,\frac {x^{-n}}{a}\right )}{2 n} \]

[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 6032

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

Rule 6036

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\frac {a x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};a^2 x^{2 n}\right )}{n}+\left (\coth ^{-1}\left (a x^n\right )-\text {arctanh}\left (a x^n\right )\right ) \log (x) \]

[In]

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

[Out]

(a*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, a^2*x^(2*n)])/n + (ArcCoth[a*x^n] - ArcTanh[a*x^n])*Log[x]

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18

method	result	size
risch	\(-\frac {\ln \left (a \,x^{n}-1\right ) \ln \left (a \,x^{n}\right )}{2 n}-\frac {\operatorname {dilog}\left (a \,x^{n}\right )}{2 n}-\frac {\operatorname {dilog}\left (a \,x^{n}+1\right )}{2 n}\)	\(45\)
derivativedivides	\(\frac {\ln \left (a \,x^{n}\right ) \operatorname {arccoth}\left (a \,x^{n}\right )-\frac {\operatorname {dilog}\left (a \,x^{n}\right )}{2}-\frac {\operatorname {dilog}\left (a \,x^{n}+1\right )}{2}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2}}{n}\)	\(53\)
default	\(\frac {\ln \left (a \,x^{n}\right ) \operatorname {arccoth}\left (a \,x^{n}\right )-\frac {\operatorname {dilog}\left (a \,x^{n}\right )}{2}-\frac {\operatorname {dilog}\left (a \,x^{n}+1\right )}{2}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2}}{n}\)	\(53\)

[In]

int(arccoth(a*x^n)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (32) = 64\).

Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.37 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=-\frac {n \log \left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (\frac {a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1}{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - {\rm Li}_2\left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right )\right ) + {\rm Li}_2\left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \]

[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]

\[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\operatorname {acoth}{\left (a x^{n} \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(acoth(a*x**n)/x, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (32) = 64\).

Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 3.87 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=-\frac {1}{2} \, a n {\left (\frac {\log \left (\frac {a x^{n} + 1}{a}\right )}{a n} - \frac {\log \left (\frac {a x^{n} - 1}{a}\right )}{a n}\right )} \log \left (x\right ) + \frac {1}{2} \, a n {\left (\frac {\log \left (a x^{n} + 1\right ) \log \left (x\right ) - \log \left (a x^{n} - 1\right ) \log \left (x\right )}{a n} - \frac {n \log \left (a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x^{n}\right )}{a n^{2}} + \frac {n \log \left (-a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x^{n}\right )}{a n^{2}}\right )} + \operatorname {arcoth}\left (a x^{n}\right ) \log \left (x\right ) \]

[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)

Giac [F]

\[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x^{n}\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(arccoth(a*x^n)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\mathrm {acoth}\left (a\,x^n\right )}{x} \,d x \]

[In]

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

[Out]

int(acoth(a*x^n)/x, x)