\(\int \frac {\coth ^{-1}(a x^n)}{x} \, dx\) [49]

Optimal result

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

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

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 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\frac {a x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};a^2 x^{2 n}\right )}{n}+\left (\coth ^{-1}\left (a x^n\right )-\text {arctanh}\left (a x^n\right )\right ) \log (x) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6451, 6447}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6451

\(\displaystyle \frac {\int x^{-n} \coth ^{-1}\left (a x^n\right )dx^n}{n}\)

\(\Big \downarrow \) 6447

\(\displaystyle \frac {\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {x^{-n}}{a}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {x^{-n}}{a}\right )}{n}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 6447

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 6451

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[ 
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]
 

Maple [A] (verified)

Time = 0.27 (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	\(\frac {\ln \left (a \,x^{n}\right ) \operatorname {arccoth}\left (a \,x^{n}\right )-\frac {\operatorname {dilog}\left (a \,x^{n}\right )}{2}-\frac {\operatorname {dilog}\left (a \,x^{n}+1\right )}{2}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2}}{n}\)	\(53\)

Input:

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

Output:

-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 = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.37 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=-\frac {n \log \left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (\frac {a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1}{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - {\rm Li}_2\left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right )\right ) + {\rm Li}_2\left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \] Input:

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

Output:

-1/2*(n*log(a*cosh(n*log(x)) + a*sinh(n*log(x)) + 1)*log(x) - n*log(-a*cos 
h(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)) - dil 
og(a*cosh(n*log(x)) + a*sinh(n*log(x))) + dilog(-a*cosh(n*log(x)) - a*sinh 
(n*log(x))))/n
 

Sympy [F]

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

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

Output:

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 = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 3.87 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=-\frac {1}{2} \, a n {\left (\frac {\log \left (\frac {a x^{n} + 1}{a}\right )}{a n} - \frac {\log \left (\frac {a x^{n} - 1}{a}\right )}{a n}\right )} \log \left (x\right ) + \frac {1}{2} \, a n {\left (\frac {\log \left (a x^{n} + 1\right ) \log \left (x\right ) - \log \left (a x^{n} - 1\right ) \log \left (x\right )}{a n} - \frac {n \log \left (a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x^{n}\right )}{a n^{2}} + \frac {n \log \left (-a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x^{n}\right )}{a n^{2}}\right )} + \operatorname {arcoth}\left (a x^{n}\right ) \log \left (x\right ) \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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