\(\int \frac {\coth ^{-1}(a x^5)}{x} \, dx\) [93]

Optimal result

Integrand size = 10, antiderivative size = 28 \[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} \operatorname {PolyLog}\left (2,-\frac {1}{a x^5}\right )-\frac {1}{10} \operatorname {PolyLog}\left (2,\frac {1}{a x^5}\right ) \]

[Out]

1/10*polylog(2,-1/a/x^5)-1/10*polylog(2,1/a/x^5)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, 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} \[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} \operatorname {PolyLog}\left (2,-\frac {1}{a x^5}\right )-\frac {1}{10} \operatorname {PolyLog}\left (2,\frac {1}{a x^5}\right ) \]

[In]

Int[ArcCoth[a*x^5]/x,x]

[Out]

PolyLog[2, -(1/(a*x^5))]/10 - PolyLog[2, 1/(a*x^5)]/10

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 {1}{5} \text {Subst}\left (\int \frac {\coth ^{-1}(a x)}{x} \, dx,x,x^5\right ) \\ & = \frac {1}{10} \operatorname {PolyLog}\left (2,-\frac {1}{a x^5}\right )-\frac {1}{10} \operatorname {PolyLog}\left (2,\frac {1}{a x^5}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=\frac {1}{10} \left (\operatorname {PolyLog}\left (2,-\frac {1}{a x^5}\right )-\operatorname {PolyLog}\left (2,\frac {1}{a x^5}\right )\right ) \]

[In]

Integrate[ArcCoth[a*x^5]/x,x]

[Out]

(PolyLog[2, -(1/(a*x^5))] - PolyLog[2, 1/(a*x^5)])/10

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.18 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.39

method	result	size
default	\(\ln \left (x \right ) \operatorname {arccoth}\left (a \,x^{5}\right )+5 a \left (-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{5}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )}{10 a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{5}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )}{10 a}\right )\)	\(95\)
parts	\(\ln \left (x \right ) \operatorname {arccoth}\left (a \,x^{5}\right )+5 a \left (-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{5}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )}{10 a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{5}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )}{10 a}\right )\)	\(95\)
risch	\(\frac {\ln \left (x \right ) \ln \left (a \,x^{5}+1\right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{5}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}-\frac {\ln \left (x \right ) \ln \left (a \,x^{5}-1\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{5}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}\)	\(100\)

[In]

int(arccoth(a*x^5)/x,x,method=_RETURNVERBOSE)

[Out]

ln(x)*arccoth(a*x^5)+5*a*(-1/10/a*sum(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1),_R1=RootOf(_Z^5*a+1))+1/10/a*su
m(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1),_R1=RootOf(_Z^5*a-1)))

Fricas [F]

\[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x^{5}\right )}{x} \,d x } \]

[In]

integrate(arccoth(a*x^5)/x,x, algorithm="fricas")

[Out]

integral(arccoth(a*x^5)/x, x)

Sympy [F]

\[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=\int \frac {\operatorname {acoth}{\left (a x^{5} \right )}}{x}\, dx \]

[In]

integrate(acoth(a*x**5)/x,x)

[Out]

Integral(acoth(a*x**5)/x, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (22) = 44\).

Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.71 \[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=-\frac {1}{2} \, a {\left (\frac {\log \left (a x^{5} + 1\right )}{a} - \frac {\log \left (a x^{5} - 1\right )}{a}\right )} \log \left (x\right ) - \frac {1}{10} \, a {\left (\frac {\log \left (a x^{5} - 1\right ) \log \left (a x^{5}\right ) + {\rm Li}_2\left (-a x^{5} + 1\right )}{a} - \frac {\log \left (a x^{5} + 1\right ) \log \left (-a x^{5}\right ) + {\rm Li}_2\left (a x^{5} + 1\right )}{a}\right )} + \operatorname {arcoth}\left (a x^{5}\right ) \log \left (x\right ) \]

[In]

integrate(arccoth(a*x^5)/x,x, algorithm="maxima")

[Out]

-1/2*a*(log(a*x^5 + 1)/a - log(a*x^5 - 1)/a)*log(x) - 1/10*a*((log(a*x^5 - 1)*log(a*x^5) + dilog(-a*x^5 + 1))/
a - (log(a*x^5 + 1)*log(-a*x^5) + dilog(a*x^5 + 1))/a) + arccoth(a*x^5)*log(x)

Giac [F]

\[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x^{5}\right )}{x} \,d x } \]

[In]

integrate(arccoth(a*x^5)/x,x, algorithm="giac")

[Out]

integrate(arccoth(a*x^5)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx=\int \frac {\mathrm {acoth}\left (a\,x^5\right )}{x} \,d x \]

[In]

int(acoth(a*x^5)/x,x)

[Out]

int(acoth(a*x^5)/x, x)