\(\int \frac {\coth ^{-1}(\sqrt {x})}{x} \, dx\) [86]

Optimal result

Integrand size = 10, antiderivative size = 19 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \, dx=\operatorname {PolyLog}\left (2,-\frac {1}{\sqrt {x}}\right )-\operatorname {PolyLog}\left (2,\frac {1}{\sqrt {x}}\right ) \]

[Out]

polylog(2,-1/x^(1/2))-polylog(2,1/x^(1/2))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, 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} \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \, dx=\operatorname {PolyLog}\left (2,-\frac {1}{\sqrt {x}}\right )-\operatorname {PolyLog}\left (2,\frac {1}{\sqrt {x}}\right ) \]

[In]

Int[ArcCoth[Sqrt[x]]/x,x]

[Out]

PolyLog[2, -(1/Sqrt[x])] - PolyLog[2, 1/Sqrt[x]]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \, dx=\operatorname {PolyLog}\left (2,-\frac {1}{\sqrt {x}}\right )-\operatorname {PolyLog}\left (2,\frac {1}{\sqrt {x}}\right ) \]

[In]

Integrate[ArcCoth[Sqrt[x]]/x,x]

[Out]

PolyLog[2, -(1/Sqrt[x])] - PolyLog[2, 1/Sqrt[x]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(15)=30\).

Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74

method	result	size
derivativedivides	\(\ln \left (x \right ) \operatorname {arccoth}\left (\sqrt {x}\right )-\operatorname {dilog}\left (\sqrt {x}\right )-\operatorname {dilog}\left (\sqrt {x}+1\right )-\frac {\ln \left (x \right ) \ln \left (\sqrt {x}+1\right )}{2}\)	\(33\)
default	\(\ln \left (x \right ) \operatorname {arccoth}\left (\sqrt {x}\right )-\operatorname {dilog}\left (\sqrt {x}\right )-\operatorname {dilog}\left (\sqrt {x}+1\right )-\frac {\ln \left (x \right ) \ln \left (\sqrt {x}+1\right )}{2}\)	\(33\)
parts	\(\ln \left (x \right ) \operatorname {arccoth}\left (\sqrt {x}\right )-\operatorname {dilog}\left (\sqrt {x}\right )-\operatorname {dilog}\left (\sqrt {x}+1\right )-\frac {\ln \left (x \right ) \ln \left (\sqrt {x}+1\right )}{2}\)	\(33\)

[In]

int(arccoth(x^(1/2))/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

integrate(arccoth(x^(1/2))/x,x, algorithm="fricas")

[Out]

integral(arccoth(sqrt(x))/x, x)

Sympy [F]

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

[In]

integrate(acoth(x**(1/2))/x,x)

[Out]

Integral(acoth(sqrt(x))/x, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (13) = 26\).

Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.47 \[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x} \, dx=-\frac {1}{2} \, {\left (\log \left (\sqrt {x} + 1\right ) - \log \left (\sqrt {x} - 1\right )\right )} \log \left (x\right ) + \operatorname {arcoth}\left (\sqrt {x}\right ) \log \left (x\right ) + \log \left (-\sqrt {x}\right ) \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (x\right ) \log \left (\sqrt {x} - 1\right ) + {\rm Li}_2\left (\sqrt {x} + 1\right ) - {\rm Li}_2\left (-\sqrt {x} + 1\right ) \]

[In]

integrate(arccoth(x^(1/2))/x,x, algorithm="maxima")

[Out]

-1/2*(log(sqrt(x) + 1) - log(sqrt(x) - 1))*log(x) + arccoth(sqrt(x))*log(x) + log(-sqrt(x))*log(sqrt(x) + 1) -
1/2*log(x)*log(sqrt(x) - 1) + dilog(sqrt(x) + 1) - dilog(-sqrt(x) + 1)

Giac [F]

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

[In]

integrate(arccoth(x^(1/2))/x,x, algorithm="giac")

[Out]

integrate(arccoth(sqrt(x))/x, x)

Mupad [F(-1)]

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

[In]

int(acoth(x^(1/2))/x,x)

[Out]

int(acoth(x^(1/2))/x, x)