3.86 \(\int \frac {\coth ^{-1}(\sqrt {x})}{x} \, dx\)

Optimal. Leaf size=19 \[ \text {Li}_2\left (-\frac {1}{\sqrt {x}}\right )-\text {Li}_2\left (\frac {1}{\sqrt {x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 19, 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} \[ \text {PolyLog}\left (2,-\frac {1}{\sqrt {x}}\right )-\text {PolyLog}\left (2,\frac {1}{\sqrt {x}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (\sqrt {x}\right )}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{x} \, dx,x,\sqrt {x}\right )\\ &=\text {Li}_2\left (-\frac {1}{\sqrt {x}}\right )-\text {Li}_2\left (\frac {1}{\sqrt {x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \text {Li}_2\left (-\frac {1}{\sqrt {x}}\right )-\text {Li}_2\left (\frac {1}{\sqrt {x}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccoth(sqrt(x))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(sqrt(x))/x, x)

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maple [B]  time = 0.06, size = 33, normalized size = 1.74 \[ \ln \relax (x ) \mathrm {arccoth}\left (\sqrt {x}\right )-\dilog \left (\sqrt {x}\right )-\dilog \left (1+\sqrt {x}\right )-\frac {\ln \relax (x ) \ln \left (1+\sqrt {x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.31, size = 66, normalized size = 3.47 \[ -\frac {1}{2} \, {\left (\log \left (\sqrt {x} + 1\right ) - \log \left (\sqrt {x} - 1\right )\right )} \log \relax (x) + \operatorname {arcoth}\left (\sqrt {x}\right ) \log \relax (x) + \log \left (-\sqrt {x}\right ) \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \relax (x) \log \left (\sqrt {x} - 1\right ) + {\rm Li}_2\left (\sqrt {x} + 1\right ) - {\rm Li}_2\left (-\sqrt {x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\mathrm {acoth}\left (\sqrt {x}\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acoth(x^(1/2))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(sqrt(x))/x, x)

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