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

Optimal. Leaf size=19 \[ \text {PolyLog}\left (2,-\frac {1}{\sqrt {x}}\right )-\text {PolyLog}\left (2,\frac {1}{\sqrt {x}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, 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 = {6036, 6032} \begin {gather*} \text {Li}_2\left (-\frac {1}{\sqrt {x}}\right )-\text {Li}_2\left (\frac {1}{\sqrt {x}}\right ) \end {gather*}

Antiderivative was successfully verified.

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

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(15)=30\).
time = 0.05, size = 33, normalized size = 1.74

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (13) = 26\).
time = 0.25, size = 66, normalized size = 3.47 \begin {gather*} -\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 ) \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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