3.88 \(\int \frac{\cot ^{-1}(\sqrt{x})}{x} \, dx\)

Optimal. Leaf size=31 \[ i \text{PolyLog}\left (2,\frac{i}{\sqrt{x}}\right )-i \text{PolyLog}\left (2,-\frac{i}{\sqrt{x}}\right ) \]

[Out]

(-I)*PolyLog[2, (-I)/Sqrt[x]] + I*PolyLog[2, I/Sqrt[x]]

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Rubi [A]  time = 0.0322822, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5032, 4849, 2391} \[ i \text{PolyLog}\left (2,\frac{i}{\sqrt{x}}\right )-i \text{PolyLog}\left (2,-\frac{i}{\sqrt{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[Sqrt[x]]/x,x]

[Out]

(-I)*PolyLog[2, (-I)/Sqrt[x]] + I*PolyLog[2, I/Sqrt[x]]

Rule 5032

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcCot[c*x])^p
/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

Rule 4849

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I/(c*
x)]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I/(c*x)]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [A]  time = 0.0051538, size = 31, normalized size = 1. \[ i \text{PolyLog}\left (2,\frac{i}{\sqrt{x}}\right )-i \text{PolyLog}\left (2,-\frac{i}{\sqrt{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Sqrt[x]]/x,x]

[Out]

(-I)*PolyLog[2, (-I)/Sqrt[x]] + I*PolyLog[2, I/Sqrt[x]]

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Maple [B]  time = 0.033, size = 61, normalized size = 2. \begin{align*} \ln \left ( x \right ){\rm arccot} \left (\sqrt{x}\right )-{\frac{i}{2}}\ln \left ( x \right ) \ln \left ( 1+i\sqrt{x} \right ) +{\frac{i}{2}}\ln \left ( x \right ) \ln \left ( 1-i\sqrt{x} \right ) -i{\it dilog} \left ( 1+i\sqrt{x} \right ) +i{\it dilog} \left ( 1-i\sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.58263, size = 47, normalized size = 1.52 \begin{align*} \frac{1}{2} \, \pi \log \left (x + 1\right ) + \operatorname{arccot}\left (\sqrt{x}\right ) \log \left (x\right ) + i \,{\rm Li}_2\left (i \, \sqrt{x} + 1\right ) - i \,{\rm Li}_2\left (-i \, \sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*pi*log(x + 1) + arccot(sqrt(x))*log(x) + I*dilog(I*sqrt(x) + 1) - I*dilog(-I*sqrt(x) + 1)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (\sqrt{x}\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(sqrt(x))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(sqrt(x))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (\sqrt{x}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(sqrt(x))/x, x)