3.1.88 \(\int \frac {\cot ^{-1}(\sqrt {x})}{x} \, dx\) [88]

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/x^(1/2))+I*polylog(2,I/x^(1/2))

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4945, 4941, 2438} \begin {gather*} i \text {Li}_2\left (\frac {i}{\sqrt {x}}\right )-i \text {Li}_2\left (-\frac {i}{\sqrt {x}}\right ) \end {gather*}

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 2438

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]

Rule 4941

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 4945

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} -i \text {PolyLog}\left (2,-\frac {i}{\sqrt {x}}\right )+i \text {PolyLog}\left (2,\frac {i}{\sqrt {x}}\right ) \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (23 ) = 46\).
time = 0.02, size = 61, normalized size = 1.97

method	result	size
derivativedivides	\(\ln \left (x \right ) \mathrm {arccot}\left (\sqrt {x}\right )-\frac {i \ln \left (x \right ) \ln \left (1+i \sqrt {x}\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (1-i \sqrt {x}\right )}{2}-i \dilog \left (1+i \sqrt {x}\right )+i \dilog \left (1-i \sqrt {x}\right )\)	\(61\)
default	\(\ln \left (x \right ) \mathrm {arccot}\left (\sqrt {x}\right )-\frac {i \ln \left (x \right ) \ln \left (1+i \sqrt {x}\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (1-i \sqrt {x}\right )}{2}-i \dilog \left (1+i \sqrt {x}\right )+i \dilog \left (1-i \sqrt {x}\right )\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.49, size = 35, normalized size = 1.13 \begin {gather*} \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 {gather*}

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 = 0.00, 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(arccot(x^(1/2))/x,x, algorithm="fricas")

[Out]

integral(arccot(sqrt(x))/x, x)

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

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 [A]
time = 0.42, size = 19, normalized size = 0.61 \begin {gather*} -x \arctan \left (\frac {1}{\sqrt {x}}\right ) - \sqrt {x} - \arctan \left (\frac {1}{\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*arctan(1/sqrt(x)) - sqrt(x) - arctan(1/sqrt(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {acot}\left (\sqrt {x}\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acot(x^(1/2))/x, x)

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