3.77 \(\int \frac {\cot ^{-1}(a x^2)}{x} \, dx\)

Optimal. Leaf size=37 \[ \frac {1}{4} i \text {Li}_2\left (\frac {i}{a x^2}\right )-\frac {1}{4} i \text {Li}_2\left (-\frac {i}{a x^2}\right ) \]

[Out]

-1/4*I*polylog(2,-I/a/x^2)+1/4*I*polylog(2,I/a/x^2)

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Rubi [A]  time = 0.03, antiderivative size = 37, 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 = {5032, 4849, 2391} \[ \frac {1}{4} i \text {PolyLog}\left (2,\frac {i}{a x^2}\right )-\frac {1}{4} i \text {PolyLog}\left (2,-\frac {i}{a x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x^2]/x,x]

[Out]

(-I/4)*PolyLog[2, (-I)/(a*x^2)] + (I/4)*PolyLog[2, I/(a*x^2)]

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]

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 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]

Rubi steps

\begin {align*} \int \frac {\cot ^{-1}\left (a x^2\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\cot ^{-1}(a x)}{x} \, dx,x,x^2\right )\\ &=\frac {1}{4} i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i}{a x}\right )}{x} \, dx,x,x^2\right )-\frac {1}{4} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i}{a x}\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {1}{4} i \text {Li}_2\left (-\frac {i}{a x^2}\right )+\frac {1}{4} i \text {Li}_2\left (\frac {i}{a x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 1.00 \[ \frac {1}{4} i \text {Li}_2\left (\frac {i}{a x^2}\right )-\frac {1}{4} i \text {Li}_2\left (-\frac {i}{a x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x^2]/x,x]

[Out]

(-1/4*I)*PolyLog[2, (-I)/(a*x^2)] + (I/4)*PolyLog[2, I/(a*x^2)]

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x^{2}\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(a*x^2)/x, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (a x^{2}\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x^2)/x, x)

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maple [C]  time = 0.16, size = 57, normalized size = 1.54 \[ \ln \relax (x ) \mathrm {arccot}\left (a \,x^{2}\right )+\frac {\munderset {\textit {\_R1} =\RootOf \left (a^{2} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \relax (x ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x^2)/x,x)

[Out]

ln(x)*arccot(a*x^2)+1/2/a*sum(1/_R1^2*(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1)),_R1=RootOf(_Z^4*a^2+1))

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maxima [B]  time = 0.48, size = 68, normalized size = 1.84 \[ \frac {1}{8} \, \pi \log \left (a^{2} x^{4} + 1\right ) - \frac {1}{2} \, \arctan \left (a x^{2}\right ) \log \left (a x^{2}\right ) + \operatorname {arccot}\left (a x^{2}\right ) \log \relax (x) + \arctan \left (a x^{2}\right ) \log \relax (x) + \frac {1}{4} i \, {\rm Li}_2\left (i \, a x^{2} + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-i \, a x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*pi*log(a^2*x^4 + 1) - 1/2*arctan(a*x^2)*log(a*x^2) + arccot(a*x^2)*log(x) + arctan(a*x^2)*log(x) + 1/4*I*d
ilog(I*a*x^2 + 1) - 1/4*I*dilog(-I*a*x^2 + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acot}\left (a\,x^2\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x^2)/x,x)

[Out]

int(acot(a*x^2)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}{\left (a x^{2} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(a*x**2)/x, x)

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