3.1.77 \(\int \frac {\cot ^{-1}(a x^2)}{x} \, dx\) [77]

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

[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 = {4945, 4941, 2438} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.08, size = 57, normalized size = 1.54

method	result	size
default	\(\ln \left (x \right ) \mathrm {arccot}\left (a \,x^{2}\right )+\frac {\munderset {\textit {\_R1} =\RootOf \left (a^{2} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}}}{2 a}\)	\(57\)
risch	\(\frac {\pi \ln \left (x \right )}{2}+\frac {i \ln \left (1+i x \sqrt {-i a}\right ) \ln \left (x \right )}{2}+\frac {i \ln \left (1-i x \sqrt {-i a}\right ) \ln \left (x \right )}{2}+\frac {i \dilog \left (1-i x \sqrt {-i a}\right )}{2}+\frac {i \dilog \left (1+i x \sqrt {-i a}\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (-i a \,x^{2}+1\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (1+i x \sqrt {i a}\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (1-i x \sqrt {i a}\right )}{2}-\frac {i \dilog \left (1+i x \sqrt {i a}\right )}{2}-\frac {i \dilog \left (1-i x \sqrt {i a}\right )}{2}+\frac {i \ln \left (i a \,x^{2}+1\right ) \ln \left (x \right )}{2}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (23) = 46\).
time = 0.49, size = 68, normalized size = 1.84 \begin {gather*} \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 \left (x\right ) + \arctan \left (a x^{2}\right ) \log \left (x\right ) + \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 ) \end {gather*}

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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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(a*x^2)/x,x, algorithm="fricas")

[Out]

integral(arccot(a*x^2)/x, x)

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

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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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(arccot(a*x^2)/x,x, algorithm="giac")

[Out]

integrate(arccot(a*x^2)/x, x)

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

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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