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

Optimal. Leaf size=116 \[ -\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 i}{a x+i}\right )+\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 a x}{a x+i}\right )-i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2 i}{a x+i}\right )+i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2 a x}{a x+i}\right )+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

[Out]

2*ArcCot[a*x]^2*ArcCoth[1 - 2/(1 + I*a*x)] - I*ArcCot[a*x]*PolyLog[2, 1 - (2*I)/(I + a*x)] + I*ArcCot[a*x]*Pol
yLog[2, 1 - (2*a*x)/(I + a*x)] - PolyLog[3, 1 - (2*I)/(I + a*x)]/2 + PolyLog[3, 1 - (2*a*x)/(I + a*x)]/2

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Rubi [A]  time = 0.212925, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4851, 4989, 4885, 4993, 6610} \[ -\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 i}{a x+i}\right )+\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 a x}{a x+i}\right )-i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2 i}{a x+i}\right )+i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2 a x}{a x+i}\right )+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcCot[a*x]^2*ArcCoth[1 - 2/(1 + I*a*x)] - I*ArcCot[a*x]*PolyLog[2, 1 - (2*I)/(I + a*x)] + I*ArcCot[a*x]*Pol
yLog[2, 1 - (2*a*x)/(I + a*x)] - PolyLog[3, 1 - (2*I)/(I + a*x)]/2 + PolyLog[3, 1 - (2*a*x)/(I + a*x)]/2

Rule 4851

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

Rule 4989

Int[(ArcCoth[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[(
Log[SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[SimplifyInt
egrand[1 - 1/u, x]]*(a + b*ArcCot[c*x])^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && E
qQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^2, 0]

Rule 4885

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(a + b*ArcCot[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 4993

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a + b*ArcC
ot[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u]
)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*I
)/(I + c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(a x)^2}{x} \, dx &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )+(4 a) \int \frac{\cot ^{-1}(a x) \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )-(2 a) \int \frac{\cot ^{-1}(a x) \log \left (\frac{2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(2 a) \int \frac{\cot ^{-1}(a x) \log \left (\frac{2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )-i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2 i}{i+a x}\right )+i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2 a x}{i+a x}\right )-(i a) \int \frac{\text{Li}_2\left (1-\frac{2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(i a) \int \frac{\text{Li}_2\left (1-\frac{2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )-i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2 i}{i+a x}\right )+i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2 a x}{i+a x}\right )-\frac{1}{2} \text{Li}_3\left (1-\frac{2 i}{i+a x}\right )+\frac{1}{2} \text{Li}_3\left (1-\frac{2 a x}{i+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0604744, size = 132, normalized size = 1.14 \[ -i \cot ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-i \cot ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )-\frac{2}{3} i \cot ^{-1}(a x)^3-\cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*I)/3)*ArcCot[a*x]^3 - ArcCot[a*x]^2*Log[1 - E^((-2*I)*ArcCot[a*x])] + ArcCot[a*x]^2*Log[1 + E^((2*I)*ArcC
ot[a*x])] - I*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a*x])] - I*ArcCot[a*x]*PolyLog[2, -E^((2*I)*ArcCot[a*x])
] - PolyLog[3, E^((-2*I)*ArcCot[a*x])]/2 + PolyLog[3, -E^((2*I)*ArcCot[a*x])]/2

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Maple [C]  time = 0.601, size = 959, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(a*x)*arccot(a*x)^2+1/2*I*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^3*arccot(a*x)^2-1/2
*I*Pi*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^3*arccot(a*x)^2+1/2*I*Pi*csgn(I/((a*x+I)^2/(
a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))*arccot(a*
x)^2+2*I*arccot(a*x)*polylog(2,-(a*x+I)/(a^2*x^2+1)^(1/2))-1/2*I*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^
2/(a^2*x^2+1)+1))^2*csgn(I*((a*x+I)^2/(a^2*x^2+1)+1))*arccot(a*x)^2-1/2*I*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*
((a*x+I)^2/(a^2*x^2+1)+1))*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^2*arccot(a*x)^2-1/2*I*P
i*arccot(a*x)^2+1/2*I*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1))*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^
2+1)+1))*csgn(I*((a*x+I)^2/(a^2*x^2+1)+1))*arccot(a*x)^2+2*I*arccot(a*x)*polylog(2,(a*x+I)/(a^2*x^2+1)^(1/2))+
arccot(a*x)^2*ln((a*x+I)^2/(a^2*x^2+1)-1)-arccot(a*x)^2*ln(1-(a*x+I)/(a^2*x^2+1)^(1/2))-1/2*I*Pi*csgn(I/((a*x+
I)^2/(a^2*x^2+1)-1))*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^2*arccot(a*x)^2-2*polylog(3,(
a*x+I)/(a^2*x^2+1)^(1/2))-arccot(a*x)^2*ln(1+(a*x+I)/(a^2*x^2+1)^(1/2))-I*arccot(a*x)*polylog(2,-(a*x+I)^2/(a^
2*x^2+1))-2*polylog(3,-(a*x+I)/(a^2*x^2+1)^(1/2))+1/2*I*Pi*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^
2+1)+1))^2*arccot(a*x)^2+1/2*polylog(3,-(a*x+I)^2/(a^2*x^2+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x\right )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x)^2/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (a x\right )^{2}}{x}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}^{2}{\left (a x \right )}}{x}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x\right )^{2}}{x}\,{d x} \end{align*}

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)