3.29 \(\int \frac{\cot ^{-1}(a x)^3}{x} \, dx\)

Optimal. Leaf size=178 \[ \frac{3}{4} i \text{PolyLog}\left (4,1-\frac{2 i}{a x+i}\right )-\frac{3}{4} i \text{PolyLog}\left (4,1-\frac{2 a x}{a x+i}\right )-\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2 i}{a x+i}\right )+\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2 a x}{a x+i}\right )-\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2 i}{a x+i}\right )+\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2 a x}{a x+i}\right )+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

[Out]

2*ArcCot[a*x]^3*ArcCoth[1 - 2/(1 + I*a*x)] - ((3*I)/2)*ArcCot[a*x]^2*PolyLog[2, 1 - (2*I)/(I + a*x)] + ((3*I)/
2)*ArcCot[a*x]^2*PolyLog[2, 1 - (2*a*x)/(I + a*x)] - (3*ArcCot[a*x]*PolyLog[3, 1 - (2*I)/(I + a*x)])/2 + (3*Ar
cCot[a*x]*PolyLog[3, 1 - (2*a*x)/(I + a*x)])/2 + ((3*I)/4)*PolyLog[4, 1 - (2*I)/(I + a*x)] - ((3*I)/4)*PolyLog
[4, 1 - (2*a*x)/(I + a*x)]

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Rubi [A]  time = 0.326695, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4851, 4989, 4885, 4993, 4997, 6610} \[ \frac{3}{4} i \text{PolyLog}\left (4,1-\frac{2 i}{a x+i}\right )-\frac{3}{4} i \text{PolyLog}\left (4,1-\frac{2 a x}{a x+i}\right )-\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2 i}{a x+i}\right )+\frac{3}{2} i \cot ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2 a x}{a x+i}\right )-\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2 i}{a x+i}\right )+\frac{3}{2} \cot ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2 a x}{a x+i}\right )+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]^3/x,x]

[Out]

2*ArcCot[a*x]^3*ArcCoth[1 - 2/(1 + I*a*x)] - ((3*I)/2)*ArcCot[a*x]^2*PolyLog[2, 1 - (2*I)/(I + a*x)] + ((3*I)/
2)*ArcCot[a*x]^2*PolyLog[2, 1 - (2*a*x)/(I + a*x)] - (3*ArcCot[a*x]*PolyLog[3, 1 - (2*I)/(I + a*x)])/2 + (3*Ar
cCot[a*x]*PolyLog[3, 1 - (2*a*x)/(I + a*x)])/2 + ((3*I)/4)*PolyLog[4, 1 - (2*I)/(I + a*x)] - ((3*I)/4)*PolyLog
[4, 1 - (2*a*x)/(I + a*x)]

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 4997

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a
+ b*ArcCot[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcCot[c*x])^(p - 1)*PolyLog[
k + 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[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)^3}{x} \, dx &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )+(6 a) \int \frac{\cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )-(3 a) \int \frac{\cot ^{-1}(a x)^2 \log \left (\frac{2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(3 a) \int \frac{\cot ^{-1}(a x)^2 \log \left (\frac{2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i \cot ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 i}{i+a x}\right )+\frac{3}{2} i \cot ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 a x}{i+a x}\right )-(3 i a) \int \frac{\cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(3 i a) \int \frac{\cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i \cot ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 i}{i+a x}\right )+\frac{3}{2} i \cot ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 a x}{i+a x}\right )-\frac{3}{2} \cot ^{-1}(a x) \text{Li}_3\left (1-\frac{2 i}{i+a x}\right )+\frac{3}{2} \cot ^{-1}(a x) \text{Li}_3\left (1-\frac{2 a x}{i+a x}\right )-\frac{1}{2} (3 a) \int \frac{\text{Li}_3\left (1-\frac{2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{2} (3 a) \int \frac{\text{Li}_3\left (1-\frac{2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i \cot ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 i}{i+a x}\right )+\frac{3}{2} i \cot ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2 a x}{i+a x}\right )-\frac{3}{2} \cot ^{-1}(a x) \text{Li}_3\left (1-\frac{2 i}{i+a x}\right )+\frac{3}{2} \cot ^{-1}(a x) \text{Li}_3\left (1-\frac{2 a x}{i+a x}\right )+\frac{3}{4} i \text{Li}_4\left (1-\frac{2 i}{i+a x}\right )-\frac{3}{4} i \text{Li}_4\left (1-\frac{2 a x}{i+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0757724, size = 180, normalized size = 1.01 \[ \frac{1}{64} i \left (-96 \cot ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-96 \cot ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+96 i \cot ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )-96 i \cot ^{-1}(a x) \text{PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,e^{-2 i \cot ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,-e^{2 i \cot ^{-1}(a x)}\right )-32 \cot ^{-1}(a x)^4+64 i \cot ^{-1}(a x)^3 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-64 i \cot ^{-1}(a x)^3 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+\pi ^4\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[a*x]^3/x,x]

[Out]

(I/64)*(Pi^4 - 32*ArcCot[a*x]^4 + (64*I)*ArcCot[a*x]^3*Log[1 - E^((-2*I)*ArcCot[a*x])] - (64*I)*ArcCot[a*x]^3*
Log[1 + E^((2*I)*ArcCot[a*x])] - 96*ArcCot[a*x]^2*PolyLog[2, E^((-2*I)*ArcCot[a*x])] - 96*ArcCot[a*x]^2*PolyLo
g[2, -E^((2*I)*ArcCot[a*x])] + (96*I)*ArcCot[a*x]*PolyLog[3, E^((-2*I)*ArcCot[a*x])] - (96*I)*ArcCot[a*x]*Poly
Log[3, -E^((2*I)*ArcCot[a*x])] + 48*PolyLog[4, E^((-2*I)*ArcCot[a*x])] + 48*PolyLog[4, -E^((2*I)*ArcCot[a*x])]
)

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Maple [C]  time = 0.412, size = 1050, normalized size = 5.9 \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)^3/x,x)

[Out]

ln(a*x)*arccot(a*x)^3+arccot(a*x)^3*ln((a*x+I)^2/(a^2*x^2+1)-1)-arccot(a*x)^3*ln(1-(a*x+I)/(a^2*x^2+1)^(1/2))+
3/4*I*polylog(4,-(a*x+I)^2/(a^2*x^2+1))-6*arccot(a*x)*polylog(3,(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))*csgn(I*((a*x+I)^2/(a^2*x^2
+1)+1))*arccot(a*x)^3-arccot(a*x)^3*ln(1+(a*x+I)/(a^2*x^2+1)^(1/2))-6*I*polylog(4,-(a*x+I)/(a^2*x^2+1)^(1/2))-
6*arccot(a*x)*polylog(3,-(a*x+I)/(a^2*x^2+1)^(1/2))+3*I*arccot(a*x)^2*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))*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)^3-1/2*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)^3-3/2*I*arccot(a*x)^2*polylog(2,-(a*x+I)^2/(a^2*x^2+1))-1/2*I*Pi*arccot(a
*x)^3+3/2*arccot(a*x)*polylog(3,-(a*x+I)^2/(a^2*x^2+1))-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)^3-6*I*polylog(4,(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))^3*arccot(a*x)^3+3*I
*arccot(a*x)^2*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)^3+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)^3+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*ar
ccot(a*x)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^3/x,x, algorithm="maxima")

[Out]

integrate(arccot(a*x)^3/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 )^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^3/x,x, algorithm="fricas")

[Out]

integral(arccot(a*x)^3/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}^{3}{\left (a x \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)**3/x,x)

[Out]

Integral(acot(a*x)**3/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 )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^3/x,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^3/x, x)