\(\int \frac {\cot ^{-1}(a x)^3}{x} \, dx\) [29]

Optimal result

Integrand size = 10, antiderivative size = 178 \[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{i+a x}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 i}{i+a x}\right )-\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 a x}{i+a x}\right ) \]

[Out]

2*arccot(a*x)^3*arccoth(1-2/(1+I*a*x))-3/2*I*arccot(a*x)^2*polylog(2,1-2*I/(I+a*x))+3/2*I*arccot(a*x)^2*polylo
g(2,1-2*a*x/(I+a*x))-3/2*arccot(a*x)*polylog(3,1-2*I/(I+a*x))+3/2*arccot(a*x)*polylog(3,1-2*a*x/(I+a*x))+3/4*I
*polylog(4,1-2*I/(I+a*x))-3/4*I*polylog(4,1-2*a*x/(I+a*x))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4943, 5109, 5005, 5113, 5117, 6745} \[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 i}{a x+i}\right )-\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 a x}{a x+i}\right )-\frac {3}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)^2+\frac {3}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)^2-\frac {3}{2} \operatorname {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)+\frac {3}{2} \operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]

[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 4943

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 5005

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 5109

Int[(ArcCoth[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L
og[SimplifyIntegrand[1 + 1/u, x]]*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[SimplifyInte
grand[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 5113

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcCo
t[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 5117

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)*(PolyLo
g[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 6745

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*} \text {integral}& = 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 \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-(3 i a) \int \frac {\cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(3 i a) \int \frac {\cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\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 \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{i+a x}\right )-\frac {1}{2} (3 a) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{2} (3 a) \int \frac {\operatorname {PolyLog}\left (3,1-\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 \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{i+a x}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 i}{i+a x}\right )-\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 a x}{i+a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\frac {1}{64} i \left (\pi ^4-32 \cot ^{-1}(a x)^4+64 i \cot ^{-1}(a x)^3 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-64 i \cot ^{-1}(a x)^3 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )-96 \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-96 \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+96 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )-96 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \cot ^{-1}(a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \cot ^{-1}(a x)}\right )\right ) \]

[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])]
)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 6.61 (sec) , antiderivative size = 982, normalized size of antiderivative = 5.52

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(982\)
default	\(\text {Expression too large to display}\)	\(982\)
parts	\(\text {Expression too large to display}\)	\(1417\)

[In]

int(arccot(a*x)^3/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

integral(arccot(a*x)^3/x, x)

Sympy [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(acot(a*x)**3/x, x)

Maxima [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

integrate(arccot(a*x)^3/x, x)

Giac [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

integrate(arccot(a*x)^3/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x} \,d x \]

[In]

int(acot(a*x)^3/x,x)

[Out]

int(acot(a*x)^3/x, x)