\(\int x^4 \cot ^{-1}(a x)^3 \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 205 \[ \int x^4 \cot ^{-1}(a x)^3 \, dx=\frac {x^2}{20 a^3}-\frac {9 x \cot ^{-1}(a x)}{10 a^4}+\frac {x^3 \cot ^{-1}(a x)}{10 a^2}-\frac {9 \cot ^{-1}(a x)^2}{20 a^5}-\frac {3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac {i \cot ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^5}-\frac {\log \left (1+a^2 x^2\right )}{2 a^5}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^5}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{10 a^5} \]

[Out]

1/20*x^2/a^3-9/10*x*arccot(a*x)/a^4+1/10*x^3*arccot(a*x)/a^2-9/20*arccot(a*x)^2/a^5-3/10*x^2*arccot(a*x)^2/a^3
+3/20*x^4*arccot(a*x)^2/a+1/5*I*arccot(a*x)^3/a^5+1/5*x^5*arccot(a*x)^3-3/5*arccot(a*x)^2*ln(2/(1+I*a*x))/a^5-
1/2*ln(a^2*x^2+1)/a^5+3/5*I*arccot(a*x)*polylog(2,1-2/(1+I*a*x))/a^5-3/10*polylog(3,1-2/(1+I*a*x))/a^5

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {4947, 5037, 272, 45, 4931, 266, 5005, 5041, 4965, 5115, 6745} \[ \int x^4 \cot ^{-1}(a x)^3 \, dx=-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{10 a^5}+\frac {3 i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right ) \cot ^{-1}(a x)}{5 a^5}+\frac {i \cot ^{-1}(a x)^3}{5 a^5}-\frac {9 \cot ^{-1}(a x)^2}{20 a^5}-\frac {3 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{5 a^5}-\frac {9 x \cot ^{-1}(a x)}{10 a^4}+\frac {x^2}{20 a^3}-\frac {3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac {x^3 \cot ^{-1}(a x)}{10 a^2}-\frac {\log \left (a^2 x^2+1\right )}{2 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3+\frac {3 x^4 \cot ^{-1}(a x)^2}{20 a} \]

[In]

Int[x^4*ArcCot[a*x]^3,x]

[Out]

x^2/(20*a^3) - (9*x*ArcCot[a*x])/(10*a^4) + (x^3*ArcCot[a*x])/(10*a^2) - (9*ArcCot[a*x]^2)/(20*a^5) - (3*x^2*A
rcCot[a*x]^2)/(10*a^3) + (3*x^4*ArcCot[a*x]^2)/(20*a) + ((I/5)*ArcCot[a*x]^3)/a^5 + (x^5*ArcCot[a*x]^3)/5 - (3
*ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^5) - Log[1 + a^2*x^2]/(2*a^5) + (((3*I)/5)*ArcCot[a*x]*PolyLog[2, 1 -
2/(1 + I*a*x)])/a^5 - (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(10*a^5)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4931

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Dist[b*c
*n*p, Int[x^n*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rule 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4965

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

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 5037

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcCot[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 5041

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

Rule 5115

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cCot[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 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}& = \frac {1}{5} x^5 \cot ^{-1}(a x)^3+\frac {1}{5} (3 a) \int \frac {x^5 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{5} x^5 \cot ^{-1}(a x)^3+\frac {3 \int x^3 \cot ^{-1}(a x)^2 \, dx}{5 a}-\frac {3 \int \frac {x^3 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a} \\ & = \frac {3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3+\frac {3}{10} \int \frac {x^4 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {3 \int x \cot ^{-1}(a x)^2 \, dx}{5 a^3}+\frac {3 \int \frac {x \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a^3} \\ & = -\frac {3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac {i \cot ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3-\frac {3 \int \frac {\cot ^{-1}(a x)^2}{i-a x} \, dx}{5 a^4}+\frac {3 \int x^2 \cot ^{-1}(a x) \, dx}{10 a^2}-\frac {3 \int \frac {x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{10 a^2}-\frac {3 \int \frac {x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2} \\ & = \frac {x^3 \cot ^{-1}(a x)}{10 a^2}-\frac {3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac {i \cot ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^5}-\frac {3 \int \cot ^{-1}(a x) \, dx}{10 a^4}+\frac {3 \int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{10 a^4}-\frac {3 \int \cot ^{-1}(a x) \, dx}{5 a^4}+\frac {3 \int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^4}-\frac {6 \int \frac {\cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^4}+\frac {\int \frac {x^3}{1+a^2 x^2} \, dx}{10 a} \\ & = -\frac {9 x \cot ^{-1}(a x)}{10 a^4}+\frac {x^3 \cot ^{-1}(a x)}{10 a^2}-\frac {9 \cot ^{-1}(a x)^2}{20 a^5}-\frac {3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac {i \cot ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^5}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^5}+\frac {(3 i) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^4}-\frac {3 \int \frac {x}{1+a^2 x^2} \, dx}{10 a^3}-\frac {3 \int \frac {x}{1+a^2 x^2} \, dx}{5 a^3}+\frac {\text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )}{20 a} \\ & = -\frac {9 x \cot ^{-1}(a x)}{10 a^4}+\frac {x^3 \cot ^{-1}(a x)}{10 a^2}-\frac {9 \cot ^{-1}(a x)^2}{20 a^5}-\frac {3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac {i \cot ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^5}-\frac {9 \log \left (1+a^2 x^2\right )}{20 a^5}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^5}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{10 a^5}+\frac {\text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{20 a} \\ & = \frac {x^2}{20 a^3}-\frac {9 x \cot ^{-1}(a x)}{10 a^4}+\frac {x^3 \cot ^{-1}(a x)}{10 a^2}-\frac {9 \cot ^{-1}(a x)^2}{20 a^5}-\frac {3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac {i \cot ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^5}-\frac {\log \left (1+a^2 x^2\right )}{2 a^5}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^5}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{10 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.91 \[ \int x^4 \cot ^{-1}(a x)^3 \, dx=\frac {2+i \pi ^3+2 a^2 x^2-36 a x \cot ^{-1}(a x)+4 a^3 x^3 \cot ^{-1}(a x)-18 \cot ^{-1}(a x)^2-12 a^2 x^2 \cot ^{-1}(a x)^2+6 a^4 x^4 \cot ^{-1}(a x)^2-8 i \cot ^{-1}(a x)^3+8 a^5 x^5 \cot ^{-1}(a x)^3-24 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )+40 \log \left (\frac {1}{\sqrt {1+\frac {1}{a^2 x^2}}}\right )+40 \log \left (\frac {1}{a x}\right )-24 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-12 \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{40 a^5} \]

[In]

Integrate[x^4*ArcCot[a*x]^3,x]

[Out]

(2 + I*Pi^3 + 2*a^2*x^2 - 36*a*x*ArcCot[a*x] + 4*a^3*x^3*ArcCot[a*x] - 18*ArcCot[a*x]^2 - 12*a^2*x^2*ArcCot[a*
x]^2 + 6*a^4*x^4*ArcCot[a*x]^2 - (8*I)*ArcCot[a*x]^3 + 8*a^5*x^5*ArcCot[a*x]^3 - 24*ArcCot[a*x]^2*Log[1 - E^((
-2*I)*ArcCot[a*x])] + 40*Log[1/Sqrt[1 + 1/(a^2*x^2)]] + 40*Log[1/(a*x)] - (24*I)*ArcCot[a*x]*PolyLog[2, E^((-2
*I)*ArcCot[a*x])] - 12*PolyLog[3, E^((-2*I)*ArcCot[a*x])])/(40*a^5)

Maple [C] (warning: unable to verify)

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

Time = 8.10 (sec) , antiderivative size = 1108, normalized size of antiderivative = 5.40

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1108\)
default	\(\text {Expression too large to display}\)	\(1108\)
parts	\(\text {Expression too large to display}\)	\(1110\)

[In]

int(x^4*arccot(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(1/5*a^5*x^5*arccot(a*x)^3+3/20*a^4*x^4*arccot(a*x)^2-3/10*a^2*x^2*arccot(a*x)^2+3/10*arccot(a*x)^2*ln(a
^2*x^2+1)-3/5*arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))+3/5*arccot(a*x)^2*ln((I+a*x)^2/(a^2*x^2+1)-1)+1/20*I
*(-3*arccot(a*x)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^3*Pi-3*arccot(a*x)^2*csgn(I*(I+a*
x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1))*Pi-3*arccot(a*x)^2*csgn(I*(I+a*x
)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)^2)+3*arccot(a*x)^2*csgn(I*(
I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)*csgn(I*(I+a*x)^2/(a^2*x^2+1))*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+
1)-1)^2)+3*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*arccot(a*x)^2-6*Pi*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(I+a
*x)^2/(a^2*x^2+1))^2*arccot(a*x)^2+3*Pi*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(I+a*x)^2/(a^2*x^2+1))*arcc
ot(a*x)^2-3*arccot(a*x)^2*Pi*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1))^2*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)+6*arccot(
a*x)^2*Pi*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^2-3*arccot(a*x)^2*Pi*csgn(I*((
I+a*x)^2/(a^2*x^2+1)-1)^2)^3+6*arccot(a*x)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*Pi+4*
arccot(a*x)^3-6*Pi*arccot(a*x)^2-I*a^2*x^2+12*I*arccot(a*x)^2*ln(2)+18*I*arccot(a*x)*a*x-20*arccot(a*x)+9*I*ar
ccot(a*x)^2-2*I*arccot(a*x)*a^3*x^3-I)+ln((I+a*x)/(a^2*x^2+1)^(1/2)-1)+ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))-3/5*arc
cot(a*x)^2*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+6/5*I*arccot(a*x)*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-6/5*polylog(
3,(I+a*x)/(a^2*x^2+1)^(1/2))-3/5*arccot(a*x)^2*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+6/5*I*arccot(a*x)*polylog(2,-(I
+a*x)/(a^2*x^2+1)^(1/2))-6/5*polylog(3,-(I+a*x)/(a^2*x^2+1)^(1/2)))

Fricas [F]

\[ \int x^4 \cot ^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^4*arccot(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^4*arccot(a*x)^3, x)

Sympy [F]

\[ \int x^4 \cot ^{-1}(a x)^3 \, dx=\int x^{4} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]

[In]

integrate(x**4*acot(a*x)**3,x)

[Out]

Integral(x**4*acot(a*x)**3, x)

Maxima [F]

\[ \int x^4 \cot ^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^4*arccot(a*x)^3,x, algorithm="maxima")

[Out]

1/40*x^5*arctan2(1, a*x)^3 - 3/160*x^5*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + integrate(1/160*(140*a^2*x^6*arcta
n2(1, a*x)^3 + 12*a^2*x^6*arctan2(1, a*x)*log(a^2*x^2 + 1) + 12*a*x^5*arctan2(1, a*x)^2 + 140*x^4*arctan2(1, a
*x)^3 + 3*(5*a^2*x^6*arctan2(1, a*x) - a*x^5 + 5*x^4*arctan2(1, a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)

Giac [F]

\[ \int x^4 \cot ^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^4*arccot(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^4*arccot(a*x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^4 \cot ^{-1}(a x)^3 \, dx=\int x^4\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]

[In]

int(x^4*acot(a*x)^3,x)

[Out]

int(x^4*acot(a*x)^3, x)