3.1.32 \(\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx\) [32]

3.1.32.1 Optimal result

Integrand size = 10, antiderivative size = 167 \[ \int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}-a^3 \log (x)+\frac {1}{2} a^3 \log \left (1+a^2 x^2\right )+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {1}{2} a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right ) \]

output

-a^2*arccot(a*x)/x+1/2*a^3*arccot(a*x)^2+1/2*a*arccot(a*x)^2/x^2+1/3*I*a^3 
*arccot(a*x)^3-1/3*arccot(a*x)^3/x^3-a^3*ln(x)+1/2*a^3*ln(a^2*x^2+1)+a^3*a 
rccot(a*x)^2*ln(2-2/(1-I*a*x))+I*a^3*arccot(a*x)*polylog(2,-1+2/(1-I*a*x)) 
+1/2*a^3*polylog(3,-1+2/(1-I*a*x))
 

3.1.32.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\frac {1}{6} \left (-\frac {6 a^2 \cot ^{-1}(a x)}{x}+3 a^3 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{x^2}-2 i a^3 \cot ^{-1}(a x)^3-\frac {2 \cot ^{-1}(a x)^3}{x^3}+6 a^3 \cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )-6 a^3 \log \left (\frac {1}{\sqrt {1+\frac {1}{a^2 x^2}}}\right )-6 i a^3 \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+3 a^3 \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )\right ) \]

input

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

output

((-6*a^2*ArcCot[a*x])/x + 3*a^3*ArcCot[a*x]^2 + (3*a*ArcCot[a*x]^2)/x^2 - 
(2*I)*a^3*ArcCot[a*x]^3 - (2*ArcCot[a*x]^3)/x^3 + 6*a^3*ArcCot[a*x]^2*Log[ 
1 + E^((2*I)*ArcCot[a*x])] - 6*a^3*Log[1/Sqrt[1 + 1/(a^2*x^2)]] - (6*I)*a^ 
3*ArcCot[a*x]*PolyLog[2, -E^((2*I)*ArcCot[a*x])] + 3*a^3*PolyLog[3, -E^((2 
*I)*ArcCot[a*x])])/6
 

3.1.32.3 Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {5362, 5454, 5362, 5454, 5362, 243, 47, 14, 16, 5420, 5460, 5404, 5528, 7164}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx\)

\(\Big \downarrow \) 5362

\(\displaystyle -a \int \frac {\cot ^{-1}(a x)^2}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 5454

\(\displaystyle -a \left (\int \frac {\cot ^{-1}(a x)^2}{x^3}dx-a^2 \int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 5362

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \int \frac {\cot ^{-1}(a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 5454

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \left (\int \frac {\cot ^{-1}(a x)}{x^2}dx-a^2 \int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 5362

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \left (-a \int \frac {1}{x \left (a^2 x^2+1\right )}dx+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 243

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \left (-\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 47

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \left (-\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 14

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 16

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 5420

\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x \left (a^2 x^2+1\right )}dx\right )-a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )+\frac {1}{2} a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}\)

\(\Big \downarrow \) 5460

\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{3 x^3}-a \left (-\left (a^2 \left (i \int \frac {\cot ^{-1}(a x)^2}{x (a x+i)}dx+\frac {1}{3} i \cot ^{-1}(a x)^3\right )\right )-a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )+\frac {1}{2} a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )\)

\(\Big \downarrow \) 5404

\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{3 x^3}-a \left (-\left (a^2 \left (i \left (-2 i a \int \frac {\cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)^2\right )+\frac {1}{3} i \cot ^{-1}(a x)^3\right )\right )-a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )+\frac {1}{2} a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )\)

\(\Big \downarrow \) 5528

\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{3 x^3}-a \left (-\left (a^2 \left (i \left (-2 i a \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx+\frac {i \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right ) \cot ^{-1}(a x)}{2 a}\right )-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)^2\right )+\frac {1}{3} i \cot ^{-1}(a x)^3\right )\right )-a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )+\frac {1}{2} a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{3 x^3}-a \left (-\left (a^2 \left (i \left (-2 i a \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}+\frac {i \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right ) \cot ^{-1}(a x)}{2 a}\right )-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)^2\right )+\frac {1}{3} i \cot ^{-1}(a x)^3\right )\right )-a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )+\frac {1}{2} a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\right )\)

input

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

output

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

3.1.32.3.1 Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 5362

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] + Simp[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 5404

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcCot[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] + Si 
mp[b*c*(p/d)   Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[2 - 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 5420

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> 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 5454

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], 
 x] - Simp[e/(d*f^2)   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] && LtQ[m, -1]
 

rule 5460

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

rule 5528

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 
), x_Symbol] :> Simp[I*(a + b*ArcCot[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] + Simp[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 7164

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]
 

3.1.32.4 Maple [C] (warning: unable to verify)

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

Time = 25.04 (sec) , antiderivative size = 1622, normalized size of antiderivative = 9.71

method	result	size
parts	\(\text {Expression too large to display}\)	\(1622\)
derivativedivides	\(\text {Expression too large to display}\)	\(1623\)
default	\(\text {Expression too large to display}\)	\(1623\)

input

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

output

-1/3*arccot(a*x)^3/x^3-a^3*(-1/2/a^2/x^2*arccot(a*x)^2-arccot(a*x)^2*ln(a* 
x)+1/2*arccot(a*x)^2*ln(a^2*x^2+1)-arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1 
/2))+I*arccot(a*x)*polylog(2,-(I+a*x)^2/(a^2*x^2+1))-1/2*polylog(3,-(I+a*x 
)^2/(a^2*x^2+1))+1/12*arccot(a*x)*(4*I*arccot(a*x)^2*a*x-3*I*arccot(a*x)*P 
i*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)* 
a*x+6*I*arccot(a*x)*Pi*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*a*x+3*I*arccot(a*x)*Pi*csgn(I 
*(I+a*x)^2/(a^2*x^2+1))^3*a*x-6*I*arccot(a*x)*Pi*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)))*a*x-3*I*arccot(a*x)*Pi*csgn(I*((I+a*x)^2/(a^2*x^2+1)- 
1)^2)^3*a*x-6*I*arccot(a*x)*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^2*csgn(I*(I+a 
*x)/(a^2*x^2+1)^(1/2))*a*x+6*I*arccot(a*x)*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*a*x+6*I*arccot(a*x)*Pi*csgn(1/ 
((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^3*a*x-12*I*a*x-6*I*ar 
ccot(a*x)*Pi*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))*a*x+6*I 
*arccot(a*x)*Pi*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))*a*x-3*I*arccot(a*x)*Pi*csgn(I*(I+a*x 
)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^3*a*x-6*I*arccot(a*x)*Pi*csgn 
(1/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^2*a*x+6*I*arcco...
 

3.1.32.5 Fricas [F]

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

input

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

output

integral(arccot(a*x)^3/x^4, x)
 

3.1.32.6 Sympy [F]

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

input

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

output

Integral(acot(a*x)**3/x**4, x)
 

3.1.32.7 Maxima [F]

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

input

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

output

1/96*(96*x^3*integrate(1/32*(28*a^2*x^2*arctan2(1, a*x)^3 - 4*a^2*x^2*arct 
an2(1, a*x)*log(a^2*x^2 + 1) - 4*a*x*arctan2(1, a*x)^2 + 28*arctan2(1, a*x 
)^3 + (3*a^2*x^2*arctan2(1, a*x) + a*x + 3*arctan2(1, a*x))*log(a^2*x^2 + 
1)^2)/(a^2*x^6 + x^4), x) - 4*arctan2(1, a*x)^3 + 3*arctan2(1, a*x)*log(a^ 
2*x^2 + 1)^2)/x^3
 

3.1.32.8 Giac [F]

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

input

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

output

integrate(arccot(a*x)^3/x^4, x)
 

3.1.32.9 Mupad [F(-1)]

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

input

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

output

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