\(\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx\) [32]
Optimal result
Integrand size = 10, antiderivative size = 167 \[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{3 x^3}-a^3 \log (x)+\frac {1}{2} a^3 \log \left (1+a^2 x^2\right )+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {1}{2} a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )
\]
[Out]
-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*arccot(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))
Rubi [A] (verified)
Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of
steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {4947, 5039, 272, 36, 29, 31,
5005, 5045, 4989, 5113, 6745} \[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\frac {1}{2} a^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+i a^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right ) \cot ^{-1}(a x)-a^3 \log (x)+\frac {1}{3} i a^3 \cot ^{-1}(a x)^3+\frac {1}{2} a^3 \cot ^{-1}(a x)^2+a^3 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)^2-\frac {a^2 \cot ^{-1}(a x)}{x}+\frac {1}{2} a^3 \log \left (a^2 x^2+1\right )-\frac {\cot ^{-1}(a x)^3}{3 x^3}+\frac {a \cot ^{-1}(a x)^2}{2 x^2}
\]
[In]
Int[ArcCot[a*x]^3/x^4,x]
[Out]
-((a^2*ArcCot[a*x])/x) + (a^3*ArcCot[a*x]^2)/2 + (a*ArcCot[a*x]^2)/(2*x^2) + (I/3)*a^3*ArcCot[a*x]^3 - ArcCot[
a*x]^3/(3*x^3) - a^3*Log[x] + (a^3*Log[1 + a^2*x^2])/2 + a^3*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] + I*a^3*ArcC
ot[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/2
Rule 29
Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 36
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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 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 4989
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] + Dist[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 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 5039
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[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 5045
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] + Dist[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 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 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 {\cot ^{-1}(a x)^3}{3 x^3}-a \int \frac {\cot ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a x)^3}{3 x^3}-a \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx+a^3 \int \frac {\cot ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx \\ & = \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^2 \int \frac {\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx+\left (i a^3\right ) \int \frac {\cot ^{-1}(a x)^2}{x (i+a x)} \, dx \\ & = \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 \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+a^2 \int \frac {\cot ^{-1}(a x)}{x^2} \, dx-a^4 \int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^4\right ) \int \frac {\cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, 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 \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 )-a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx+\left (i a^4\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, 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 \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 )-\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\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 \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 )-\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\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 ) \\
\end{align*}
Mathematica [A] (verified)
Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90
\[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\frac {1}{6} \left (-\frac {6 a^2 \cot ^{-1}(a x)}{x}+3 a^3 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{x^2}-2 i a^3 \cot ^{-1}(a x)^3-\frac {2 \cot ^{-1}(a x)^3}{x^3}+6 a^3 \cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )-6 a^3 \log \left (\frac {1}{\sqrt {1+\frac {1}{a^2 x^2}}}\right )-6 i a^3 \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+3 a^3 \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )\right )
\]
[In]
Integrate[ArcCot[a*x]^3/x^4,x]
[Out]
((-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
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 |
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| parts |
\(\text {Expression too large to display}\) |
\(1622\) |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(1623\) |
| default |
\(\text {Expression too large to display}\) |
\(1623\) |
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[In]
int(arccot(a*x)^3/x^4,x,method=_RETURNVERBOSE)
[Out]
-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)-a
rccot(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)*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)*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*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)))*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*c
sgn(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*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)))^2*a*x+3*I*arccot(a*x)*Pi*csgn(I*(I+a*x)^2/(
a^2*x^2+1))*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))^2*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)))^3*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)^2*csgn
(I/((I+a*x)^2/(a^2*x^2+1)-1)^2)*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)^2*csgn(I*(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)/((I+a*x)^2/(a^2*x^2+1)-
1)^2)*csgn(I*(I+a*x)^2/(a^2*x^2+1))*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)^2)*a*x-12*arccot(a*x)*ln(2)*a*x-6*arccot(
a*x)*a*x+12+6*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)^2*a*x)/a/x+ln(1+(I+a*
x)^2/(a^2*x^2+1)))
Fricas [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{4}} \,d x }
\]
[In]
integrate(arccot(a*x)^3/x^4,x, algorithm="fricas")
[Out]
integral(arccot(a*x)^3/x^4, x)
Sympy [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{4}}\, dx
\]
[In]
integrate(acot(a*x)**3/x**4,x)
[Out]
Integral(acot(a*x)**3/x**4, x)
Maxima [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{4}} \,d x }
\]
[In]
integrate(arccot(a*x)^3/x^4,x, algorithm="maxima")
[Out]
1/96*(96*x^3*integrate(1/32*(28*a^2*x^2*arctan2(1, a*x)^3 - 4*a^2*x^2*arctan2(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
Giac [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{4}} \,d x }
\]
[In]
integrate(arccot(a*x)^3/x^4,x, algorithm="giac")
[Out]
integrate(arccot(a*x)^3/x^4, x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\cot ^{-1}(a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^4} \,d x
\]
[In]
int(acot(a*x)^3/x^4,x)
[Out]
int(acot(a*x)^3/x^4, x)