\(\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx\) [30]
Optimal result
Integrand size = 10, antiderivative size = 93 \[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=-i a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )
\]
[Out]
-I*a*arccot(a*x)^3-arccot(a*x)^3/x-3*a*arccot(a*x)^2*ln(2-2/(1-I*a*x))-3*I*a*arccot(a*x)*polylog(2,-1+2/(1-I*a
*x))-3/2*a*polylog(3,-1+2/(1-I*a*x))
Rubi [A] (verified)
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4947, 5045, 4989, 5005, 5113,
6745} \[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=-\frac {3}{2} a \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )-3 i a \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right ) \cot ^{-1}(a x)-i a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{x}-3 a \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)^2
\]
[In]
Int[ArcCot[a*x]^3/x^2,x]
[Out]
(-I)*a*ArcCot[a*x]^3 - ArcCot[a*x]^3/x - 3*a*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*ArcCot[a*x]*PolyLo
g[2, -1 + 2/(1 - I*a*x)] - (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/2
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 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}{x}-(3 a) \int \frac {\cot ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx \\ & = -i a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{x}-(3 i a) \int \frac {\cot ^{-1}(a x)^2}{x (i+a x)} \, dx \\ & = -i a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-\left (6 a^2\right ) \int \frac {\cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx \\ & = -i a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\left (3 i a^2\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx \\ & = -i a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right ) \\
\end{align*}
Mathematica [A] (verified)
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.89
\[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\frac {(-1+i a x) \cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+3 i a \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )
\]
[In]
Integrate[ArcCot[a*x]^3/x^2,x]
[Out]
((-1 + I*a*x)*ArcCot[a*x]^3)/x - 3*a*ArcCot[a*x]^2*Log[1 + E^((2*I)*ArcCot[a*x])] + (3*I)*a*ArcCot[a*x]*PolyLo
g[2, -E^((2*I)*ArcCot[a*x])] - (3*a*PolyLog[3, -E^((2*I)*ArcCot[a*x])])/2
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 8.94 (sec) , antiderivative size = 1441, normalized size of antiderivative =
15.49
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| method | result | size |
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| parts |
\(\text {Expression too large to display}\) |
\(1441\) |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(1444\) |
| default |
\(\text {Expression too large to display}\) |
\(1444\) |
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[In]
int(arccot(a*x)^3/x^2,x,method=_RETURNVERBOSE)
[Out]
-arccot(a*x)^3/x-3*a*(-1/2*arccot(a*x)^2*ln(a^2*x^2+1)+arccot(a*x)^2*ln(a*x)+arccot(a*x)^2*ln((I+a*x)/(a^2*x^2
+1)^(1/2))-1/3*I*arccot(a*x)^3+1/4*(-2*I*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)
*(1+(I+a*x)^2/(a^2*x^2+1)))^2+I*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)-I*P
i*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3+2*I*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*I*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+2*I*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+I*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)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2-2*
I*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)^2-2*I*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+2*I*Pi*csgn(1/
((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^2-I*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))*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)+I*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))*c
sgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2+2*I*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^
2/(a^2*x^2+1)))^3+I*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^3+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)))-I*
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))+I*Pi*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^
3-2*I*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2-2*I*Pi*csgn(1/((I+a*x)^2/(a^2*x^2+1)-1)*(
1+(I+a*x)^2/(a^2*x^2+1)))^3+4*ln(2))*arccot(a*x)^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)))
Fricas [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{2}} \,d x }
\]
[In]
integrate(arccot(a*x)^3/x^2,x, algorithm="fricas")
[Out]
integral(arccot(a*x)^3/x^2, x)
Sympy [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{2}}\, dx
\]
[In]
integrate(acot(a*x)**3/x**2,x)
[Out]
Integral(acot(a*x)**3/x**2, x)
Maxima [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{2}} \,d x }
\]
[In]
integrate(arccot(a*x)^3/x^2,x, algorithm="maxima")
[Out]
-1/32*(4*arctan2(1, a*x)^3 - 3*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 - (28*a*arctan(a*x)*arctan(1/(a*x))^3 + 7*(6
*arctan(a*x)^2*arctan(1/(a*x))^2/a + (a*arctan(a*x)^4 + 4*a*arctan(a*x)^3*arctan(1/(a*x)))/a^2)*a^2 + 96*a^2*i
ntegrate(1/32*x^2*arctan(1/(a*x))*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 384*a^2*integrate(1/32*x^2*arctan(1
/(a*x))*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 384*a*integrate(1/32*x*arctan(1/(a*x))^2/(a^2*x^4 + x^2), x) +
96*a*integrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 896*integrate(1/32*arctan(1/(a*x))^3/(a^2*x^4 +
x^2), x) + 96*integrate(1/32*arctan(1/(a*x))*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x
Giac [F]
\[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{2}} \,d x }
\]
[In]
integrate(arccot(a*x)^3/x^2,x, algorithm="giac")
[Out]
integrate(arccot(a*x)^3/x^2, x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^2} \,d x
\]
[In]
int(acot(a*x)^3/x^2,x)
[Out]
int(acot(a*x)^3/x^2, x)