\(\int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx\) [31]

Optimal result

Integrand size = 10, antiderivative size = 105 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+3 a^2 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4947, 5039, 5045, 4989, 2497, 5005} \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-\frac {1}{2} a^2 \cot ^{-1}(a x)^3+\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\frac {3 a \cot ^{-1}(a x)^2}{2 x} \]

[In]

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

[Out]

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

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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]

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=-\frac {\cot ^{-1}(a x) \left (3 i a x (i+a x) \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2-6 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )}{2 x^2}-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right ) \]

[In]

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

[Out]

-1/2*(ArcCot[a*x]*((3*I)*a*x*(I + a*x)*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 - 6*a^2*x^2*Log[1 + E^((2*I)*
ArcCot[a*x])]))/x^2 - ((3*I)/2)*a^2*PolyLog[2, -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 = 6.76 (sec) , antiderivative size = 2956, normalized size of antiderivative = 28.15

method	result	size
parts	\(\text {Expression too large to display}\)	\(2956\)
derivativedivides	\(\text {Expression too large to display}\)	\(2957\)
default	\(\text {Expression too large to display}\)	\(2957\)

[In]

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

[Out]

-1/2*arccot(a*x)^3/x^2-3/2*a^2*(1/2*Pi*arccot(a*x)^2-1/2*I*Pi*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))-1/
2*I*Pi*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+1/2*I*Pi*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+1/8*Pi*c
sgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*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*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^2+1)))-1/4*P
i*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*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))*(I*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+d
ilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))+1/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/
((I+a*x)^2/(a^2*x^2+1)-1))^3*(2*I*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2
/(a^2*x^2+1)))-1/4*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^3*(I*arccot(a*x)*ln(1+I*(I+a*x)/
(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(
1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))+1/2*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2*(I*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*
x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(
I+a*x)/(a^2*x^2+1)^(1/2)))+1/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*(2*I*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2
+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^2+1)))+1/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*(2*I*arccot(a*x
)*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^2+1)))-1/2*Pi*csgn(I/((I+a*x)^2/(a^2
*x^2+1)-1))^3*(I*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))
+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))-1/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+
1)-1))^2*(2*I*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^2+1)))-arcco
t(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))-arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))-arccot(a*x)*ln(1-I*(I+a*x)/(a
^2*x^2+1)^(1/2))-1/2*Pi*dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))-1/2*Pi*dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+1/4*P
i*polylog(2,-(I+a*x)^2/(a^2*x^2+1))+I*arccot(a*x)^2+1/3*arccot(a*x)^3-1/4*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*(
I*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1+I*(I+a
*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))+1/8*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))*(2*I*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2
*x^2+1)))+1/2*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*arccot(a*x)*ln(1+I*(I+a*
x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dil
og(1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))+1/4*Pi*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*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^
(1/2))+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))-1/4*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*(2*I*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+poly
log(2,-(I+a*x)^2/(a^2*x^2+1)))-1/4*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*arc
cot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1+I*(I+a*x)/(
a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))-1/x*arccot(a*x)^2/a+I*arccot(a*x)^2*ln((I+a*x)/(a^2*x^
2+1)^(1/2))-1/8*Pi*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*(2*
I*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^2+1)))-1/8*Pi*csgn(I/((I
+a*x)^2/(a^2*x^2+1)-1))*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^2*(2*I*arccot(a*x)*ln(1+(I+a*x
)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^2+1)))+1/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*c
sgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^2*(I*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*ar
ccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2*x^2+1)
^(1/2)))+I*dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+1/2*I*polylog(2,-(I+a*x
)^2/(a^2*x^2+1))-arccot(a*x)^2*arctan(a*x))

Fricas [F]

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

[In]

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

[Out]

integral(arccot(a*x)^3/x^3, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acot(a*x)**3/x**3, x)

Maxima [F]

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

[In]

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

[Out]

-1/32*(8*a^2*x^2*arctan2(1, a*x)^3 - 12*a*x*arctan2(1, a*x)^2 + 3*a*x*log(a^2*x^2 + 1)^2 + 4*(3*a^2*arctan(a*x
)*arctan(1/(a*x))^2 + (arctan(a*x)^3/a + 3*arctan(a*x)^2*arctan(1/(a*x))/a)*a^3 + 24*a^3*integrate(1/32*x^3*lo
g(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) - 96*a^3*integrate(1/32*x^3*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 128*a^
2*integrate(1/32*x^2*arctan(1/(a*x))^3/(a^2*x^5 + x^3), x) - 192*a^2*integrate(1/32*x^2*arctan(1/(a*x))/(a^2*x
^5 + x^3), x) + 96*a*integrate(1/32*x*arctan(1/(a*x))^2/(a^2*x^5 + x^3), x) + 24*a*integrate(1/32*x*log(a^2*x^
2 + 1)^2/(a^2*x^5 + x^3), x) - 128*integrate(1/32*arctan(1/(a*x))^3/(a^2*x^5 + x^3), x))*x^2 + 8*arctan2(1, a*
x)^3)/x^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=-\frac {1}{2} \, a \arctan \left (\frac {1}{a x}\right )^{3} - \frac {\arctan \left (\frac {1}{a x}\right )^{3}}{2 \, x^{2}} \]

[In]

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

[Out]

-1/2*a*arctan(1/(a*x))^3 - 1/2*arctan(1/(a*x))^3/x^2

Mupad [F(-1)]

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

[In]

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

[Out]

int(acot(a*x)^3/x^3, x)