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

Optimal result

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

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*ar 
ccot(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))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=-\frac {\cot ^{-1}(a x) \left (3 i a x (i+a x) \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2-6 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )}{2 x^2}-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right ) \] Input:

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

Output:

-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*Po 
lyLog[2, -E^((2*I)*ArcCot[a*x])]
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5362, 5454, 5362, 5420, 5460, 5404, 2897}

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.

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

\(\Big \downarrow \) 5362

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

\(\Big \downarrow \) 5454

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

\(\Big \downarrow \) 5362

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

\(\Big \downarrow \) 5420

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

\(\Big \downarrow \) 5460

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

\(\Big \downarrow \) 5404

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2897

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[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 5.70 (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	\(\text {Expression too large to display}\)	\(2957\)

Input:

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

Output:

-1/2*arccot(a*x)^3/x^2-3/2*a^2*(-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)))+1/2*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))+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+polylog(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*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))+di 
log(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)))-a 
rccot(a*x)^2/x/a+1/4*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*(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*c 
sgn(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*arcc 
ot(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))*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*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-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*log( 
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*integra 
te(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)

Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=-\frac {1}{2} \, a \arctan \left (\frac {1}{a x}\right )^{3} - \frac {\arctan \left (\frac {1}{a x}\right )^{3}}{2 \, x^{2}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\frac {-\mathit {acot} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acot} \left (a x \right )^{3}+3 \mathit {acot} \left (a x \right )^{2} a x -3 \mathit {acot} \left (a x \right ) a^{2} x^{2}-3 \mathit {acot} \left (a x \right )-6 \left (\int \frac {\mathit {acot} \left (a x \right )}{a^{2} x^{5}+x^{3}}d x \right ) x^{2}+3 a x}{2 x^{2}} \] Input:

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

Output:

( - acot(a*x)**3*a**2*x**2 - acot(a*x)**3 + 3*acot(a*x)**2*a*x - 3*acot(a* 
x)*a**2*x**2 - 3*acot(a*x) - 6*int(acot(a*x)/(a**2*x**5 + x**3),x)*x**2 + 
3*a*x)/(2*x**2)