\(\int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 93 \[ \int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=-i a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right ) \] Output:

-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))
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\frac {(-1+i a x) \cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+3 i a \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right ) \] Input:

Integrate[ArcCot[a*x]^3/x^2,x]
 

Output:

((-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]*PolyLog[2, -E^((2*I)*ArcCot[a*x])] - (3*a*Po 
lyLog[3, -E^((2*I)*ArcCot[a*x])])/2
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5362, 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.

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

\(\Big \downarrow \) 5362

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

\(\Big \downarrow \) 5460

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

\(\Big \downarrow \) 5404

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

\(\Big \downarrow \) 5528

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

\(\Big \downarrow \) 7164

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

Input:

Int[ArcCot[a*x]^3/x^2,x]
 

Output:

-(ArcCot[a*x]^3/x) - 3*a*((I/3)*ArcCot[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))))
 

Defintions of rubi rules used

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 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]
 

Maple [C] (warning: unable to verify)

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

Time = 7.70 (sec) , antiderivative size = 1441, normalized size of antiderivative = 15.49

method	result	size
parts	\(\text {Expression too large to display}\)	\(1441\)
derivativedivides	\(\text {Expression too large to display}\)	\(1444\)
default	\(\text {Expression too large to display}\)	\(1444\)

Input:

int(arccot(a*x)^3/x^2,x,method=_RETURNVERBOSE)
 

Output:

-arccot(a*x)^3/x-3*a*(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))-1/3*I*arccot(a*x)^3+1/4*(I*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)+ 
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)))-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)*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2) 
^2+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(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*(I+a*x)^2/(a^2*x^2+1))^3+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))*csgn(I*(I+a*x)^2/(a^2 
*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^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+2*I*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*c 
sgn(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*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(1/((I+a*x)^2/( 
a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^3+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)))^2-2*I*Pi*csgn(I/((I+a*x)^2/(a^2*x^2...
 

Fricas [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{2}} \,d x } \] Input:

integrate(arccot(a*x)^3/x^2,x, algorithm="fricas")
 

Output:

integral(arccot(a*x)^3/x^2, x)
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(acot(a*x)**3/x**2,x)
 

Output:

Integral(acot(a*x)**3/x**2, x)
 

Maxima [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{2}} \,d x } \] Input:

integrate(arccot(a*x)^3/x^2,x, algorithm="maxima")
 

Output:

-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*int 
egrate(1/32*x^2*arctan(1/(a*x))*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 3 
84*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]

\[ \int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{2}} \,d x } \] Input:

integrate(arccot(a*x)^3/x^2,x, algorithm="giac")
 

Output:

integrate(arccot(a*x)^3/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)^3}{x^2} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^2} \,d x \] Input:

int(acot(a*x)^3/x^2,x)
 

Output:

int(acot(a*x)^3/x^2, x)
 

Reduce [F]

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

int(acot(a*x)^3/x^2,x)
 

Output:

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