3.1.33 \(\int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx\) [33]

3.1.33.1 Optimal result

Integrand size = 10, antiderivative size = 152 \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\frac {a^3}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \arctan (a x)-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]

output

1/4*a^3/x-1/4*a^2*arccot(a*x)/x^2-I*a^4*arccot(a*x)^2+1/4*a*arccot(a*x)^2/ 
x^3-3/4*a^3*arccot(a*x)^2/x+1/4*a^4*arccot(a*x)^3-1/4*arccot(a*x)^3/x^4+1/ 
4*a^4*arctan(a*x)-2*a^4*arccot(a*x)*ln(2-2/(1-I*a*x))-I*a^4*polylog(2,-1+2 
/(1-I*a*x))
 

3.1.33.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\frac {a^3 x^3+\left (a x-3 a^3 x^3+4 i a^4 x^4\right ) \cot ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^3-a^2 x^2 \cot ^{-1}(a x) \left (1+a^2 x^2+8 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )+4 i a^4 x^4 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )}{4 x^4} \]

input

Integrate[ArcCot[a*x]^3/x^5,x]
 

output

(a^3*x^3 + (a*x - 3*a^3*x^3 + (4*I)*a^4*x^4)*ArcCot[a*x]^2 + (-1 + a^4*x^4 
)*ArcCot[a*x]^3 - a^2*x^2*ArcCot[a*x]*(1 + a^2*x^2 + 8*a^2*x^2*Log[1 + E^( 
(2*I)*ArcCot[a*x])]) + (4*I)*a^4*x^4*PolyLog[2, -E^((2*I)*ArcCot[a*x])])/( 
4*x^4)
 

3.1.33.3 Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.49, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5362, 5454, 5362, 5454, 5362, 264, 216, 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.

\(\displaystyle \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx\)

\(\Big \downarrow \) 5362

\(\displaystyle -\frac {3}{4} a \int \frac {\cot ^{-1}(a x)^2}{x^4 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 5454

\(\displaystyle -\frac {3}{4} a \left (\int \frac {\cot ^{-1}(a x)^2}{x^4}dx-a^2 \int \frac {\cot ^{-1}(a x)^2}{x^2 \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 5362

\(\displaystyle -\frac {3}{4} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{x^2 \left (a^2 x^2+1\right )}dx\right )-\frac {2}{3} a \int \frac {\cot ^{-1}(a x)}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 5454

\(\displaystyle -\frac {3}{4} a \left (-\left (a^2 \left (\int \frac {\cot ^{-1}(a x)^2}{x^2}dx-a^2 \int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx\right )\right )-\frac {2}{3} a \left (\int \frac {\cot ^{-1}(a x)}{x^3}dx-a^2 \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 5362

\(\displaystyle -\frac {3}{4} a \left (-\left (a^2 \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx\right )-2 a \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{x}\right )\right )-\frac {2}{3} a \left (-\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {3}{4} a \left (-\left (a^2 \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx\right )-2 a \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{x}\right )\right )-\frac {2}{3} a \left (-\frac {1}{2} a \left (a^2 \left (-\int \frac {1}{a^2 x^2+1}dx\right )-\frac {1}{x}\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {3}{4} a \left (-\frac {2}{3} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\left (a^2 \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx\right )-2 a \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{x}\right )\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 5420

\(\displaystyle -\frac {3}{4} a \left (-\frac {2}{3} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\left (a^2 \left (-2 a \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx+\frac {1}{3} a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^2}{x}\right )\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^3}{4 x^4}\)

\(\Big \downarrow \) 5460

\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{4 x^4}-\frac {3}{4} a \left (-\frac {2}{3} a \left (-\left (a^2 \left (i \int \frac {\cot ^{-1}(a x)}{x (a x+i)}dx+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\left (a^2 \left (-2 a \left (i \int \frac {\cot ^{-1}(a x)}{x (a x+i)}dx+\frac {1}{2} i \cot ^{-1}(a x)^2\right )+\frac {1}{3} a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^2}{x}\right )\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )\)

\(\Big \downarrow \) 5404

\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{4 x^4}-\frac {3}{4} a \left (-\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\left (a^2 \left (-2 a \left (i \left (-i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )+\frac {1}{3} a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^2}{x}\right )\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\right )\)

\(\Big \downarrow \) 2897

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

input

Int[ArcCot[a*x]^3/x^5,x]
 

output

-1/4*ArcCot[a*x]^3/x^4 - (3*a*(-1/3*ArcCot[a*x]^2/x^3 - a^2*(-(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)] + PolyLog[2, -1 + 2/(1 - I*a*x)]/2))) - (2*a*(-1 
/2*ArcCot[a*x]/x^2 - (a*(-x^(-1) - a*ArcTan[a*x]))/2 - a^2*((I/2)*ArcCot[a 
*x]^2 + I*((-I)*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] + PolyLog[2, -1 + 2/(1 
- I*a*x)]/2))))/3))/4
 

3.1.33.3.1 Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

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]
 

3.1.33.4 Maple [C] (warning: unable to verify)

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

Time = 10.74 (sec) , antiderivative size = 852, normalized size of antiderivative = 5.61

method	result	size
parts	\(\text {Expression too large to display}\)	\(852\)
derivativedivides	\(\text {Expression too large to display}\)	\(855\)
default	\(\text {Expression too large to display}\)	\(855\)

input

int(arccot(a*x)^3/x^5,x,method=_RETURNVERBOSE)
 

output

-1/4*arccot(a*x)^3/x^4-3/4*a^4*(-1/3/a^3/x^3*arccot(a*x)^2+1/x*arccot(a*x) 
^2/a+arccot(a*x)^2*arctan(a*x)-1/3*arccot(a*x)^3-1/2*Pi*arccot(a*x)^2+8/3* 
arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+8/3*arccot(a*x)*ln(1-I*(I+a* 
x)/(a^2*x^2+1)^(1/2))-1/3*I/a/x*(a*x-I)-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))*arccot(a*x)^2+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*arccot(a*x)^2+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*arccot(a*x)^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*arccot(a*x)^2-1/4*Pi*c 
sgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^3*arccot(a*x)^2-I*a 
rccot(a*x)^2*ln((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))^3*arccot(a*x)^2-1/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I* 
(I+a*x)^2/(a^2*x^2+1))*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1) 
-1))*arccot(a*x)^2+1/2*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2*arccot(a*x)^ 
2-1/4*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*arccot(a*x)^2-8/3*I*dilog(1+I*(I+ 
a*x)/(a^2*x^2+1)^(1/2))-2/3*arccot(a*x)*(I+a*x)/a/x+1/3*arccot(a*x)*(I+a*x 
)^2/a^2/x^2+2/3*arccot(a*x)*(I+a*x)*(a*x-I)/a^2/x^2-8/3*I*dilog(1-I*(I+a*x 
)/(a^2*x^2+1)^(1/2))-4/3*I*arccot(a*x)^2)
 

3.1.33.5 Fricas [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{5}} \,d x } \]

input

integrate(arccot(a*x)^3/x^5,x, algorithm="fricas")
 

output

integral(arccot(a*x)^3/x^5, x)
 

3.1.33.6 Sympy [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{5}}\, dx \]

input

integrate(acot(a*x)**3/x**5,x)
 

output

Integral(acot(a*x)**3/x**5, x)
 

3.1.33.7 Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\text {Timed out} \]

input

integrate(arccot(a*x)^3/x^5,x, algorithm="maxima")
 

output

Timed out
 

3.1.33.8 Giac [F]

\[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{5}} \,d x } \]

input

integrate(arccot(a*x)^3/x^5,x, algorithm="giac")
 

output

integrate(arccot(a*x)^3/x^5, x)
 

3.1.33.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^5} \,d x \]

input

int(acot(a*x)^3/x^5,x)
 

output

int(acot(a*x)^3/x^5, x)