3.1.29 \(\int \frac {\cot ^{-1}(a x)^3}{x} \, dx\) [29]

3.1.29.1 Optimal result

Integrand size = 10, antiderivative size = 178 \[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} i \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )+\frac {3}{2} \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{i+a x}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 i}{i+a x}\right )-\frac {3}{4} i \operatorname {PolyLog}\left (4,1-\frac {2 a x}{i+a x}\right ) \]

output

2*arccot(a*x)^3*arccoth(1-2/(1+I*a*x))-3/2*I*arccot(a*x)^2*polylog(2,1-2*I 
/(I+a*x))+3/2*I*arccot(a*x)^2*polylog(2,1-2*a*x/(I+a*x))-3/2*arccot(a*x)*p 
olylog(3,1-2*I/(I+a*x))+3/2*arccot(a*x)*polylog(3,1-2*a*x/(I+a*x))+3/4*I*p 
olylog(4,1-2*I/(I+a*x))-3/4*I*polylog(4,1-2*a*x/(I+a*x))
 

3.1.29.2 Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^{-1}(a x)^3}{x} \, dx=\frac {1}{64} i \left (\pi ^4-32 \cot ^{-1}(a x)^4+64 i \cot ^{-1}(a x)^3 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-64 i \cot ^{-1}(a x)^3 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )-96 \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-96 \cot ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+96 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )-96 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \cot ^{-1}(a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \cot ^{-1}(a x)}\right )\right ) \]

input

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

output

(I/64)*(Pi^4 - 32*ArcCot[a*x]^4 + (64*I)*ArcCot[a*x]^3*Log[1 - E^((-2*I)*A 
rcCot[a*x])] - (64*I)*ArcCot[a*x]^3*Log[1 + E^((2*I)*ArcCot[a*x])] - 96*Ar 
cCot[a*x]^2*PolyLog[2, E^((-2*I)*ArcCot[a*x])] - 96*ArcCot[a*x]^2*PolyLog[ 
2, -E^((2*I)*ArcCot[a*x])] + (96*I)*ArcCot[a*x]*PolyLog[3, E^((-2*I)*ArcCo 
t[a*x])] - (96*I)*ArcCot[a*x]*PolyLog[3, -E^((2*I)*ArcCot[a*x])] + 48*Poly 
Log[4, E^((-2*I)*ArcCot[a*x])] + 48*PolyLog[4, -E^((2*I)*ArcCot[a*x])])
 

3.1.29.3 Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5358, 5524, 5528, 5532, 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.

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

\(\Big \downarrow \) 5358

\(\displaystyle 6 a \int \frac {\cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\)

\(\Big \downarrow \) 5524

\(\displaystyle 6 a \left (\frac {1}{2} \int \frac {\cot ^{-1}(a x)^2 \log \left (\frac {2 a x}{a x+i}\right )}{a^2 x^2+1}dx-\frac {1}{2} \int \frac {\cot ^{-1}(a x)^2 \log \left (\frac {2 i}{a x+i}\right )}{a^2 x^2+1}dx\right )+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\)

\(\Big \downarrow \) 5528

\(\displaystyle 6 a \left (\frac {1}{2} \left (-i \int \frac {\cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right )}{a^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)^2}{2 a}\right )+\frac {1}{2} \left (i \int \frac {\cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right )}{a^2 x^2+1}dx+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)^2}{2 a}\right )\right )+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\)

\(\Big \downarrow \) 5532

\(\displaystyle 6 a \left (\frac {1}{2} \left (-i \left (-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right )}{a^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)^2}{2 a}\right )+\frac {1}{2} \left (i \left (-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right )}{a^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)^2}{2 a}\right )\right )+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle 6 a \left (\frac {1}{2} \left (-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)^2}{2 a}-i \left (-\frac {\operatorname {PolyLog}\left (4,1-\frac {2 i}{a x+i}\right )}{4 a}-\frac {i \operatorname {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )\right )+\frac {1}{2} \left (\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)^2}{2 a}+i \left (-\frac {\operatorname {PolyLog}\left (4,1-\frac {2 a x}{a x+i}\right )}{4 a}-\frac {i \operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )\right )\right )+2 \cot ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\)

input

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

output

2*ArcCot[a*x]^3*ArcCoth[1 - 2/(1 + I*a*x)] + 6*a*((((-1/2*I)*ArcCot[a*x]^2 
*PolyLog[2, 1 - (2*I)/(I + a*x)])/a - I*(((-1/2*I)*ArcCot[a*x]*PolyLog[3, 
1 - (2*I)/(I + a*x)])/a - PolyLog[4, 1 - (2*I)/(I + a*x)]/(4*a)))/2 + (((I 
/2)*ArcCot[a*x]^2*PolyLog[2, 1 - (2*a*x)/(I + a*x)])/a + I*(((-1/2*I)*ArcC 
ot[a*x]*PolyLog[3, 1 - (2*a*x)/(I + a*x)])/a - PolyLog[4, 1 - (2*a*x)/(I + 
 a*x)]/(4*a)))/2)
 

3.1.29.3.1 Defintions of rubi rules used

rule 5358

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + 
b*ArcCot[c*x])^p*ArcCoth[1 - 2/(1 + I*c*x)], x] + Simp[2*b*c*p   Int[(a + b 
*ArcCot[c*x])^(p - 1)*(ArcCoth[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2)), x], x] /; 
 FreeQ[{a, b, c}, x] && IGtQ[p, 1]
 

rule 5524

Int[(ArcCoth[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x 
_)^2), x_Symbol] :> Simp[1/2   Int[Log[SimplifyIntegrand[1 + 1/u, x]]*((a + 
 b*ArcCot[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2   Int[Log[SimplifyIntegra 
nd[1 - 1/u, x]]*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, 
 c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c* 
x)))^2, 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 5532

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_. 
)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*ArcCot[c*x])^p*(PolyLog[k + 1, u]/ 
(2*c*d)), x] - Simp[b*p*(I/2)   Int[(a + b*ArcCot[c*x])^(p - 1)*(PolyLog[k 
+ 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] 
&& EqQ[e, c^2*d] && EqQ[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]
 

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

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

Time = 6.61 (sec) , antiderivative size = 982, normalized size of antiderivative = 5.52

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(982\)
default	\(\text {Expression too large to display}\)	\(982\)
parts	\(\text {Expression too large to display}\)	\(1417\)

input

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

output

ln(a*x)*arccot(a*x)^3+1/2*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)))-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-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+csgn(I/((I+a*x)^2/ 
(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^3-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+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)))-csgn(1/((I 
+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^3+csgn(1/((I+a*x)^2/(a^2 
*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^2-1)*arccot(a*x)^3+arccot(a*x)^3*ln( 
(I+a*x)^2/(a^2*x^2+1)-1)-arccot(a*x)^3*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+3*I 
*arccot(a*x)^2*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-6*arccot(a*x)*polylog( 
3,(I+a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog(4,(I+a*x)/(a^2*x^2+1)^(1/2))-arcc 
ot(a*x)^3*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+3*I*arccot(a*x)^2*polylog(2,-(I+ 
a*x)/(a^2*x^2+1)^(1/2))-6*arccot(a*x)*polylog(3,-(I+a*x)/(a^2*x^2+1)^(1/2) 
)-6*I*polylog(4,-(I+a*x)/(a^2*x^2+1)^(1/2))-3/2*I*arccot(a*x)^2*polylog(2, 
-(I+a*x)^2/(a^2*x^2+1))+3/2*arccot(a*x)*polylog(3,-(I+a*x)^2/(a^2*x^2+1))+ 
3/4*I*polylog(4,-(I+a*x)^2/(a^2*x^2+1))
 

3.1.29.5 Fricas [F]

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

input

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

output

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

3.1.29.6 Sympy [F]

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

input

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

output

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

3.1.29.7 Maxima [F]

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

input

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

output

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

3.1.29.8 Giac [F]

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

input

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

output

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

3.1.29.9 Mupad [F(-1)]

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

input

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

output

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