\(\int \cot ^{-1}(a x)^3 \, dx\) [28]

Optimal result

Integrand size = 6, antiderivative size = 96 \[ \int \cot ^{-1}(a x)^3 \, dx=\frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \cot ^{-1}(a x)^3 \, dx=-\frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )}{a}-\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )}{a}-\frac {3 \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{2 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5346, 5456, 5380, 5530, 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.

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

\(\Big \downarrow \) 5346

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

\(\Big \downarrow \) 5456

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

\(\Big \downarrow \) 5380

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

\(\Big \downarrow \) 5530

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 5346

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p   Int[x^n*((a + b*ArcCot[c*x^n])^(p 
 - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && 
 (EqQ[n, 1] || EqQ[p, 1])
 

rule 5380

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[b*c*( 
p/e)   Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[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 5456

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

rule 5530

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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (89 ) = 178\).

Time = 0.82 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.95

method	result	size
derivativedivides	\(\frac {\operatorname {arccot}\left (a x \right )^{3} \left (a x -i\right )+2 i \operatorname {arccot}\left (a x \right )^{3}-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}\)	\(187\)
default	\(\frac {\operatorname {arccot}\left (a x \right )^{3} \left (a x -i\right )+2 i \operatorname {arccot}\left (a x \right )^{3}-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}\)	\(187\)

Input:

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

Output:

1/a*(arccot(a*x)^3*(a*x-I)+2*I*arccot(a*x)^3-3*arccot(a*x)^2*ln(1+(I+a*x)/ 
(a^2*x^2+1)^(1/2))+6*I*arccot(a*x)*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-6 
*polylog(3,-(I+a*x)/(a^2*x^2+1)^(1/2))-3*arccot(a*x)^2*ln(1-(I+a*x)/(a^2*x 
^2+1)^(1/2))+6*I*arccot(a*x)*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-6*polylo 
g(3,(I+a*x)/(a^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/8*x*arctan2(1, a*x)^3 - 3/32*x*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + 21/1 
6*arctan(a*x)^2*arctan(1/(a*x))^2/a + 7/8*arctan(a*x)*arctan(1/(a*x))^3/a 
+ 28*a^2*integrate(1/32*x^2*arctan(1/(a*x))^3/(a^2*x^2 + 1), x) + 3*a^2*in 
tegrate(1/32*x^2*arctan(1/(a*x))*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 12 
*a^2*integrate(1/32*x^2*arctan(1/(a*x))*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) 
 + 12*a*integrate(1/32*x*arctan(1/(a*x))^2/(a^2*x^2 + 1), x) - 3*a*integra 
te(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 7/32*(a*arctan(a*x)^4 + 4 
*a*arctan(a*x)^3*arctan(1/(a*x)))/a^2 + 3*integrate(1/32*arctan(1/(a*x))*l 
og(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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