\(\int x^2 \cot ^{-1}(a x)^3 \, dx\) [26]

Optimal result

Integrand size = 10, antiderivative size = 157 \[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\frac {x \cot ^{-1}(a x)}{a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \cot ^{-1}(a x)^2}{2 a}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {\log \left (1+a^2 x^2\right )}{2 a^3}-\frac {i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\frac {-i \pi ^3+24 a x \cot ^{-1}(a x)+12 \cot ^{-1}(a x)^2+12 a^2 x^2 \cot ^{-1}(a x)^2+8 i \cot ^{-1}(a x)^3+8 a^3 x^3 \cot ^{-1}(a x)^3+24 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-24 \log \left (\frac {1}{a \sqrt {1+\frac {1}{a^2 x^2}} x}\right )+24 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{24 a^3} \] Input:

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

Output:

((-I)*Pi^3 + 24*a*x*ArcCot[a*x] + 12*ArcCot[a*x]^2 + 12*a^2*x^2*ArcCot[a*x 
]^2 + (8*I)*ArcCot[a*x]^3 + 8*a^3*x^3*ArcCot[a*x]^3 + 24*ArcCot[a*x]^2*Log 
[1 - E^((-2*I)*ArcCot[a*x])] - 24*Log[1/(a*Sqrt[1 + 1/(a^2*x^2)]*x)] + (24 
*I)*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a*x])] + 12*PolyLog[3, E^((-2* 
I)*ArcCot[a*x])])/(24*a^3)
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5362, 5452, 5362, 5452, 5346, 240, 5420, 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.

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

\(\Big \downarrow \) 5362

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

\(\Big \downarrow \) 5452

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

\(\Big \downarrow \) 5362

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

\(\Big \downarrow \) 5452

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

\(\Big \downarrow \) 5346

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 5420

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

\(\Big \downarrow \) 5456

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

\(\Big \downarrow \) 5380

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

\(\Big \downarrow \) 5530

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

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 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 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 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 5452

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[f^2/e   Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x] 
)^p, x], x] - Simp[d*(f^2/e)   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] && GtQ[m, 1]
 

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 [C] (warning: unable to verify)

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

Time = 5.87 (sec) , antiderivative size = 1108, normalized size of antiderivative = 7.06

method	result	size
parts	\(\text {Expression too large to display}\)	\(1108\)
derivativedivides	\(\text {Expression too large to display}\)	\(1110\)
default	\(\text {Expression too large to display}\)	\(1110\)

Input:

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

Output:

1/3*x^3*arccot(a*x)^3+1/a^3*(-1/4*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)*arccot(a*x)^2+1/2*arccot(a*x)*(a*x+I+(a^2*x^2+1)^(1/2))+arc 
cot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))+ln(2)*arccot(a*x)^2+1/2*a^2*x^2*a 
rccot(a*x)^2-ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))-1/2*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*arccot(a*x)^2+1/4*I*Pi*csgn(I*((I+ 
a*x)^2/(a^2*x^2+1)-1)^2)^3*arccot(a*x)^2-1/4*I*Pi*csgn(I*(I+a*x)^2/(a^2*x^ 
2+1))^3*arccot(a*x)^2+1/4*I*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^ 
2*x^2+1)-1)^2)^3*arccot(a*x)^2-1/3*I*arccot(a*x)^3-1/2*arccot(a*x)^2*ln(a^ 
2*x^2+1)+2*polylog(3,(I+a*x)/(a^2*x^2+1)^(1/2))+2*polylog(3,-(I+a*x)/(a^2* 
x^2+1)^(1/2))-ln((I+a*x)/(a^2*x^2+1)^(1/2)-1)-1/4*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))*arccot(a*x)^2+1/2*arccot(a* 
x)*(a*x+I-(a^2*x^2+1)^(1/2))+1/2*I*Pi*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))*cs 
gn(I*(I+a*x)^2/(a^2*x^2+1))^2*arccot(a*x)^2+1/4*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 
*arccot(a*x)^2-1/2*I*Pi*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*((I+a*x)^ 
2/(a^2*x^2+1)-1)^2)^2*arccot(a*x)^2+1/4*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)-1)^2)*arccot(a*x)^2+1/4*I*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*arccot(a*x)^2+1/2*arccot(a*x)^2-arccot(a*x)^2*ln((I+a*x)^2/(a^2...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/24*x^3*arctan2(1, a*x)^3 - 1/32*x^3*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + 
 integrate(1/32*(28*a^2*x^4*arctan2(1, a*x)^3 + 4*a^2*x^4*arctan2(1, a*x)* 
log(a^2*x^2 + 1) + 4*a*x^3*arctan2(1, a*x)^2 + 28*x^2*arctan2(1, a*x)^3 + 
(3*a^2*x^4*arctan2(1, a*x) - a*x^3 + 3*x^2*arctan2(1, a*x))*log(a^2*x^2 + 
1)^2)/(a^2*x^2 + 1), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*acot(a*x)**3*a**3*x**3 + 2*acot(a*x)**3*a*x + 3*acot(a*x)**2*a**2*x**2 
+ 3*acot(a*x)**2 + 6*acot(a*x)*a*x - 2*int(acot(a*x)**3,x)*a + 3*log(a**2* 
x**2 + 1))/(6*a**3)