\(\int x^2 \cot ^{-1}(a x^2) \, dx\) [44]

Optimal result

Integrand size = 10, antiderivative size = 117 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {2 x}{3 a}+\frac {1}{3} x^3 \cot ^{-1}\left (a x^2\right )+\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{3 \sqrt {2} a^{3/2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{3 \sqrt {2} a^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} x}{1+a x^2}\right )}{3 \sqrt {2} a^{3/2}} \] Output:

2/3*x/a+1/3*x^3*arccot(a*x^2)-1/6*arctan(-1+2^(1/2)*a^(1/2)*x)*2^(1/2)/a^( 
3/2)-1/6*arctan(1+2^(1/2)*a^(1/2)*x)*2^(1/2)/a^(3/2)-1/6*arctanh(2^(1/2)*a 
^(1/2)*x/(a*x^2+1))*2^(1/2)/a^(3/2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {8 \sqrt {a} x+4 a^{3/2} x^3 \cot ^{-1}\left (a x^2\right )+2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {a} x\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {a} x\right )+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{12 a^{3/2}} \] Input:

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

Output:

(8*Sqrt[a]*x + 4*a^(3/2)*x^3*ArcCot[a*x^2] + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]* 
Sqrt[a]*x] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] + Sqrt[2]*Log[1 - Sqr 
t[2]*Sqrt[a]*x + a*x^2] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(12* 
a^(3/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.40, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5362, 843, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 5362

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

\(\Big \downarrow \) 843

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 843

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

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]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {x^{3} \operatorname {arccot}\left (a \,x^{2}\right )}{3}+\frac {2 a \left (\frac {x}{a^{2}}-\frac {\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{2}}\right )}{3}\)	\(109\)
parts	\(\frac {x^{3} \operatorname {arccot}\left (a \,x^{2}\right )}{3}+\frac {2 a \left (\frac {x}{a^{2}}-\frac {\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{2}}\right )}{3}\)	\(109\)

Input:

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

Output:

1/3*x^3*arccot(a*x^2)+2/3*a*(x/a^2-1/8/a^2*(1/a^2)^(1/4)*2^(1/2)*(ln((x^2+ 
(1/a^2)^(1/4)*x*2^(1/2)+(1/a^2)^(1/2))/(x^2-(1/a^2)^(1/4)*x*2^(1/2)+(1/a^2 
)^(1/2)))+2*arctan(2^(1/2)/(1/a^2)^(1/4)*x+1)+2*arctan(2^(1/2)/(1/a^2)^(1/ 
4)*x-1)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {4 \, a x^{3} \operatorname {arccot}\left (a x^{2}\right ) + 8 \, x - \frac {2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}} - \frac {2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {a} x - 1\right )}{\sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}}}{12 \, a} \] Input:

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

Output:

1/12*(4*a*x^3*arccot(a*x^2) + 8*x - 2*sqrt(2)*arctan(sqrt(2)*sqrt(a)*x + 1 
)/sqrt(a) - 2*sqrt(2)*arctan(sqrt(2)*sqrt(a)*x - 1)/sqrt(a) - sqrt(2)*log( 
a*x^2 + sqrt(2)*sqrt(a)*x + 1)/sqrt(a) + sqrt(2)*log(a*x^2 - sqrt(2)*sqrt( 
a)*x + 1)/sqrt(a))/a
 

Sympy [A] (verification not implemented)

Time = 6.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.08 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\begin {cases} \frac {x^{3} \operatorname {acot}{\left (a x^{2} \right )}}{3} + \frac {\left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \operatorname {acot}{\left (a x^{2} \right )}}{3} + \frac {2 x}{3 a} + \frac {\sqrt [4]{- \frac {1}{a^{2}}} \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{3 a} - \frac {\sqrt [4]{- \frac {1}{a^{2}}} \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{6 a} - \frac {\sqrt [4]{- \frac {1}{a^{2}}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{3 a} & \text {for}\: a \neq 0 \\\frac {\pi x^{3}}{6} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((x**3*acot(a*x**2)/3 + (-1/a**2)**(3/4)*acot(a*x**2)/3 + 2*x/(3* 
a) + (-1/a**2)**(1/4)*log(x - (-1/a**2)**(1/4))/(3*a) - (-1/a**2)**(1/4)*l 
og(x**2 + sqrt(-1/a**2))/(6*a) - (-1/a**2)**(1/4)*atan(x/(-1/a**2)**(1/4)) 
/(3*a), Ne(a, 0)), (pi*x**3/6, True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.15 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{12} \, a {\left (\frac {8 \, x}{a^{2}} - \frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}}}{a^{2}}\right )} \] Input:

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

Output:

1/3*x^3*arccot(a*x^2) + 1/12*a*(8*x/a^2 - (2*sqrt(2)*arctan(1/2*sqrt(2)*(2 
*a*x + sqrt(2)*sqrt(a))/sqrt(a))/sqrt(a) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2 
*a*x - sqrt(2)*sqrt(a))/sqrt(a))/sqrt(a) + sqrt(2)*log(a*x^2 + sqrt(2)*sqr 
t(a)*x + 1)/sqrt(a) - sqrt(2)*log(a*x^2 - sqrt(2)*sqrt(a)*x + 1)/sqrt(a))/ 
a^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.31 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{3} \, x^{3} \arctan \left (\frac {1}{a x^{2}}\right ) + \frac {1}{12} \, a {\left (\frac {8 \, x}{a^{2}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}} - \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}} + \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}}\right )} \] Input:

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

Output:

1/3*x^3*arctan(1/(a*x^2)) + 1/12*a*(8*x/a^2 - 2*sqrt(2)*arctan(1/2*sqrt(2) 
*(2*x + sqrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/(a^2*sqrt(abs(a))) - 2*sqrt(2) 
*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/(a^2*sqrt(a 
bs(a))) - sqrt(2)*log(x^2 + sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/(a^2*sqrt(a 
bs(a))) + sqrt(2)*log(x^2 - sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/(a^2*sqrt(a 
bs(a))))
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.44 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {x^3\,\mathrm {acot}\left (a\,x^2\right )}{3}+\frac {2\,x}{3\,a}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i}}{3\,a^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\,1{}\mathrm {i}\right )}{3\,a^{3/2}} \] Input:

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

Output:

(x^3*acot(a*x^2))/3 + (2*x)/(3*a) + ((-1)^(1/4)*atan((-1)^(1/4)*a^(1/2)*x) 
*1i)/(3*a^(3/2)) + ((-1)^(1/4)*atan((-1)^(1/4)*a^(1/2)*x*1i))/(3*a^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.87 \[ \int x^2 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {-2 \sqrt {a}\, \sqrt {2}\, \mathit {acot} \left (a \,x^{2}\right )+4 \mathit {acot} \left (a \,x^{2}\right ) a^{2} x^{3}+4 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {a}\, \sqrt {2}-2 a x}{\sqrt {a}\, \sqrt {2}}\right )+\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {2}\, x +a \,x^{2}+1\right )-\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {2}\, x +a \,x^{2}+1\right )+8 a x}{12 a^{2}} \] Input:

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

Output:

( - 2*sqrt(a)*sqrt(2)*acot(a*x**2) + 4*acot(a*x**2)*a**2*x**3 + 4*sqrt(a)* 
sqrt(2)*atan((sqrt(a)*sqrt(2) - 2*a*x)/(sqrt(a)*sqrt(2))) + sqrt(a)*sqrt(2 
)*log( - sqrt(a)*sqrt(2)*x + a*x**2 + 1) - sqrt(a)*sqrt(2)*log(sqrt(a)*sqr 
t(2)*x + a*x**2 + 1) + 8*a*x)/(12*a**2)