\(\int \cot ^{-1}(a x^2) \, dx\) [45]

Optimal result

Integrand size = 6, antiderivative size = 97 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x \cot ^{-1}\left (a x^2\right )-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} x}{1+a x^2}\right )}{\sqrt {2} \sqrt {a}} \] Output:

x*arccot(a*x^2)+1/2*arctan(-1+2^(1/2)*a^(1/2)*x)*2^(1/2)/a^(1/2)+1/2*arcta 
n(1+2^(1/2)*a^(1/2)*x)*2^(1/2)/a^(1/2)-1/2*arctanh(2^(1/2)*a^(1/2)*x/(a*x^ 
2+1))*2^(1/2)/a^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.05 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x \cot ^{-1}\left (a x^2\right )+\frac {-2 \arctan \left (1-\sqrt {2} \sqrt {a} x\right )+2 \arctan \left (1+\sqrt {2} \sqrt {a} x\right )+\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )-\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {5346, 826, 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.

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

\(\Big \downarrow \) 5346

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

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 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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 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 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])
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00

method	result	size
default	\(x \,\operatorname {arccot}\left (a \,x^{2}\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\)	\(97\)
parts	\(x \,\operatorname {arccot}\left (a \,x^{2}\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\)	\(97\)

Input:

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

Output:

x*arccot(a*x^2)+1/4/a/(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 = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x \operatorname {arccot}\left (a x^{2}\right ) + \frac {\sqrt {2} \arctan \left (\sqrt {2} \sqrt {a} x + 1\right )}{2 \, \sqrt {a}} + \frac {\sqrt {2} \arctan \left (\sqrt {2} \sqrt {a} x - 1\right )}{2 \, \sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{4 \, \sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{4 \, \sqrt {a}} \] Input:

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

Output:

x*arccot(a*x^2) + 1/2*sqrt(2)*arctan(sqrt(2)*sqrt(a)*x + 1)/sqrt(a) + 1/2* 
sqrt(2)*arctan(sqrt(2)*sqrt(a)*x - 1)/sqrt(a) - 1/4*sqrt(2)*log(a*x^2 + sq 
rt(2)*sqrt(a)*x + 1)/sqrt(a) + 1/4*sqrt(2)*log(a*x^2 - sqrt(2)*sqrt(a)*x + 
 1)/sqrt(a)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.69 (sec) , antiderivative size = 615, normalized size of antiderivative = 6.34 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((pi*x/2, Eq(a, 0)), (oo*I*x, Eq(a, -I/x**2)), (-oo*I*x, Eq(a, I/ 
x**2)), (2*a**5*x**5*(-1/a**2)**(7/4)*acot(a*x**2)/(2*a**5*x**4*(-1/a**2)* 
*(7/4) + 2*a**3*(-1/a**2)**(7/4)) + 2*a**4*x**4*(-1/a**2)**(3/2)*log(x - ( 
-1/a**2)**(1/4))/(2*a**5*x**4*(-1/a**2)**(7/4) + 2*a**3*(-1/a**2)**(7/4)) 
- a**4*x**4*(-1/a**2)**(3/2)*log(x**2 + sqrt(-1/a**2))/(2*a**5*x**4*(-1/a* 
*2)**(7/4) + 2*a**3*(-1/a**2)**(7/4)) + 2*a**4*x**4*(-1/a**2)**(3/2)*atan( 
x/(-1/a**2)**(1/4))/(2*a**5*x**4*(-1/a**2)**(7/4) + 2*a**3*(-1/a**2)**(7/4 
)) + 2*a**3*x*(-1/a**2)**(7/4)*acot(a*x**2)/(2*a**5*x**4*(-1/a**2)**(7/4) 
+ 2*a**3*(-1/a**2)**(7/4)) + 2*a**2*(-1/a**2)**(3/2)*log(x - (-1/a**2)**(1 
/4))/(2*a**5*x**4*(-1/a**2)**(7/4) + 2*a**3*(-1/a**2)**(7/4)) - a**2*(-1/a 
**2)**(3/2)*log(x**2 + sqrt(-1/a**2))/(2*a**5*x**4*(-1/a**2)**(7/4) + 2*a* 
*3*(-1/a**2)**(7/4)) + 2*a**2*(-1/a**2)**(3/2)*atan(x/(-1/a**2)**(1/4))/(2 
*a**5*x**4*(-1/a**2)**(7/4) + 2*a**3*(-1/a**2)**(7/4)) + 2*a*x**4*acot(a*x 
**2)/(2*a**5*x**4*(-1/a**2)**(7/4) + 2*a**3*(-1/a**2)**(7/4)) + 2*acot(a*x 
**2)/(2*a**6*x**4*(-1/a**2)**(7/4) + 2*a**4*(-1/a**2)**(7/4)), True))
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.24 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )} + x \operatorname {arccot}\left (a x^{2}\right ) \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.48 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} - \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}} + \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}}\right )} + x \arctan \left (\frac {1}{a x^{2}}\right ) \] Input:

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

Output:

1/4*a*(2*sqrt(2)*sqrt(abs(a))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(a 
)))*sqrt(abs(a)))/a^2 + 2*sqrt(2)*sqrt(abs(a))*arctan(1/2*sqrt(2)*(2*x - s 
qrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/a^2 - sqrt(2)*sqrt(abs(a))*log(x^2 + sq 
rt(2)*x/sqrt(abs(a)) + 1/abs(a))/a^2 + sqrt(2)*sqrt(abs(a))*log(x^2 - sqrt 
(2)*x/sqrt(abs(a)) + 1/abs(a))/a^2) + x*arctan(1/(a*x^2))
 

Mupad [B] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.43 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x\,\mathrm {acot}\left (a\,x^2\right )+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=\frac {2 \sqrt {a}\, \sqrt {2}\, \mathit {acot} \left (a \,x^{2}\right )+4 \mathit {acot} \left (a \,x^{2}\right ) a x -4 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {a}\, \sqrt {2}-2 a x}{\sqrt {a}\, \sqrt {2}}\right )+\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {2}\, x +a \,x^{2}+1\right )-\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {2}\, x +a \,x^{2}+1\right )}{4 a} \] Input:

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

Output:

(2*sqrt(a)*sqrt(2)*acot(a*x**2) + 4*acot(a*x**2)*a*x - 4*sqrt(a)*sqrt(2)*a 
tan((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)*sqrt(2)*x + 
a*x**2 + 1))/(4*a)