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

Optimal. Leaf size=132 \[ \frac{\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2} \sqrt{a}}-\frac{\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2} \sqrt{a}}+x \cot ^{-1}\left (a x^2\right )-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{\sqrt{2} \sqrt{a}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{\sqrt{2} \sqrt{a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0760943, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.167, Rules used = {5028, 297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2} \sqrt{a}}-\frac{\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2} \sqrt{a}}+x \cot ^{-1}\left (a x^2\right )-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{\sqrt{2} \sqrt{a}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{\sqrt{2} \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5028

Int[ArcCot[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcCot[c*x^n], x] + Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x],
x] /; FreeQ[{c, n}, x]

Rule 297

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

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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]

Rubi steps

\begin{align*} \int \cot ^{-1}\left (a x^2\right ) \, dx &=x \cot ^{-1}\left (a x^2\right )+(2 a) \int \frac{x^2}{1+a^2 x^4} \, dx\\ &=x \cot ^{-1}\left (a x^2\right )-\int \frac{1-a x^2}{1+a^2 x^4} \, dx+\int \frac{1+a x^2}{1+a^2 x^4} \, dx\\ &=x \cot ^{-1}\left (a x^2\right )+\frac{\int \frac{1}{\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx}{2 a}+\frac{\int \frac{1}{\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx}{2 a}+\frac{\int \frac{\frac{\sqrt{2}}{\sqrt{a}}+2 x}{-\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{2 \sqrt{2} \sqrt{a}}+\frac{\int \frac{\frac{\sqrt{2}}{\sqrt{a}}-2 x}{-\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{2 \sqrt{2} \sqrt{a}}\\ &=x \cot ^{-1}\left (a x^2\right )+\frac{\log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{2 \sqrt{2} \sqrt{a}}-\frac{\log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{2 \sqrt{2} \sqrt{a}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{a} x\right )}{\sqrt{2} \sqrt{a}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{a} x\right )}{\sqrt{2} \sqrt{a}}\\ &=x \cot ^{-1}\left (a x^2\right )-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{\sqrt{2} \sqrt{a}}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{a} x\right )}{\sqrt{2} \sqrt{a}}+\frac{\log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{2 \sqrt{2} \sqrt{a}}-\frac{\log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{2 \sqrt{2} \sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0337439, size = 102, normalized size = 0.77 \[ x \cot ^{-1}\left (a x^2\right )+\frac{\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )-\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2} \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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])

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 118, normalized size = 0.9 \begin{align*} x{\rm arccot} \left (a{x}^{2}\right )+{\frac{\sqrt{2}}{4\,a}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{a}^{-2}}x\sqrt{2}+\sqrt{{a}^{-2}} \right ) \left ({x}^{2}+\sqrt [4]{{a}^{-2}}x\sqrt{2}+\sqrt{{a}^{-2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}}+{\frac{\sqrt{2}}{2\,a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}}+{\frac{\sqrt{2}}{2\,a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)))+1/2/a/(1/a^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x+1)+1/2/a/(1/a^2)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x-1)

________________________________________________________________________________________

Maxima [B]  time = 1.51113, size = 339, normalized size = 2.57 \begin{align*} -\frac{1}{4} \, a{\left (\frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (a^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (a^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{a^{2}} x - \sqrt{2} \sqrt{-\sqrt{a^{2}}} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{a^{2}} x + \sqrt{2} \sqrt{-\sqrt{a^{2}}} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{a^{2}} \sqrt{-\sqrt{a^{2}}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{a^{2}} x - \sqrt{2} \sqrt{-\sqrt{a^{2}}} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{a^{2}} x + \sqrt{2} \sqrt{-\sqrt{a^{2}}} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{a^{2}} \sqrt{-\sqrt{a^{2}}}}\right )} + x \operatorname{arccot}\left (a x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.24999, size = 647, normalized size = 4.9 \begin{align*} x \arctan \left (\frac{1}{a x^{2}}\right ) - \sqrt{2} \frac{1}{a^{2}}^{\frac{1}{4}} \arctan \left (-\sqrt{2} a \frac{1}{a^{2}}^{\frac{1}{4}} x + \sqrt{2} \sqrt{\sqrt{2} a \frac{1}{a^{2}}^{\frac{3}{4}} x + x^{2} + \sqrt{\frac{1}{a^{2}}}} a \frac{1}{a^{2}}^{\frac{1}{4}} - 1\right ) - \sqrt{2} \frac{1}{a^{2}}^{\frac{1}{4}} \arctan \left (-\sqrt{2} a \frac{1}{a^{2}}^{\frac{1}{4}} x + \sqrt{2} \sqrt{-\sqrt{2} a \frac{1}{a^{2}}^{\frac{3}{4}} x + x^{2} + \sqrt{\frac{1}{a^{2}}}} a \frac{1}{a^{2}}^{\frac{1}{4}} + 1\right ) - \frac{1}{4} \, \sqrt{2} \frac{1}{a^{2}}^{\frac{1}{4}} \log \left (\sqrt{2} a \frac{1}{a^{2}}^{\frac{3}{4}} x + x^{2} + \sqrt{\frac{1}{a^{2}}}\right ) + \frac{1}{4} \, \sqrt{2} \frac{1}{a^{2}}^{\frac{1}{4}} \log \left (-\sqrt{2} a \frac{1}{a^{2}}^{\frac{3}{4}} x + x^{2} + \sqrt{\frac{1}{a^{2}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 14.9019, size = 440, normalized size = 3.33 \begin{align*} \begin{cases} \frac{\pi x}{2} & \text{for}\: a = 0 \\\infty i x & \text{for}\: a = - \frac{i}{x^{2}} \\- \infty i x & \text{for}\: a = \frac{i}{x^{2}} \\- \frac{2 \left (-1\right )^{\frac{3}{4}} a^{7} x^{4} \left (\frac{1}{a^{2}}\right )^{\frac{11}{4}} \log{\left (x - \sqrt [4]{-1} \sqrt [4]{\frac{1}{a^{2}}} \right )}}{2 a^{2} x^{4} + 2} + \frac{\left (-1\right )^{\frac{3}{4}} a^{7} x^{4} \left (\frac{1}{a^{2}}\right )^{\frac{11}{4}} \log{\left (x^{2} + i \sqrt{\frac{1}{a^{2}}} \right )}}{2 a^{2} x^{4} + 2} + \frac{2 \left (-1\right )^{\frac{3}{4}} a^{7} x^{4} \left (\frac{1}{a^{2}}\right )^{\frac{11}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} x}{\sqrt [4]{\frac{1}{a^{2}}}} \right )}}{2 a^{2} x^{4} + 2} - \frac{2 \left (-1\right )^{\frac{3}{4}} a^{5} \left (\frac{1}{a^{2}}\right )^{\frac{11}{4}} \log{\left (x - \sqrt [4]{-1} \sqrt [4]{\frac{1}{a^{2}}} \right )}}{2 a^{2} x^{4} + 2} + \frac{\left (-1\right )^{\frac{3}{4}} a^{5} \left (\frac{1}{a^{2}}\right )^{\frac{11}{4}} \log{\left (x^{2} + i \sqrt{\frac{1}{a^{2}}} \right )}}{2 a^{2} x^{4} + 2} + \frac{2 \left (-1\right )^{\frac{3}{4}} a^{5} \left (\frac{1}{a^{2}}\right )^{\frac{11}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} x}{\sqrt [4]{\frac{1}{a^{2}}}} \right )}}{2 a^{2} x^{4} + 2} + \frac{2 \sqrt [4]{-1} a^{4} x^{4} \left (\frac{1}{a^{2}}\right )^{\frac{5}{4}} \operatorname{acot}{\left (a x^{2} \right )}}{2 a^{2} x^{4} + 2} + \frac{2 \sqrt [4]{-1} a^{4} \left (\frac{1}{a^{2}}\right )^{\frac{9}{4}} \operatorname{acot}{\left (a x^{2} \right )}}{2 a^{2} x^{4} + 2} + \frac{2 a^{2} x^{5} \operatorname{acot}{\left (a x^{2} \right )}}{2 a^{2} x^{4} + 2} + \frac{2 x \operatorname{acot}{\left (a x^{2} \right )}}{2 a^{2} x^{4} + 2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((pi*x/2, Eq(a, 0)), (oo*I*x, Eq(a, -I/x**2)), (-oo*I*x, Eq(a, I/x**2)), (-2*(-1)**(3/4)*a**7*x**4*(a
**(-2))**(11/4)*log(x - (-1)**(1/4)*(a**(-2))**(1/4))/(2*a**2*x**4 + 2) + (-1)**(3/4)*a**7*x**4*(a**(-2))**(11
/4)*log(x**2 + I*sqrt(a**(-2)))/(2*a**2*x**4 + 2) + 2*(-1)**(3/4)*a**7*x**4*(a**(-2))**(11/4)*atan((-1)**(3/4)
*x/(a**(-2))**(1/4))/(2*a**2*x**4 + 2) - 2*(-1)**(3/4)*a**5*(a**(-2))**(11/4)*log(x - (-1)**(1/4)*(a**(-2))**(
1/4))/(2*a**2*x**4 + 2) + (-1)**(3/4)*a**5*(a**(-2))**(11/4)*log(x**2 + I*sqrt(a**(-2)))/(2*a**2*x**4 + 2) + 2
*(-1)**(3/4)*a**5*(a**(-2))**(11/4)*atan((-1)**(3/4)*x/(a**(-2))**(1/4))/(2*a**2*x**4 + 2) + 2*(-1)**(1/4)*a**
4*x**4*(a**(-2))**(5/4)*acot(a*x**2)/(2*a**2*x**4 + 2) + 2*(-1)**(1/4)*a**4*(a**(-2))**(9/4)*acot(a*x**2)/(2*a
**2*x**4 + 2) + 2*a**2*x**5*acot(a*x**2)/(2*a**2*x**4 + 2) + 2*x*acot(a*x**2)/(2*a**2*x**4 + 2), True))

________________________________________________________________________________________

Giac [A]  time = 1.09722, size = 194, normalized size = 1.47 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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