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

Optimal. Leaf size=135 \[ \frac{\sqrt{a} \log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2}}-\frac{\sqrt{a} \log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2}}-\frac{\cot ^{-1}\left (a x^2\right )}{x}+\frac{\sqrt{a} \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{\sqrt{2}}-\frac{\sqrt{a} \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{\sqrt{2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0786399, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5034, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt{a} \log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2}}-\frac{\sqrt{a} \log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{2 \sqrt{2}}-\frac{\cot ^{-1}\left (a x^2\right )}{x}+\frac{\sqrt{a} \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{\sqrt{2}}-\frac{\sqrt{a} \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5034

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCot
[c*x^n]))/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 211

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

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])

Rubi steps

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

Mathematica [A]  time = 0.0397873, size = 105, normalized size = 0.78 \[ \frac{\sqrt{a} \left (\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )-\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )\right )}{2 \sqrt{2}}-\frac{\cot ^{-1}\left (a x^2\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 115, normalized size = 0.9 \begin{align*} -{\frac{{\rm arccot} \left (a{x}^{2}\right )}{x}}-{\frac{a\sqrt{2}}{2}\sqrt [4]{{a}^{-2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{a}^{-2}}}}}-1 \right ) }-{\frac{a\sqrt{2}}{4}\sqrt [4]{{a}^{-2}}\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{a\sqrt{2}}{2}\sqrt [4]{{a}^{-2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{a}^{-2}}}}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.49246, size = 327, normalized size = 2.42 \begin{align*} -\frac{1}{4} \,{\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{1}{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{1}{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{-\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{-\sqrt{a^{2}}}}\right )} a - \frac{\operatorname{arccot}\left (a x^{2}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.31341, size = 633, normalized size = 4.69 \begin{align*} \frac{4 \, \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x \arctan \left (-\frac{\sqrt{2}{\left (a^{2}\right )}^{\frac{3}{4}} a x + a^{2} - \sqrt{2} \sqrt{a^{2} x^{2} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} a x + \sqrt{a^{2}}}{\left (a^{2}\right )}^{\frac{3}{4}}}{a^{2}}\right ) + 4 \, \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x \arctan \left (-\frac{\sqrt{2}{\left (a^{2}\right )}^{\frac{3}{4}} a x - a^{2} - \sqrt{2} \sqrt{a^{2} x^{2} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} a x + \sqrt{a^{2}}}{\left (a^{2}\right )}^{\frac{3}{4}}}{a^{2}}\right ) - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x \log \left (a^{2} x^{2} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} a x + \sqrt{a^{2}}\right ) + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x \log \left (a^{2} x^{2} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} a x + \sqrt{a^{2}}\right ) - 4 \, \arctan \left (\frac{1}{a x^{2}}\right )}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 30.0655, size = 462, normalized size = 3.42 \begin{align*} \begin{cases} - \frac{\infty i}{x} & \text{for}\: a = - \frac{i}{x^{2}} \\\frac{\infty i}{x} & \text{for}\: a = \frac{i}{x^{2}} \\- \frac{\pi }{2 x} & \text{for}\: a = 0 \\\frac{2 \left (-1\right )^{\frac{3}{4}} a^{5} x^{5} \left (\frac{1}{a^{2}}\right )^{\frac{7}{4}} \operatorname{acot}{\left (a x^{2} \right )}}{2 a x^{5} + \frac{2 x}{a}} + \frac{2 \sqrt [4]{-1} a^{4} x^{5} \left (\frac{1}{a^{2}}\right )^{\frac{5}{4}} \log{\left (x - \sqrt [4]{-1} \sqrt [4]{\frac{1}{a^{2}}} \right )}}{2 a x^{5} + \frac{2 x}{a}} - \frac{\sqrt [4]{-1} a^{4} x^{5} \left (\frac{1}{a^{2}}\right )^{\frac{5}{4}} \log{\left (x^{2} + i \sqrt{\frac{1}{a^{2}}} \right )}}{2 a x^{5} + \frac{2 x}{a}} + \frac{2 \sqrt [4]{-1} a^{4} x^{5} \left (\frac{1}{a^{2}}\right )^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} x}{\sqrt [4]{\frac{1}{a^{2}}}} \right )}}{2 a x^{5} + \frac{2 x}{a}} + \frac{2 \left (-1\right )^{\frac{3}{4}} a^{3} x \left (\frac{1}{a^{2}}\right )^{\frac{7}{4}} \operatorname{acot}{\left (a x^{2} \right )}}{2 a x^{5} + \frac{2 x}{a}} + \frac{2 \sqrt [4]{-1} a^{2} x \left (\frac{1}{a^{2}}\right )^{\frac{5}{4}} \log{\left (x - \sqrt [4]{-1} \sqrt [4]{\frac{1}{a^{2}}} \right )}}{2 a x^{5} + \frac{2 x}{a}} - \frac{\sqrt [4]{-1} a^{2} x \left (\frac{1}{a^{2}}\right )^{\frac{5}{4}} \log{\left (x^{2} + i \sqrt{\frac{1}{a^{2}}} \right )}}{2 a x^{5} + \frac{2 x}{a}} + \frac{2 \sqrt [4]{-1} a^{2} x \left (\frac{1}{a^{2}}\right )^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} x}{\sqrt [4]{\frac{1}{a^{2}}}} \right )}}{2 a x^{5} + \frac{2 x}{a}} - \frac{2 a x^{4} \operatorname{acot}{\left (a x^{2} \right )}}{2 a x^{5} + \frac{2 x}{a}} - \frac{2 \operatorname{acot}{\left (a x^{2} \right )}}{2 a^{2} x^{5} + 2 x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12927, size = 182, normalized size = 1.35 \begin{align*} -\frac{1}{4} \, a{\left (\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 )}{\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 )}{\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 )}{\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 )}{\sqrt{{\left | a \right |}}}\right )} - \frac{\arctan \left (\frac{1}{a x^{2}}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/sqrt(abs(a)) + 2*sqrt(2)*arcta
n(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/sqrt(abs(a)) + sqrt(2)*log(x^2 + sqrt(2)*x/sqrt(abs(a
)) + 1/abs(a))/sqrt(abs(a)) - sqrt(2)*log(x^2 - sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/sqrt(abs(a))) - arctan(1/(a
*x^2))/x