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

Optimal. Leaf size=150 \[ \frac{\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2} a^{3/2}}-\frac{\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2} a^{3/2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2} a^{3/2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{3 \sqrt{2} a^{3/2}}+\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )+\frac{2 x}{3 a} \]

[Out]

(2*x)/(3*a) + (x^3*ArcCot[a*x^2])/3 + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(3*Sqrt[2]*a^(3/2)) - ArcTan[1 + Sqrt[2]*S
qrt[a]*x]/(3*Sqrt[2]*a^(3/2)) + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2)) - Log[1 + Sqrt[2]*Sqrt[
a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2))

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Rubi [A]  time = 0.0942241, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5034, 321, 211, 1165, 628, 1162, 617, 204} \[ \frac{\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2} a^{3/2}}-\frac{\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2} a^{3/2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2} a^{3/2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{3 \sqrt{2} a^{3/2}}+\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )+\frac{2 x}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/(3*a) + (x^3*ArcCot[a*x^2])/3 + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(3*Sqrt[2]*a^(3/2)) - ArcTan[1 + Sqrt[2]*S
qrt[a]*x]/(3*Sqrt[2]*a^(3/2)) + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2)) - Log[1 + Sqrt[2]*Sqrt[
a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2))

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 321

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] - Dist[(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 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 x^2 \cot ^{-1}\left (a x^2\right ) \, dx &=\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )+\frac{1}{3} (2 a) \int \frac{x^4}{1+a^2 x^4} \, dx\\ &=\frac{2 x}{3 a}+\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )-\frac{2 \int \frac{1}{1+a^2 x^4} \, dx}{3 a}\\ &=\frac{2 x}{3 a}+\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )-\frac{\int \frac{1-a x^2}{1+a^2 x^4} \, dx}{3 a}-\frac{\int \frac{1+a x^2}{1+a^2 x^4} \, dx}{3 a}\\ &=\frac{2 x}{3 a}+\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )-\frac{\int \frac{1}{\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx}{6 a^2}-\frac{\int \frac{1}{\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx}{6 a^2}+\frac{\int \frac{\frac{\sqrt{2}}{\sqrt{a}}+2 x}{-\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{6 \sqrt{2} a^{3/2}}+\frac{\int \frac{\frac{\sqrt{2}}{\sqrt{a}}-2 x}{-\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{6 \sqrt{2} a^{3/2}}\\ &=\frac{2 x}{3 a}+\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )+\frac{\log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2} a^{3/2}}-\frac{\log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2} a^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2} a^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2} a^{3/2}}\\ &=\frac{2 x}{3 a}+\frac{1}{3} x^3 \cot ^{-1}\left (a x^2\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2} a^{3/2}}-\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2} a^{3/2}}+\frac{\log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2} a^{3/2}}-\frac{\log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2} a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0273759, size = 133, normalized size = 0.89 \[ \frac{4 a^{3/2} x^3 \cot ^{-1}\left (a x^2\right )+\sqrt{2} \log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )-\sqrt{2} \log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )+8 \sqrt{a} x+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{12 a^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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 + Sq
rt[2]*Sqrt[a]*x] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(1
2*a^(3/2))

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Maple [A]  time = 0.043, size = 127, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}{\rm arccot} \left (a{x}^{2}\right )}{3}}+{\frac{2\,x}{3\,a}}-{\frac{\sqrt{2}}{12\,a}\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{\sqrt{2}}{6\,a}\sqrt [4]{{a}^{-2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{6\,a}\sqrt [4]{{a}^{-2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccot(a*x^2)+2/3*x/a-1/12/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)))-1/6/a*(1/a^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x+1)-1/6/a*(
1/a^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x-1)

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Maxima [B]  time = 1.50193, size = 342, normalized size = 2.28 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arccot}\left (a x^{2}\right ) - \frac{1}{12} \, a{\left (\frac{\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}}}}}{a^{2}} - \frac{8 \, x}{a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccot(a*x^2) - 1/12*a*((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^2 - 8*x/a^2)

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Fricas [B]  time = 2.30081, size = 713, normalized size = 4.75 \begin{align*} \frac{4 \, a x^{3} \arctan \left (\frac{1}{a x^{2}}\right ) + 4 \, \sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} \arctan \left (-\sqrt{2} a^{5} \frac{1}{a^{6}}^{\frac{3}{4}} x + \sqrt{2} \sqrt{\sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} x + a^{2} \sqrt{\frac{1}{a^{6}}} + x^{2}} a^{5} \frac{1}{a^{6}}^{\frac{3}{4}} - 1\right ) + 4 \, \sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} \arctan \left (-\sqrt{2} a^{5} \frac{1}{a^{6}}^{\frac{3}{4}} x + \sqrt{2} \sqrt{-\sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} x + a^{2} \sqrt{\frac{1}{a^{6}}} + x^{2}} a^{5} \frac{1}{a^{6}}^{\frac{3}{4}} + 1\right ) - \sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} \log \left (\sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} x + a^{2} \sqrt{\frac{1}{a^{6}}} + x^{2}\right ) + \sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} \log \left (-\sqrt{2} a \frac{1}{a^{6}}^{\frac{1}{4}} x + a^{2} \sqrt{\frac{1}{a^{6}}} + x^{2}\right ) + 8 \, x}{12 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*a*x^3*arctan(1/(a*x^2)) + 4*sqrt(2)*a*(a^(-6))^(1/4)*arctan(-sqrt(2)*a^5*(a^(-6))^(3/4)*x + sqrt(2)*sq
rt(sqrt(2)*a*(a^(-6))^(1/4)*x + a^2*sqrt(a^(-6)) + x^2)*a^5*(a^(-6))^(3/4) - 1) + 4*sqrt(2)*a*(a^(-6))^(1/4)*a
rctan(-sqrt(2)*a^5*(a^(-6))^(3/4)*x + sqrt(2)*sqrt(-sqrt(2)*a*(a^(-6))^(1/4)*x + a^2*sqrt(a^(-6)) + x^2)*a^5*(
a^(-6))^(3/4) + 1) - sqrt(2)*a*(a^(-6))^(1/4)*log(sqrt(2)*a*(a^(-6))^(1/4)*x + a^2*sqrt(a^(-6)) + x^2) + sqrt(
2)*a*(a^(-6))^(1/4)*log(-sqrt(2)*a*(a^(-6))^(1/4)*x + a^2*sqrt(a^(-6)) + x^2) + 8*x)/a

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Sympy [A]  time = 30.4211, size = 1081, normalized size = 7.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**3*acot((-sqrt(2)/2 - sqrt(2)*I/2)**(-2))/3, Eq(a, -1/(x**2*(-sqrt(2)/2 - sqrt(2)*I/2)**2))), (-
x**3*acot((-sqrt(2)/2 + sqrt(2)*I/2)**(-2))/3, Eq(a, -1/(x**2*(-sqrt(2)/2 + sqrt(2)*I/2)**2))), (-x**3*acot((s
qrt(2)/2 - sqrt(2)*I/2)**(-2))/3, Eq(a, -1/(x**2*(sqrt(2)/2 - sqrt(2)*I/2)**2))), (-x**3*acot((sqrt(2)/2 + sqr
t(2)*I/2)**(-2))/3, Eq(a, -1/(x**2*(sqrt(2)/2 + sqrt(2)*I/2)**2))), (pi*x**3/6, Eq(a, 0)), (-2*(-1)**(1/4)*a**
8*x**7*(a**(-2))**(13/4)*acot(a*x**2)/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(a**(-2
))**(13/4)) - 4*(-1)**(1/4)*a**7*x**5*(a**(-2))**(13/4)/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**
(1/4)*a**6*(a**(-2))**(13/4)) - 2*(-1)**(1/4)*a**6*x**3*(a**(-2))**(13/4)*acot(a*x**2)/(-6*(-1)**(1/4)*a**8*x*
*4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(a**(-2))**(13/4)) - 2*I*a**5*x**4*(a**(-2))**(5/2)*log(x - (-1)**(1
/4)*(a**(-2))**(1/4))/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(a**(-2))**(13/4)) + I*
a**5*x**4*(a**(-2))**(5/2)*log(x**2 + I*sqrt(a**(-2)))/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(
1/4)*a**6*(a**(-2))**(13/4)) - 2*I*a**5*x**4*(a**(-2))**(5/2)*atan((-1)**(3/4)*x/(a**(-2))**(1/4))/(-6*(-1)**(
1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(a**(-2))**(13/4)) - 4*(-1)**(1/4)*a**5*x*(a**(-2))**(13
/4)/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(a**(-2))**(13/4)) - 2*I*a**3*(a**(-2))**
(5/2)*log(x - (-1)**(1/4)*(a**(-2))**(1/4))/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(
a**(-2))**(13/4)) + I*a**3*(a**(-2))**(5/2)*log(x**2 + I*sqrt(a**(-2)))/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(
13/4) - 6*(-1)**(1/4)*a**6*(a**(-2))**(13/4)) - 2*I*a**3*(a**(-2))**(5/2)*atan((-1)**(3/4)*x/(a**(-2))**(1/4))
/(-6*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(a**(-2))**(13/4)) + 2*x**4*acot(a*x**2)/(-6
*(-1)**(1/4)*a**8*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**6*(a**(-2))**(13/4)) + 2*acot(a*x**2)/(-6*(-1)**(1
/4)*a**10*x**4*(a**(-2))**(13/4) - 6*(-1)**(1/4)*a**8*(a**(-2))**(13/4)), True))

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Giac [A]  time = 1.11955, size = 207, normalized size = 1.38 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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(a
bs(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
(abs(a))) - sqrt(2)*log(x^2 + sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/(a^2*sqrt(abs(a))) + sqrt(2)*log(x^2 - sqrt(2
)*x/sqrt(abs(a)) + 1/abs(a))/(a^2*sqrt(abs(a))))