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]

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

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Rubi [A]  time = 0.09, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, 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 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [B]  time = 0.81, size = 228, normalized size = 1.52 \[ \frac {4 \, a x^{3} \arctan \left (\frac {1}{a x^{2}}\right ) + 4 \, \sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a^{5} \frac {1}{a^{6}}^{\frac {3}{4}} x + \sqrt {2} \sqrt {\sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} x + a^{2} \sqrt {\frac {1}{a^{6}}} + x^{2}} a^{5} \frac {1}{a^{6}}^{\frac {3}{4}} - 1\right ) + 4 \, \sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a^{5} \frac {1}{a^{6}}^{\frac {3}{4}} x + \sqrt {2} \sqrt {-\sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} x + a^{2} \sqrt {\frac {1}{a^{6}}} + x^{2}} a^{5} \frac {1}{a^{6}}^{\frac {3}{4}} + 1\right ) - \sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} \log \left (\sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} x + a^{2} \sqrt {\frac {1}{a^{6}}} + x^{2}\right ) + \sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} \log \left (-\sqrt {2} a \frac {1}{a^{6}}^{\frac {1}{4}} x + a^{2} \sqrt {\frac {1}{a^{6}}} + x^{2}\right ) + 8 \, x}{12 \, a} \]

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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giac [A]  time = 0.14, size = 153, normalized size = 1.02 \[ \frac {1}{3} \, x^{3} \arctan \left (\frac {1}{a x^{2}}\right ) + \frac {1}{12} \, a {\left (\frac {8 \, x}{a^{2}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}} - \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}} + \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2} \sqrt {{\left | a \right |}}}\right )} \]

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))))

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maple [A]  time = 0.04, size = 127, normalized size = 0.85 \[ \frac {x^{3} \mathrm {arccot}\left (a \,x^{2}\right )}{3}+\frac {2 x}{3 a}-\frac {\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )}{12 a}-\frac {\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )}{6 a}-\frac {\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )}{6 a} \]

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 [A]  time = 0.41, size = 135, normalized size = 0.90 \[ \frac {1}{3} \, x^{3} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{12} \, a {\left (\frac {8 \, x}{a^{2}} - \frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}}}{a^{2}}\right )} \]

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*(8*x/a^2 - (2*sqrt(2)*arctan(1/2*sqrt(2)*(2*a*x + sqrt(2)*sqrt(a))/sqrt(a))/sqr
t(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)*s
qrt(a)*x + 1)/sqrt(a) - sqrt(2)*log(a*x^2 - sqrt(2)*sqrt(a)*x + 1)/sqrt(a))/a^2)

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mupad [B]  time = 0.70, size = 52, normalized size = 0.35 \[ \frac {x^3\,\mathrm {acot}\left (a\,x^2\right )}{3}+\frac {2\,x}{3\,a}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i}}{3\,a^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\,1{}\mathrm {i}\right )}{3\,a^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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))

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sympy [A]  time = 17.16, size = 156, normalized size = 1.04 \[ \begin {cases} \frac {x^{3} \operatorname {acot}{\left (a x^{2} \right )}}{3} + \frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{a^{2}}\right )^{\frac {3}{4}} \operatorname {acot}{\left (a x^{2} \right )}}{3} + \frac {2 x}{3 a} + \frac {\sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \right )}}{3 a} - \frac {\sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \log {\left (x^{2} + i \sqrt {\frac {1}{a^{2}}} \right )}}{6 a} + \frac {\sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{a^{2}}}} \right )}}{3 a} & \text {for}\: a \neq 0 \\\frac {\pi x^{3}}{6} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*acot(a*x**2)/3 + (-1)**(3/4)*(a**(-2))**(3/4)*acot(a*x**2)/3 + 2*x/(3*a) + (-1)**(1/4)*(a**(-2
))**(1/4)*log(x - (-1)**(1/4)*(a**(-2))**(1/4))/(3*a) - (-1)**(1/4)*(a**(-2))**(1/4)*log(x**2 + I*sqrt(a**(-2)
))/(6*a) + (-1)**(1/4)*(a**(-2))**(1/4)*atan((-1)**(3/4)*x/(a**(-2))**(1/4))/(3*a), Ne(a, 0)), (pi*x**3/6, Tru
e))

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