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

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Rubi [A]  time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.00, 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.04, 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])

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fricas [A]  time = 0.62, size = 189, normalized size = 1.43 \[ x \arctan \left (\frac {1}{a x^{2}}\right ) - \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a \frac {1}{a^{2}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}} a \frac {1}{a^{2}}^{\frac {1}{4}} - 1\right ) - \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a \frac {1}{a^{2}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {-\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}} a \frac {1}{a^{2}}^{\frac {1}{4}} + 1\right ) - \frac {1}{4} \, \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \log \left (\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}\right ) + \frac {1}{4} \, \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \log \left (-\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}\right ) \]

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)))

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giac [A]  time = 0.14, size = 144, normalized size = 1.09 \[ \frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} - \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}} + \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}}\right )} + x \arctan \left (\frac {1}{a x^{2}}\right ) \]

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))

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maple [A]  time = 0.04, size = 118, normalized size = 0.89 \[ x \,\mathrm {arccot}\left (a \,x^{2}\right )+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )}{4 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )}{2 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )}{2 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}} \]

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)

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maxima [A]  time = 0.41, size = 120, normalized size = 0.91 \[ \frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )} + x \operatorname {arccot}\left (a x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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mupad [B]  time = 0.13, size = 42, normalized size = 0.32 \[ x\,\mathrm {acot}\left (a\,x^2\right )+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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sympy [A]  time = 9.45, size = 139, normalized size = 1.05 \[ \begin {cases} x \operatorname {acot}{\left (a x^{2} \right )} + \sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )} - \frac {\left (-1\right )^{\frac {3}{4}} \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \right )}}{a \sqrt [4]{\frac {1}{a^{2}}}} + \frac {\left (-1\right )^{\frac {3}{4}} \log {\left (x^{2} + i \sqrt {\frac {1}{a^{2}}} \right )}}{2 a \sqrt [4]{\frac {1}{a^{2}}}} + \frac {\left (-1\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{a^{2}}}} \right )}}{a \sqrt [4]{\frac {1}{a^{2}}}} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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