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

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Rubi [A]  time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, 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 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 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 \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*}

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Mathematica [A]  time = 0.05, 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])

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fricas [B]  time = 1.53, size = 227, normalized size = 1.68 \[ \frac {4 \, \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {\sqrt {2} {\left (a^{2}\right )}^{\frac {3}{4}} a x + a^{2} - \sqrt {2} \sqrt {a^{2} x^{2} + \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}} {\left (a^{2}\right )}^{\frac {3}{4}}}{a^{2}}\right ) + 4 \, \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {\sqrt {2} {\left (a^{2}\right )}^{\frac {3}{4}} a x - a^{2} - \sqrt {2} \sqrt {a^{2} x^{2} - \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}} {\left (a^{2}\right )}^{\frac {3}{4}}}{a^{2}}\right ) - \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \log \left (a^{2} x^{2} + \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}\right ) + \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \log \left (a^{2} x^{2} - \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}\right ) - 4 \, \arctan \left (\frac {1}{a x^{2}}\right )}{4 \, x} \]

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

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giac [A]  time = 0.14, size = 135, normalized size = 1.00 \[ -\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} + \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} - \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}}\right )} - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{x} \]

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

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maple [A]  time = 0.04, size = 115, normalized size = 0.85 \[ -\frac {\mathrm {arccot}\left (a \,x^{2}\right )}{x}-\frac {a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )}{2}-\frac {a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )}{4}-\frac {a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )}{2} \]

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)

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maxima [A]  time = 0.41, size = 123, normalized size = 0.91 \[ -\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}}\right )} - \frac {\operatorname {arccot}\left (a x^{2}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 0.68, size = 44, normalized size = 0.33 \[ -\frac {\mathrm {acot}\left (a\,x^2\right )}{x}+{\left (-1\right )}^{1/4}\,\sqrt {a}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i}+{\left (-1\right )}^{1/4}\,\sqrt {a}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1)^(1/4)*a^(1/2)*atan((-1)^(1/4)*a^(1/2)*x)*1i - acot(a*x^2)/x + (-1)^(1/4)*a^(1/2)*atanh((-1)^(1/4)*a^(1/2)
*x)*1i

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sympy [A]  time = 19.96, size = 144, normalized size = 1.07 \[ \begin {cases} \left (-1\right )^{\frac {3}{4}} a^{2} \left (\frac {1}{a^{2}}\right )^{\frac {3}{4}} \operatorname {acot}{\left (a x^{2} \right )} + \sqrt [4]{-1} a \sqrt [4]{\frac {1}{a^{2}}} \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \right )} - \frac {\sqrt [4]{-1} a \sqrt [4]{\frac {1}{a^{2}}} \log {\left (x^{2} + i \sqrt {\frac {1}{a^{2}}} \right )}}{2} + \sqrt [4]{-1} a \sqrt [4]{\frac {1}{a^{2}}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{a^{2}}}} \right )} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{x} & \text {for}\: a \neq 0 \\- \frac {\pi }{2 x} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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