3.89 \(\int \frac{\cot ^{-1}(\sqrt{x})}{x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{\sqrt{x}}+\tan ^{-1}\left (\sqrt{x}\right )-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{x} \]

[Out]

1/Sqrt[x] - ArcCot[Sqrt[x]]/x + ArcTan[Sqrt[x]]

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Rubi [A]  time = 0.0109688, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5034, 51, 63, 203} \[ \frac{1}{\sqrt{x}}+\tan ^{-1}\left (\sqrt{x}\right )-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[Sqrt[x]]/x^2,x]

[Out]

1/Sqrt[x] - ArcCot[Sqrt[x]]/x + ArcTan[Sqrt[x]]

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{x}-\frac{1}{2} \int \frac{1}{x^{3/2} (1+x)} \, dx\\ &=\frac{1}{\sqrt{x}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{x}+\frac{1}{2} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=\frac{1}{\sqrt{x}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{x}+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{\sqrt{x}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{x}+\tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0087166, size = 29, normalized size = 1.26 \[ \frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-x\right )}{\sqrt{x}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Sqrt[x]]/x^2,x]

[Out]

-(ArcCot[Sqrt[x]]/x) + Hypergeometric2F1[-1/2, 1, 1/2, -x]/Sqrt[x]

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Maple [A]  time = 0.024, size = 18, normalized size = 0.8 \begin{align*} -{\frac{1}{x}{\rm arccot} \left (\sqrt{x}\right )}+\arctan \left ( \sqrt{x} \right ) +{\frac{1}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x^(1/2))/x^2,x)

[Out]

-arccot(x^(1/2))/x+arctan(x^(1/2))+1/x^(1/2)

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Maxima [A]  time = 1.46619, size = 23, normalized size = 1. \begin{align*} -\frac{\operatorname{arccot}\left (\sqrt{x}\right )}{x} + \frac{1}{\sqrt{x}} + \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^2,x, algorithm="maxima")

[Out]

-arccot(sqrt(x))/x + 1/sqrt(x) + arctan(sqrt(x))

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Fricas [A]  time = 2.26882, size = 54, normalized size = 2.35 \begin{align*} -\frac{{\left (x + 1\right )} \operatorname{arccot}\left (\sqrt{x}\right ) - \sqrt{x}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^2,x, algorithm="fricas")

[Out]

-((x + 1)*arccot(sqrt(x)) - sqrt(x))/x

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Sympy [B]  time = 3.07875, size = 92, normalized size = 4. \begin{align*} - \frac{x^{\frac{5}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} - \frac{2 x^{\frac{3}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} - \frac{\sqrt{x} \operatorname{acot}{\left (\sqrt{x} \right )}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} + \frac{x^{2}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} + \frac{x}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x**(1/2))/x**2,x)

[Out]

-x**(5/2)*acot(sqrt(x))/(x**(5/2) + x**(3/2)) - 2*x**(3/2)*acot(sqrt(x))/(x**(5/2) + x**(3/2)) - sqrt(x)*acot(
sqrt(x))/(x**(5/2) + x**(3/2)) + x**2/(x**(5/2) + x**(3/2)) + x/(x**(5/2) + x**(3/2))

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Giac [A]  time = 1.10118, size = 26, normalized size = 1.13 \begin{align*} -\frac{\arctan \left (\frac{1}{\sqrt{x}}\right )}{x} + \frac{1}{\sqrt{x}} - \arctan \left (\frac{1}{\sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^2,x, algorithm="giac")

[Out]

-arctan(1/sqrt(x))/x + 1/sqrt(x) - arctan(1/sqrt(x))