3.87 \(\int \cot ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=22 \[ \sqrt{x}-\tan ^{-1}\left (\sqrt{x}\right )+x \cot ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0055551, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5028, 50, 63, 203} \[ \sqrt{x}-\tan ^{-1}\left (\sqrt{x}\right )+x \cot ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[Sqrt[x]],x]

[Out]

Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]

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]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 \cot ^{-1}\left (\sqrt{x}\right ) \, dx &=x \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} \int \frac{\sqrt{x}}{1+x} \, dx\\ &=\sqrt{x}+x \cot ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=\sqrt{x}+x \cot ^{-1}\left (\sqrt{x}\right )-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\sqrt{x}+x \cot ^{-1}\left (\sqrt{x}\right )-\tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.005846, size = 22, normalized size = 1. \[ \sqrt{x}-\tan ^{-1}\left (\sqrt{x}\right )+x \cot ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Sqrt[x]],x]

[Out]

Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 17, normalized size = 0.8 \begin{align*} x{\rm arccot} \left (\sqrt{x}\right )-\arctan \left ( \sqrt{x} \right ) +\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.45385, size = 22, normalized size = 1. \begin{align*} x \operatorname{arccot}\left (\sqrt{x}\right ) + \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, algorithm="maxima")

[Out]

x*arccot(sqrt(x)) + sqrt(x) - arctan(sqrt(x))

________________________________________________________________________________________

Fricas [A]  time = 2.21215, size = 47, normalized size = 2.14 \begin{align*}{\left (x + 1\right )} \operatorname{arccot}\left (\sqrt{x}\right ) + \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*arccot(sqrt(x)) + sqrt(x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acot}{\left (\sqrt{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(sqrt(x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.10771, size = 22, normalized size = 1. \begin{align*} x \arctan \left (\frac{1}{\sqrt{x}}\right ) + \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, algorithm="giac")

[Out]

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