3.95 \(\int \tan ^{-1}(\sqrt{x}) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0051936, 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 = {5027, 50, 63, 203} \[ -\sqrt{x}+x \tan ^{-1}\left (\sqrt{x}\right )+\tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Sqrt[x]],x]

[Out]

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

Rule 5027

Int[ArcTan[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTan[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 \tan ^{-1}\left (\sqrt{x}\right ) \, dx &=x \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{\sqrt{x}}{1+x} \, dx\\ &=-\sqrt{x}+x \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=-\sqrt{x}+x \tan ^{-1}\left (\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=-\sqrt{x}+\tan ^{-1}\left (\sqrt{x}\right )+x \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0051495, size = 18, normalized size = 0.82 \[ (x+1) \tan ^{-1}\left (\sqrt{x}\right )-\sqrt{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Sqrt[x]],x]

[Out]

-Sqrt[x] + (1 + x)*ArcTan[Sqrt[x]]

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Maple [A]  time = 0.003, size = 17, normalized size = 0.8 \begin{align*} \arctan \left ( \sqrt{x} \right ) +x\arctan \left ( \sqrt{x} \right ) -\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.44288, size = 22, normalized size = 1. \begin{align*} x \arctan \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(arctan(x^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.544107, size = 47, normalized size = 2.14 \begin{align*}{\left (x + 1\right )} \arctan \left (\sqrt{x}\right ) - \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*arctan(sqrt(x)) - sqrt(x)

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Sympy [A]  time = 1.35363, size = 19, normalized size = 0.86 \begin{align*} - \sqrt{x} + x \operatorname{atan}{\left (\sqrt{x} \right )} + \operatorname{atan}{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x) + x*atan(sqrt(x)) + atan(sqrt(x))

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Giac [A]  time = 1.10465, size = 22, normalized size = 1. \begin{align*} x \arctan \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(arctan(x^(1/2)),x, algorithm="giac")

[Out]

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