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

Optimal. Leaf size=42 \[ -\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[x]/2 - x^(3/2)/6 - ArcTan[Sqrt[x]]/2 + (x^2*ArcTan[Sqrt[x]])/2

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Rubi [A]  time = 0.0098464, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5033, 50, 63, 203} \[ -\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x]/2 - x^(3/2)/6 - ArcTan[Sqrt[x]]/2 + (x^2*ArcTan[Sqrt[x]])/2

Rule 5033

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTan
[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 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 x \tan ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{x^{3/2}}{1+x} \, dx\\ &=-\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{4} \int \frac{\sqrt{x}}{1+x} \, dx\\ &=\frac{\sqrt{x}}{2}-\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=\frac{\sqrt{x}}{2}-\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{2}-\frac{x^{3/2}}{6}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.01061, size = 28, normalized size = 0.67 \[ \frac{1}{6} \left (3 \left (x^2-1\right ) \tan ^{-1}\left (\sqrt{x}\right )-(x-3) \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((-3 + x)*Sqrt[x]) + 3*(-1 + x^2)*ArcTan[Sqrt[x]])/6

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Maple [A]  time = 0.025, size = 27, normalized size = 0.6 \begin{align*} -{\frac{1}{6}{x}^{{\frac{3}{2}}}}-{\frac{1}{2}\arctan \left ( \sqrt{x} \right ) }+{\frac{{x}^{2}}{2}\arctan \left ( \sqrt{x} \right ) }+{\frac{1}{2}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*x^(3/2)-1/2*arctan(x^(1/2))+1/2*x^2*arctan(x^(1/2))+1/2*x^(1/2)

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Maxima [A]  time = 1.49418, size = 35, normalized size = 0.83 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\sqrt{x}\right ) - \frac{1}{6} \, x^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arctan(sqrt(x)) - 1/6*x^(3/2) + 1/2*sqrt(x) - 1/2*arctan(sqrt(x))

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Fricas [A]  time = 2.20084, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} \arctan \left (\sqrt{x}\right ) - \frac{1}{6} \,{\left (x - 3\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^2 - 1)*arctan(sqrt(x)) - 1/6*(x - 3)*sqrt(x)

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Sympy [A]  time = 2.02125, size = 32, normalized size = 0.76 \begin{align*} - \frac{x^{\frac{3}{2}}}{6} + \frac{\sqrt{x}}{2} + \frac{x^{2} \operatorname{atan}{\left (\sqrt{x} \right )}}{2} - \frac{\operatorname{atan}{\left (\sqrt{x} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**(3/2)/6 + sqrt(x)/2 + x**2*atan(sqrt(x))/2 - atan(sqrt(x))/2

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Giac [A]  time = 1.11712, size = 35, normalized size = 0.83 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\sqrt{x}\right ) - \frac{1}{6} \, x^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arctan(sqrt(x)) - 1/6*x^(3/2) + 1/2*sqrt(x) - 1/2*arctan(sqrt(x))