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

Optimal. Leaf size=29 \[ \frac{2}{3} x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{x}{3}-\frac{1}{3} \log (x+1) \]

[Out]

x/3 + (2*x^(3/2)*ArcCot[Sqrt[x]])/3 - Log[1 + x]/3

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Rubi [A]  time = 0.0105779, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5034, 43} \[ \frac{2}{3} x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{x}{3}-\frac{1}{3} \log (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/3 + (2*x^(3/2)*ArcCot[Sqrt[x]])/3 - Log[1 + x]/3

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0108016, size = 25, normalized size = 0.86 \[ \frac{1}{3} \left (2 x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+x-\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x + 2*x^(3/2)*ArcCot[Sqrt[x]] - Log[1 + x])/3

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Maple [A]  time = 0.023, size = 20, normalized size = 0.7 \begin{align*}{\frac{x}{3}}+{\frac{2}{3}{x}^{{\frac{3}{2}}}{\rm arccot} \left (\sqrt{x}\right )}-{\frac{\ln \left ( x+1 \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x+2/3*x^(3/2)*arccot(x^(1/2))-1/3*ln(x+1)

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Maxima [A]  time = 0.973025, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \operatorname{arccot}\left (\sqrt{x}\right ) + \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arccot(sqrt(x)) + 1/3*x - 1/3*log(x + 1)

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Fricas [A]  time = 2.18903, size = 73, normalized size = 2.52 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \operatorname{arccot}\left (\sqrt{x}\right ) + \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)*arccot(sqrt(x)) + 1/3*x - 1/3*log(x + 1)

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Sympy [A]  time = 1.33226, size = 24, normalized size = 0.83 \begin{align*} \frac{2 x^{\frac{3}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{3} + \frac{x}{3} - \frac{\log{\left (x + 1 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(3/2)*acot(sqrt(x))/3 + x/3 - log(x + 1)/3

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Giac [A]  time = 1.10654, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \arctan \left (\frac{1}{\sqrt{x}}\right ) + \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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