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

Optimal. Leaf size=18 \[ \log (x+1)+2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*Sqrt[x]*ArcCot[Sqrt[x]] + Log[1 + x]

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Rubi [A]  time = 0.0068054, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5034, 31} \[ \log (x+1)+2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[x]*ArcCot[Sqrt[x]] + Log[1 + 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0068556, size = 18, normalized size = 1. \[ \log (x+1)+2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[x]*ArcCot[Sqrt[x]] + Log[1 + x]

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Maple [A]  time = 0.023, size = 15, normalized size = 0.8 \begin{align*} \ln \left ( x+1 \right ) +2\,\sqrt{x}{\rm arccot} \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955033, size = 19, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} \operatorname{arccot}\left (\sqrt{x}\right ) + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arccot(sqrt(x)) + log(x + 1)

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Fricas [A]  time = 2.20801, size = 54, normalized size = 3. \begin{align*} 2 \, \sqrt{x} \operatorname{arccot}\left (\sqrt{x}\right ) + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arccot(sqrt(x)) + log(x + 1)

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Sympy [A]  time = 0.449524, size = 17, normalized size = 0.94 \begin{align*} 2 \sqrt{x} \operatorname{acot}{\left (\sqrt{x} \right )} + \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*acot(sqrt(x)) + log(x + 1)

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Giac [A]  time = 1.09288, size = 19, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} \arctan \left (\frac{1}{\sqrt{x}}\right ) + \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arctan(1/sqrt(x)) + log(x + 1)