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]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 31

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

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ \log (x+1)+2 \sqrt {x} \cot ^{-1}\left (\sqrt {x}\right ) \]

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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fricas [A]  time = 0.79, size = 14, normalized size = 0.78 \[ 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right ) + \log \left (x + 1\right ) \]

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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giac [A]  time = 0.12, size = 18, normalized size = 1.00 \[ 2 \, \sqrt {x} \arctan \left (\frac {1}{\sqrt {x}}\right ) + \log \relax (x) + \log \left (\frac {1}{x} + 1\right ) \]

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) + log(1/x + 1)

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maple [A]  time = 0.04, size = 15, normalized size = 0.83 \[ \ln \left (x +1\right )+2 \sqrt {x}\, \mathrm {arccot}\left (\sqrt {x}\right ) \]

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 = 0.31, size = 14, normalized size = 0.78 \[ 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right ) + \log \left (x + 1\right ) \]

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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mupad [B]  time = 0.77, size = 14, normalized size = 0.78 \[ \ln \left (x+1\right )+2\,\sqrt {x}\,\mathrm {acot}\left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 1) + 2*x^(1/2)*acot(x^(1/2))

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sympy [A]  time = 0.34, size = 17, normalized size = 0.94 \[ 2 \sqrt {x} \operatorname {acot}{\left (\sqrt {x} \right )} + \log {\left (x + 1 \right )} \]

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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