3.94 \(\int \frac{\cot ^{-1}(\sqrt{x})}{x^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ -\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0076322, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5034, 36, 29, 31} \[ -\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[Sqrt[x]]/x^(3/2),x]

[Out]

(-2*ArcCot[Sqrt[x]])/Sqrt[x] - Log[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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A]  time = 0.0118346, size = 22, normalized size = 1. \[ -\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Sqrt[x]]/x^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 19, normalized size = 0.9 \begin{align*} -\ln \left ( x \right ) +\ln \left ( x+1 \right ) -2\,{\frac{{\rm arccot} \left (\sqrt{x}\right )}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x^(1/2))/x^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.978603, size = 24, normalized size = 1.09 \begin{align*} -\frac{2 \, \operatorname{arccot}\left (\sqrt{x}\right )}{\sqrt{x}} + \log \left (x + 1\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.13551, size = 77, normalized size = 3.5 \begin{align*} \frac{x \log \left (x + 1\right ) - x \log \left (x\right ) - 2 \, \sqrt{x} \operatorname{arccot}\left (\sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.76621, size = 20, normalized size = 0.91 \begin{align*} - \log{\left (x \right )} + \log{\left (x + 1 \right )} - \frac{2 \operatorname{acot}{\left (\sqrt{x} \right )}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x**(1/2))/x**(3/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12818, size = 22, normalized size = 1. \begin{align*} -\frac{2 \, \arctan \left (\frac{1}{\sqrt{x}}\right )}{\sqrt{x}} + \log \left (\frac{1}{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^(3/2),x, algorithm="giac")

[Out]

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