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]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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]

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 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 )}{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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ -\log (x)+\log (x+1)-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]

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]

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fricas [A]  time = 2.09, size = 25, normalized size = 1.14 \[ \frac {x \log \left (x + 1\right ) - x \log \relax (x) - 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right )}{x} \]

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

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giac [A]  time = 0.14, size = 16, normalized size = 0.73 \[ -\frac {2 \, \arctan \left (\frac {1}{\sqrt {x}}\right )}{\sqrt {x}} + \log \left (\frac {1}{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 19, normalized size = 0.86 \[ -\ln \relax (x )+\ln \left (x +1\right )-\frac {2 \,\mathrm {arccot}\left (\sqrt {x}\right )}{\sqrt {x}} \]

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)

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maxima [A]  time = 0.33, size = 18, normalized size = 0.82 \[ -\frac {2 \, \operatorname {arccot}\left (\sqrt {x}\right )}{\sqrt {x}} + \log \left (x + 1\right ) - \log \relax (x) \]

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)

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mupad [B]  time = 0.67, size = 20, normalized size = 0.91 \[ \ln \left (x+1\right )-2\,\ln \left (\sqrt {x}\right )-\frac {2\,\mathrm {acot}\left (\sqrt {x}\right )}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.18, size = 20, normalized size = 0.91 \[ - \log {\relax (x )} + \log {\left (x + 1 \right )} - \frac {2 \operatorname {acot}{\left (\sqrt {x} \right )}}{\sqrt {x}} \]

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)

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