3.95 \(\int \frac {\cot ^{-1}(\sqrt {x})}{x^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ -\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{3 x}+\frac {\log (x)}{3}-\frac {1}{3} \log (x+1) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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, 44} \[ -\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{3 x}+\frac {\log (x)}{3}-\frac {1}{3} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[Sqrt[x]]/x^(5/2),x]

[Out]

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 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^{5/2}} \, dx &=-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}-\frac {1}{3} \int \frac {1}{x^2 (1+x)} \, dx\\ &=-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}-\frac {1}{3} \int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx\\ &=\frac {1}{3 x}-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}+\frac {\log (x)}{3}-\frac {1}{3} \log (1+x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 29, normalized size = 0.78 \[ \frac {1}{3} \left (-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{x^{3/2}}+\frac {1}{x}+\log (x)-\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Sqrt[x]]/x^(5/2),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 33, normalized size = 0.89 \[ -\frac {x^{2} \log \left (x + 1\right ) - x^{2} \log \relax (x) + 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right ) - x}{3 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.13, size = 23, normalized size = 0.62 \[ -\frac {2 \, \arctan \left (\frac {1}{\sqrt {x}}\right )}{3 \, x^{\frac {3}{2}}} + \frac {1}{3 \, x} - \frac {1}{3} \, \log \left (\frac {1}{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^(5/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 26, normalized size = 0.70 \[ \frac {1}{3 x}-\frac {2 \,\mathrm {arccot}\left (\sqrt {x}\right )}{3 x^{\frac {3}{2}}}+\frac {\ln \relax (x )}{3}-\frac {\ln \left (x +1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x^(1/2))/x^(5/2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 25, normalized size = 0.68 \[ -\frac {2 \, \operatorname {arccot}\left (\sqrt {x}\right )}{3 \, x^{\frac {3}{2}}} + \frac {1}{3 \, x} - \frac {1}{3} \, \log \left (x + 1\right ) + \frac {1}{3} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x^(1/2))/x^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.65, size = 27, normalized size = 0.73 \[ \frac {2\,\ln \left (\sqrt {x}\right )}{3}-\frac {\ln \left (x+1\right )}{3}-\frac {2\,\mathrm {acot}\left (\sqrt {x}\right )}{3\,x^{3/2}}+\frac {1}{3\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(x^(1/2))/x^(5/2),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 4.92, size = 143, normalized size = 3.86 \[ - \frac {2 x^{\frac {3}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{3 x^{3} + 3 x^{2}} - \frac {2 \sqrt {x} \operatorname {acot}{\left (\sqrt {x} \right )}}{3 x^{3} + 3 x^{2}} + \frac {x^{3} \log {\relax (x )}}{3 x^{3} + 3 x^{2}} - \frac {x^{3} \log {\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} + \frac {x^{2} \log {\relax (x )}}{3 x^{3} + 3 x^{2}} - \frac {x^{2} \log {\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} + \frac {x^{2}}{3 x^{3} + 3 x^{2}} + \frac {x}{3 x^{3} + 3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x**(1/2))/x**(5/2),x)

[Out]

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

________________________________________________________________________________________