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

Optimal. Leaf size=42 \[ \frac{1}{6 x^{3/2}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2 \sqrt{x}}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

1/(6*x^(3/2)) - 1/(2*Sqrt[x]) - ArcCot[Sqrt[x]]/(2*x^2) - ArcTan[Sqrt[x]]/2

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Rubi [A]  time = 0.0122351, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5034, 51, 63, 203} \[ \frac{1}{6 x^{3/2}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2 \sqrt{x}}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[Sqrt[x]]/x^3,x]

[Out]

1/(6*x^(3/2)) - 1/(2*Sqrt[x]) - ArcCot[Sqrt[x]]/(2*x^2) - ArcTan[Sqrt[x]]/2

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}\left (\sqrt{x}\right )}{x^3} \, dx &=-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{4} \int \frac{1}{x^{5/2} (1+x)} \, dx\\ &=\frac{1}{6 x^{3/2}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{4} \int \frac{1}{x^{3/2} (1+x)} \, dx\\ &=\frac{1}{6 x^{3/2}}-\frac{1}{2 \sqrt{x}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{4} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=\frac{1}{6 x^{3/2}}-\frac{1}{2 \sqrt{x}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{6 x^{3/2}}-\frac{1}{2 \sqrt{x}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0095162, size = 34, normalized size = 0.81 \[ \frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-x\right )}{6 x^{3/2}}-\frac{\cot ^{-1}\left (\sqrt{x}\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Sqrt[x]]/x^3,x]

[Out]

-ArcCot[Sqrt[x]]/(2*x^2) + Hypergeometric2F1[-3/2, 1, -1/2, -x]/(6*x^(3/2))

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Maple [A]  time = 0.029, size = 27, normalized size = 0.6 \begin{align*}{\frac{1}{6}{x}^{-{\frac{3}{2}}}}-{\frac{1}{2\,{x}^{2}}{\rm arccot} \left (\sqrt{x}\right )}-{\frac{1}{2}\arctan \left ( \sqrt{x} \right ) }-{\frac{1}{2}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/x^(3/2)-1/2*arccot(x^(1/2))/x^2-1/2*arctan(x^(1/2))-1/2/x^(1/2)

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Maxima [A]  time = 1.46271, size = 35, normalized size = 0.83 \begin{align*} -\frac{3 \, x - 1}{6 \, x^{\frac{3}{2}}} - \frac{\operatorname{arccot}\left (\sqrt{x}\right )}{2 \, x^{2}} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*x - 1)/x^(3/2) - 1/2*arccot(sqrt(x))/x^2 - 1/2*arctan(sqrt(x))

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Fricas [A]  time = 2.18785, size = 80, normalized size = 1.9 \begin{align*} \frac{3 \,{\left (x^{2} - 1\right )} \operatorname{arccot}\left (\sqrt{x}\right ) -{\left (3 \, x - 1\right )} \sqrt{x}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(x^2 - 1)*arccot(sqrt(x)) - (3*x - 1)*sqrt(x))/x^2

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Sympy [B]  time = 8.74439, size = 160, normalized size = 3.81 \begin{align*} \frac{3 x^{\frac{7}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} + \frac{3 x^{\frac{5}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} - \frac{3 x^{\frac{3}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} - \frac{3 \sqrt{x} \operatorname{acot}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} - \frac{3 x^{3}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} - \frac{2 x^{2}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} + \frac{x}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**(7/2)*acot(sqrt(x))/(6*x**(7/2) + 6*x**(5/2)) + 3*x**(5/2)*acot(sqrt(x))/(6*x**(7/2) + 6*x**(5/2)) - 3*x*
*(3/2)*acot(sqrt(x))/(6*x**(7/2) + 6*x**(5/2)) - 3*sqrt(x)*acot(sqrt(x))/(6*x**(7/2) + 6*x**(5/2)) - 3*x**3/(6
*x**(7/2) + 6*x**(5/2)) - 2*x**2/(6*x**(7/2) + 6*x**(5/2)) + x/(6*x**(7/2) + 6*x**(5/2))

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Giac [A]  time = 1.1076, size = 35, normalized size = 0.83 \begin{align*} -\frac{3 \, x - 1}{6 \, x^{\frac{3}{2}}} - \frac{\arctan \left (\frac{1}{\sqrt{x}}\right )}{2 \, x^{2}} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*x - 1)/x^(3/2) - 1/2*arctan(1/sqrt(x))/x^2 - 1/2*arctan(sqrt(x))