3.1.90 \(\int \frac {\cot ^{-1}(\sqrt {x})}{x^3} \, dx\) [90]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4947, 53, 65, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\sqrt {x}\right )}{2}+\frac {1}{6 x^{3/2}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2 \sqrt {x}} \end {gather*}

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 53

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 65

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 209

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

Rule 4947

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

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} \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.01, size = 34, normalized size = 0.81 \begin {gather*} -\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-x\right )}{6 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.01, size = 27, normalized size = 0.64

method	result	size
derivativedivides	\(\frac {1}{6 x^{\frac {3}{2}}}-\frac {\mathrm {arccot}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\arctan \left (\sqrt {x}\right )}{2}-\frac {1}{2 \sqrt {x}}\)	\(27\)
default	\(\frac {1}{6 x^{\frac {3}{2}}}-\frac {\mathrm {arccot}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\arctan \left (\sqrt {x}\right )}{2}-\frac {1}{2 \sqrt {x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x^(1/2))/x^3,x,method=_RETURNVERBOSE)

[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 = 0.46, size = 26, normalized size = 0.62 \begin {gather*} -\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 {gather*}

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 = 1.91, size = 27, normalized size = 0.64 \begin {gather*} \frac {3 \, {\left (x^{2} - 1\right )} \operatorname {arccot}\left (\sqrt {x}\right ) - {\left (3 \, x - 1\right )} \sqrt {x}}{6 \, x^{2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (36) = 72\).
time = 1.21, size = 160, normalized size = 3.81 \begin {gather*} \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 {gather*}

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 = 0.41, size = 26, normalized size = 0.62 \begin {gather*} -\frac {1}{2 \, \sqrt {x}} - \frac {\arctan \left (\frac {1}{\sqrt {x}}\right )}{2 \, x^{2}} + \frac {1}{6 \, x^{\frac {3}{2}}} + \frac {1}{2} \, \arctan \left (\frac {1}{\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.65, size = 24, normalized size = 0.57 \begin {gather*} -\frac {\mathrm {atan}\left (\sqrt {x}\right )}{2}-\frac {x-\frac {1}{3}}{2\,x^{3/2}}-\frac {\mathrm {acot}\left (\sqrt {x}\right )}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan(x^(1/2))/2 - (x - 1/3)/(2*x^(3/2)) - acot(x^(1/2))/(2*x^2)

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