3.1.87 \(\int \cot ^{-1}(\sqrt {x}) \, dx\) [87]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4931, 52, 65, 209} \begin {gather*} -\text {ArcTan}\left (\sqrt {x}\right )+\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCot[Sqrt[x]],x]

[Out]

Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 4931

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

Rubi steps

\begin {align*} \int \cot ^{-1}\left (\sqrt {x}\right ) \, dx &=x \cot ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \int \frac {\sqrt {x}}{1+x} \, dx\\ &=\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\text {ArcTan}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Sqrt[x]],x]

[Out]

Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]

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Maple [A]
time = 0.01, size = 17, normalized size = 0.77

method	result	size
derivativedivides	\(x \,\mathrm {arccot}\left (\sqrt {x}\right )-\arctan \left (\sqrt {x}\right )+\sqrt {x}\)	\(17\)
default	\(x \,\mathrm {arccot}\left (\sqrt {x}\right )-\arctan \left (\sqrt {x}\right )+\sqrt {x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 16, normalized size = 0.73 \begin {gather*} x \operatorname {arccot}\left (\sqrt {x}\right ) + \sqrt {x} - \arctan \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccot(sqrt(x)) + sqrt(x) - arctan(sqrt(x))

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Fricas [A]
time = 3.13, size = 12, normalized size = 0.55 \begin {gather*} {\left (x + 1\right )} \operatorname {arccot}\left (\sqrt {x}\right ) + \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*arccot(sqrt(x)) + sqrt(x)

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Sympy [A]
time = 0.76, size = 19, normalized size = 0.86 \begin {gather*} \sqrt {x} + x \operatorname {acot}{\left (\sqrt {x} \right )} - \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x) + x*acot(sqrt(x)) - atan(sqrt(x))

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Giac [A]
time = 0.40, size = 14, normalized size = 0.64 \begin {gather*} x \arctan \left (\frac {1}{\sqrt {x}}\right ) + \sqrt {x} + \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, algorithm="giac")

[Out]

x*arctan(1/sqrt(x)) + sqrt(x) + arctan(1/sqrt(x))

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Mupad [B]
time = 0.63, size = 16, normalized size = 0.73 \begin {gather*} x\,\mathrm {acot}\left (\sqrt {x}\right )-\mathrm {atan}\left (\sqrt {x}\right )+\sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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