3.1.85 \(\int x^2 \cot ^{-1}(\sqrt {x}) \, dx\) [85]

Optimal. Leaf size=51 \[ \frac {\sqrt {x}}{3}-\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )-\frac {\text {ArcTan}\left (\sqrt {x}\right )}{3} \]

[Out]

-1/9*x^(3/2)+1/15*x^(5/2)+1/3*x^3*arccot(x^(1/2))-1/3*arctan(x^(1/2))+1/3*x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4947, 52, 65, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\sqrt {x}\right )}{3}+\frac {x^{5/2}}{15}-\frac {x^{3/2}}{9}+\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCot[Sqrt[x]],x]

[Out]

Sqrt[x]/3 - x^(3/2)/9 + x^(5/2)/15 + (x^3*ArcCot[Sqrt[x]])/3 - ArcTan[Sqrt[x]]/3

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 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 x^2 \cot ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} \int \frac {x^{5/2}}{1+x} \, dx\\ &=\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^{3/2}}{1+x} \, dx\\ &=-\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} \int \frac {\sqrt {x}}{1+x} \, dx\\ &=\frac {\sqrt {x}}{3}-\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=\frac {\sqrt {x}}{3}-\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x}}{3}-\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.78 \begin {gather*} \frac {1}{45} \left (\sqrt {x} \left (15-5 x+3 x^2\right )+15 x^3 \cot ^{-1}\left (\sqrt {x}\right )-15 \text {ArcTan}\left (\sqrt {x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCot[Sqrt[x]],x]

[Out]

(Sqrt[x]*(15 - 5*x + 3*x^2) + 15*x^3*ArcCot[Sqrt[x]] - 15*ArcTan[Sqrt[x]])/45

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Maple [A]
time = 0.01, size = 32, normalized size = 0.63

method	result	size
derivativedivides	\(-\frac {x^{\frac {3}{2}}}{9}+\frac {x^{\frac {5}{2}}}{15}+\frac {x^{3} \mathrm {arccot}\left (\sqrt {x}\right )}{3}-\frac {\arctan \left (\sqrt {x}\right )}{3}+\frac {\sqrt {x}}{3}\)	\(32\)
default	\(-\frac {x^{\frac {3}{2}}}{9}+\frac {x^{\frac {5}{2}}}{15}+\frac {x^{3} \mathrm {arccot}\left (\sqrt {x}\right )}{3}-\frac {\arctan \left (\sqrt {x}\right )}{3}+\frac {\sqrt {x}}{3}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*x^(3/2)+1/15*x^(5/2)+1/3*x^3*arccot(x^(1/2))-1/3*arctan(x^(1/2))+1/3*x^(1/2)

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Maxima [A]
time = 0.48, size = 31, normalized size = 0.61 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{15} \, x^{\frac {5}{2}} - \frac {1}{9} \, x^{\frac {3}{2}} + \frac {1}{3} \, \sqrt {x} - \frac {1}{3} \, \arctan \left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccot(sqrt(x)) + 1/15*x^(5/2) - 1/9*x^(3/2) + 1/3*sqrt(x) - 1/3*arctan(sqrt(x))

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Fricas [A]
time = 1.99, size = 27, normalized size = 0.53 \begin {gather*} \frac {1}{3} \, {\left (x^{3} + 1\right )} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{45} \, {\left (3 \, x^{2} - 5 \, x + 15\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^3 + 1)*arccot(sqrt(x)) + 1/45*(3*x^2 - 5*x + 15)*sqrt(x)

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Sympy [A]
time = 1.31, size = 39, normalized size = 0.76 \begin {gather*} \frac {x^{\frac {5}{2}}}{15} - \frac {x^{\frac {3}{2}}}{9} + \frac {\sqrt {x}}{3} + \frac {x^{3} \operatorname {acot}{\left (\sqrt {x} \right )}}{3} - \frac {\operatorname {atan}{\left (\sqrt {x} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(5/2)/15 - x**(3/2)/9 + sqrt(x)/3 + x**3*acot(sqrt(x))/3 - atan(sqrt(x))/3

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Giac [A]
time = 0.43, size = 33, normalized size = 0.65 \begin {gather*} \frac {1}{3} \, x^{3} \arctan \left (\frac {1}{\sqrt {x}}\right ) - \frac {1}{45} \, x^{\frac {5}{2}} {\left (\frac {5}{x} - \frac {15}{x^{2}} - 3\right )} + \frac {1}{3} \, \arctan \left (\frac {1}{\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arctan(1/sqrt(x)) - 1/45*x^(5/2)*(5/x - 15/x^2 - 3) + 1/3*arctan(1/sqrt(x))

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Mupad [B]
time = 0.65, size = 31, normalized size = 0.61 \begin {gather*} \frac {x^3\,\mathrm {acot}\left (\sqrt {x}\right )}{3}-\frac {\mathrm {atan}\left (\sqrt {x}\right )}{3}+\frac {\sqrt {x}}{3}-\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3*acot(x^(1/2)))/3 - atan(x^(1/2))/3 + x^(1/2)/3 - x^(3/2)/9 + x^(5/2)/15

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