3.1.92 \(\int \sqrt {x} \cot ^{-1}(\sqrt {x}) \, dx\) [92]

Optimal. Leaf size=29 \[ \frac {x}{3}+\frac {2}{3} x^{3/2} \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \log (1+x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 29, 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 = {4947, 45} \begin {gather*} \frac {2}{3} x^{3/2} \cot ^{-1}\left (\sqrt {x}\right )+\frac {x}{3}-\frac {1}{3} \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 \sqrt {x} \cot ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {2}{3} x^{3/2} \cot ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \int \frac {x}{1+x} \, dx\\ &=\frac {2}{3} x^{3/2} \cot ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \int \left (1+\frac {1}{-1-x}\right ) \, dx\\ &=\frac {x}{3}+\frac {2}{3} x^{3/2} \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \log (1+x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{3} \left (x+2 x^{3/2} \cot ^{-1}\left (\sqrt {x}\right )-\log (1+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 20, normalized size = 0.69

method	result	size
derivativedivides	\(\frac {x}{3}+\frac {2 x^{\frac {3}{2}} \mathrm {arccot}\left (\sqrt {x}\right )}{3}-\frac {\ln \left (1+x \right )}{3}\)	\(20\)
default	\(\frac {x}{3}+\frac {2 x^{\frac {3}{2}} \mathrm {arccot}\left (\sqrt {x}\right )}{3}-\frac {\ln \left (1+x \right )}{3}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 2.57, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.87, size = 24, normalized size = 0.83 \begin {gather*} \frac {2 x^{\frac {3}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{3} + \frac {x}{3} - \frac {\log {\left (x + 1 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.40, size = 30, normalized size = 1.03 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {x}}\right ) - \frac {1}{3} \, x {\left (\frac {1}{x} - 1\right )} - \frac {1}{3} \, \log \left (x\right ) - \frac {1}{3} \, \log \left (\frac {1}{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {x}\,\mathrm {acot}\left (\sqrt {x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^(1/2)*acot(x^(1/2)), x)

________________________________________________________________________________________