\(\int x \arctan (\sqrt {x}) \, dx\) [155]

Optimal result

Integrand size = 8, antiderivative size = 42 \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=\frac {\sqrt {x}}{2}-\frac {x^{3/2}}{6}-\frac {\arctan \left (\sqrt {x}\right )}{2}+\frac {1}{2} x^2 \arctan \left (\sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4946, 52, 65, 209} \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=\frac {1}{2} x^2 \arctan \left (\sqrt {x}\right )-\frac {\arctan \left (\sqrt {x}\right )}{2}-\frac {x^{3/2}}{6}+\frac {\sqrt {x}}{2} \]

[In]

Int[x*ArcTan[Sqrt[x]],x]

[Out]

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

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 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[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*} \text {integral}& = \frac {1}{2} x^2 \arctan \left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x^{3/2}}{1+x} \, dx \\ & = -\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \arctan \left (\sqrt {x}\right )+\frac {1}{4} \int \frac {\sqrt {x}}{1+x} \, dx \\ & = \frac {\sqrt {x}}{2}-\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \arctan \left (\sqrt {x}\right )-\frac {1}{4} \int \frac {1}{\sqrt {x} (1+x)} \, dx \\ & = \frac {\sqrt {x}}{2}-\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \arctan \left (\sqrt {x}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {\sqrt {x}}{2}-\frac {x^{3/2}}{6}-\frac {\arctan \left (\sqrt {x}\right )}{2}+\frac {1}{2} x^2 \arctan \left (\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=\frac {1}{6} \left (-\left ((-3+x) \sqrt {x}\right )+3 \left (-1+x^2\right ) \arctan \left (\sqrt {x}\right )\right ) \]

[In]

Integrate[x*ArcTan[Sqrt[x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.60

method	result	size
meijerg	\(\frac {\sqrt {x}\, \left (-5 x +15\right )}{30}-\frac {\left (-5 x^{2}+5\right ) \arctan \left (\sqrt {x}\right )}{10}\)	\(25\)
derivativedivides	\(-\frac {x^{\frac {3}{2}}}{6}-\frac {\arctan \left (\sqrt {x}\right )}{2}+\frac {x^{2} \arctan \left (\sqrt {x}\right )}{2}+\frac {\sqrt {x}}{2}\)	\(27\)
default	\(-\frac {x^{\frac {3}{2}}}{6}-\frac {\arctan \left (\sqrt {x}\right )}{2}+\frac {x^{2} \arctan \left (\sqrt {x}\right )}{2}+\frac {\sqrt {x}}{2}\)	\(27\)
parts	\(-\frac {x^{\frac {3}{2}}}{6}-\frac {\arctan \left (\sqrt {x}\right )}{2}+\frac {x^{2} \arctan \left (\sqrt {x}\right )}{2}+\frac {\sqrt {x}}{2}\)	\(27\)

[In]

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

[Out]

1/30*x^(1/2)*(-5*x+15)-1/10*(-5*x^2+5)*arctan(x^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.48 \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, {\left (x^{2} - 1\right )} \arctan \left (\sqrt {x}\right ) - \frac {1}{6} \, {\left (x - 3\right )} \sqrt {x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76 \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=- \frac {x^{\frac {3}{2}}}{6} + \frac {\sqrt {x}}{2} + \frac {x^{2} \operatorname {atan}{\left (\sqrt {x} \right )}}{2} - \frac {\operatorname {atan}{\left (\sqrt {x} \right )}}{2} \]

[In]

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

[Out]

-x**(3/2)/6 + sqrt(x)/2 + x**2*atan(sqrt(x))/2 - atan(sqrt(x))/2

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \arctan \left (\sqrt {x}\right ) - \frac {1}{6} \, x^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {x} - \frac {1}{2} \, \arctan \left (\sqrt {x}\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \arctan \left (\sqrt {x}\right ) - \frac {1}{6} \, x^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {x} - \frac {1}{2} \, \arctan \left (\sqrt {x}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int x \arctan \left (\sqrt {x}\right ) \, dx=\frac {x^2\,\mathrm {atan}\left (\sqrt {x}\right )}{2}-\frac {\mathrm {atan}\left (\sqrt {x}\right )}{2}+\frac {\sqrt {x}}{2}-\frac {x^{3/2}}{6} \]

[In]

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

[Out]

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