3.127 \(\int -x^2 \tan ^{-1}(\sqrt{x}-\sqrt{1+x}) \, dx\)

Optimal. Leaf size=59 \[ \frac{\pi x^3}{12}+\frac{x^{5/2}}{30}-\frac{x^{3/2}}{18}-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{6}-\frac{1}{6} \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[x]/6 - x^(3/2)/18 + x^(5/2)/30 + (Pi*x^3)/12 - ArcTan[Sqrt[x]]/6 - (x^3*ArcTan[Sqrt[x]])/6

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Rubi [A]  time = 0.0259393, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5159, 30, 5033, 50, 63, 203} \[ \frac{\pi x^3}{12}+\frac{x^{5/2}}{30}-\frac{x^{3/2}}{18}-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{6}-\frac{1}{6} \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[-(x^2*ArcTan[Sqrt[x] - Sqrt[1 + x]]),x]

[Out]

Sqrt[x]/6 - x^(3/2)/18 + x^(5/2)/30 + (Pi*x^3)/12 - ArcTan[Sqrt[x]]/6 - (x^3*ArcTan[Sqrt[x]])/6

Rule 5159

Int[ArcTan[(v_) + (s_.)*Sqrt[w_]]*(u_.), x_Symbol] :> Dist[(Pi*s)/4, Int[u, x], x] + Dist[1/2, Int[u*ArcTan[v]
, x], x] /; EqQ[s^2, 1] && EqQ[w, v^2 + 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5033

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTan
[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 50

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 63

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 203

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

Rubi steps

\begin{align*} \int -x^2 \tan ^{-1}\left (\sqrt{x}-\sqrt{1+x}\right ) \, dx &=-\left (\frac{1}{2} \int x^2 \tan ^{-1}\left (\sqrt{x}\right ) \, dx\right )+\frac{1}{4} \pi \int x^2 \, dx\\ &=\frac{\pi x^3}{12}-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{12} \int \frac{x^{5/2}}{1+x} \, dx\\ &=\frac{x^{5/2}}{30}+\frac{\pi x^3}{12}-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{12} \int \frac{x^{3/2}}{1+x} \, dx\\ &=-\frac{x^{3/2}}{18}+\frac{x^{5/2}}{30}+\frac{\pi x^3}{12}-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{12} \int \frac{\sqrt{x}}{1+x} \, dx\\ &=\frac{\sqrt{x}}{6}-\frac{x^{3/2}}{18}+\frac{x^{5/2}}{30}+\frac{\pi x^3}{12}-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{12} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=\frac{\sqrt{x}}{6}-\frac{x^{3/2}}{18}+\frac{x^{5/2}}{30}+\frac{\pi x^3}{12}-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{6}-\frac{x^{3/2}}{18}+\frac{x^{5/2}}{30}+\frac{\pi x^3}{12}-\frac{1}{6} \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} x^3 \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0321391, size = 53, normalized size = 0.9 \[ \frac{1}{90} \left (-\sqrt{x} \left (-3 x^2+30 x^{5/2} \tan ^{-1}\left (\sqrt{x}-\sqrt{x+1}\right )+5 x-15\right )-15 \tan ^{-1}\left (\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[-(x^2*ArcTan[Sqrt[x] - Sqrt[1 + x]]),x]

[Out]

(-15*ArcTan[Sqrt[x]] - Sqrt[x]*(-15 + 5*x - 3*x^2 + 30*x^(5/2)*ArcTan[Sqrt[x] - Sqrt[1 + x]]))/90

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Maple [A]  time = 0.051, size = 40, normalized size = 0.7 \begin{align*} -{\frac{{x}^{3}}{3}\arctan \left ( \sqrt{x}-\sqrt{x+1} \right ) }+{\frac{1}{30}{x}^{{\frac{5}{2}}}}-{\frac{1}{18}{x}^{{\frac{3}{2}}}}+{\frac{1}{6}\sqrt{x}}-{\frac{1}{6}\arctan \left ( \sqrt{x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^2*arctan(x^(1/2)-(x+1)^(1/2)),x)

[Out]

-1/3*x^3*arctan(x^(1/2)-(x+1)^(1/2))+1/30*x^(5/2)-1/18*x^(3/2)+1/6*x^(1/2)-1/6*arctan(x^(1/2))

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Maxima [A]  time = 1.59477, size = 53, normalized size = 0.9 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (\sqrt{x + 1} - \sqrt{x}\right ) + \frac{1}{30} \, x^{\frac{5}{2}} - \frac{1}{18} \, x^{\frac{3}{2}} + \frac{1}{6} \, \sqrt{x} - \frac{1}{6} \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arctan(sqrt(x + 1) - sqrt(x)) + 1/30*x^(5/2) - 1/18*x^(3/2) + 1/6*sqrt(x) - 1/6*arctan(sqrt(x))

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Fricas [A]  time = 1.96665, size = 107, normalized size = 1.81 \begin{align*} \frac{1}{3} \,{\left (x^{3} + 1\right )} \arctan \left (\sqrt{x + 1} - \sqrt{x}\right ) + \frac{1}{90} \,{\left (3 \, x^{2} - 5 \, x + 15\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^3 + 1)*arctan(sqrt(x + 1) - sqrt(x)) + 1/90*(3*x^2 - 5*x + 15)*sqrt(x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.13207, size = 53, normalized size = 0.9 \begin{align*} -\frac{1}{3} \, x^{3} \arctan \left (-\sqrt{x + 1} + \sqrt{x}\right ) + \frac{1}{30} \, x^{\frac{5}{2}} - \frac{1}{18} \, x^{\frac{3}{2}} + \frac{1}{6} \, \sqrt{x} - \frac{1}{6} \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x^3*arctan(-sqrt(x + 1) + sqrt(x)) + 1/30*x^(5/2) - 1/18*x^(3/2) + 1/6*sqrt(x) - 1/6*arctan(sqrt(x))