3.973 \(\int \frac {x^2}{-1+x^2+\sqrt {1-x^2}} \, dx\)

Optimal. Leaf size=4 \[ x+\sin ^{-1}(x) \]

[Out]

x+arcsin(x)

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Rubi [A]  time = 0.04, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2156, 8, 216} \[ x+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(-1 + x^2 + Sqrt[1 - x^2]),x]

[Out]

x + ArcSin[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 216

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

Rule 2156

Int[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[c, Int[u/(c^2 - a*e
^2 + c*d*x^n), x], x] - Dist[a*e, Int[u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d
, e, n}, x] && EqQ[b*c - a*d, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{-1+x^2+\sqrt {1-x^2}} \, dx &=-\int -1 \, dx+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=x+\sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 4, normalized size = 1.00 \[ x+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + ArcSin[x]

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fricas [B]  time = 0.68, size = 20, normalized size = 5.00 \[ x - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*arctan((sqrt(-x^2 + 1) - 1)/x)

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giac [A]  time = 0.33, size = 4, normalized size = 1.00 \[ x + \arcsin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + arcsin(x)

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maple [B]  time = 0.02, size = 51, normalized size = 12.75 \[ x +\arctanh \relax (x )+\arcsin \relax (x )+\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}-\frac {\sqrt {-2 x -\left (x -1\right )^{2}+2}}{2}+\frac {\sqrt {2 x -\left (x +1\right )^{2}+2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+1/2*ln(x-1)-1/2*ln(x+1)+arctanh(x)-1/2*(-(x-1)^2-2*x+2)^(1/2)+arcsin(x)+1/2*(-(x+1)^2+2*x+2)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{x^{2} + \sqrt {-x^{2} + 1} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(x^2 + sqrt(-x^2 + 1) - 1), x)

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mupad [B]  time = 0.03, size = 4, normalized size = 1.00 \[ x+\mathrm {asin}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + asin(x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{x^{2} + \sqrt {1 - x^{2}} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(x**2 + sqrt(1 - x**2) - 1), x)

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