3.10.73 \(\int \frac {x^2}{-1+x^2+\sqrt {1-x^2}} \, dx\) [973]

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

[Out]

x+arcsin(x)

________________________________________________________________________________________

Rubi [A]
time = 0.03, 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 = {2187, 8, 222} \begin {gather*} \text {ArcSin}(x)+x \end {gather*}

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 222

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 2187

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*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(22\) vs. \(2(4)=8\).
time = 0.10, size = 22, normalized size = 5.50 \begin {gather*} x+2 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + 2*ArcTan[x/(-1 + Sqrt[1 - x^2])]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(4)=8\).
time = 0.08, size = 51, normalized size = 12.75

method	result	size
trager	\(x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\)	\(29\)
default	\(x +\frac {\ln \left (-1+x \right )}{2}-\frac {\ln \left (1+x \right )}{2}+\arctanh \left (x \right )-\frac {\sqrt {-\left (-1+x \right )^{2}+2-2 x}}{2}+\arcsin \left (x \right )+\frac {\sqrt {-\left (1+x \right )^{2}+2+2 x}}{2}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-1+x^2+(-x^2+1)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (4) = 8\).
time = 0.33, size = 20, normalized size = 5.00 \begin {gather*} x - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{x^{2} + \sqrt {1 - x^{2}} - 1}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 2.95, size = 4, normalized size = 1.00 \begin {gather*} x + \arcsin \left (x\right ) \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 4, normalized size = 1.00 \begin {gather*} x+\mathrm {asin}\left (x\right ) \end {gather*}

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)

________________________________________________________________________________________