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

Optimal. Leaf size=30 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0143253, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {402, 216, 377, 203} \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(1 + x^2),x]

[Out]

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

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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 \frac{\sqrt{1-x^2}}{1+x^2} \, dx &=2 \int \frac{1}{\sqrt{1-x^2} \left (1+x^2\right )} \, dx-\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\sin ^{-1}(x)+2 \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )\\ &=-\sin ^{-1}(x)+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0276604, size = 30, normalized size = 1. \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]/(1 + x^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 33, normalized size = 1.1 \begin{align*} -\arcsin \left ( x \right ) -\sqrt{2}\arctan \left ({\frac{x\sqrt{2}}{{x}^{2}-1}\sqrt{-{x}^{2}+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(x)-2^(1/2)*arctan(2^(1/2)*(-x^2+1)^(1/2)/(x^2-1)*x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{2} + 1}}{x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.51973, size = 111, normalized size = 3.7 \begin{align*} -\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + 1}}{2 \, x}\right ) + 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{x^{2} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.21887, size = 128, normalized size = 4.27 \begin{align*} -\frac{1}{2} \, \pi \mathrm{sgn}\left (x\right ) + \frac{1}{2} \, \sqrt{2}{\left (\pi \mathrm{sgn}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} - \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*pi*sgn(x) + 1/2*sqrt(2)*(pi*sgn(x) + 2*arctan(-1/4*sqrt(2)*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2
+ 1) - 1))) - arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))