Optimal. Leaf size=30 \[ \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )-\sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
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]
[Out]
Rule 402
Rule 216
Rule 377
Rule 203
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]