Optimal. Leaf size=40 \[ \frac{\sqrt{1-2 x^2} E\left (\sin ^{-1}\left (\sqrt{2} x\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{2 x^2-1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.014525, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {427, 424} \[ \frac{\sqrt{1-2 x^2} E\left (\sin ^{-1}\left (\sqrt{2} x\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{2 x^2-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 427
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{\sqrt{-1+2 x^2}} \, dx &=\frac{\sqrt{1-2 x^2} \int \frac{\sqrt{1-x^2}}{\sqrt{1-2 x^2}} \, dx}{\sqrt{-1+2 x^2}}\\ &=\frac{\sqrt{1-2 x^2} E\left (\sin ^{-1}\left (\sqrt{2} x\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-1+2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0240141, size = 35, normalized size = 0.88 \[ \frac{\sqrt{1-2 x^2} E\left (\sin ^{-1}\left (\sqrt{2} x\right )|\frac{1}{2}\right )}{\sqrt{4 x^2-2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 32, normalized size = 0.8 \begin{align*}{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) +{\it EllipticE} \left ( x,\sqrt{2} \right ) }{2}\sqrt{-2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{2}-1}}}} \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}}{\sqrt{2 \, x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{2} + 1}}{\sqrt{2 \, x^{2} - 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 )}}{\sqrt{2 x^{2} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{2 \, x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]