Optimal. Leaf size=12 \[ -\text{EllipticF}\left (\cos ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0065863, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {420} \[ -F\left (\left .\cos ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 420
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-x^2} \sqrt{-1+x^2}} \, dx &=-F\left (\left .\cos ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )\\ \end{align*}
Mathematica [B] time = 0.0221592, size = 47, normalized size = 3.92 \[ \frac{\sqrt{1-x^2} \sqrt{1-\frac{x^2}{2}} \text{EllipticF}\left (\sin ^{-1}(x),\frac{1}{2}\right )}{\sqrt{-x^4+3 x^2-2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 28, normalized size = 2.3 \begin{align*}{{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},\sqrt{2} \right ) \sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{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{1}{\sqrt{x^{2} - 1} \sqrt{-x^{2} + 2}}\,{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{-x^{2} + 2}}{x^{4} - 3 \, x^{2} + 2}, 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{1}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \sqrt{2 - x^{2}}}\, 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{1}{\sqrt{x^{2} - 1} \sqrt{-x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]