Optimal. Leaf size=32 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),\frac{5}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0141692, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {421, 419} \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|\frac{5}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1+x^2}} \, dx &=\frac{\sqrt{1-x^2} \int \frac{1}{\sqrt{2-5 x^2} \sqrt{1-x^2}} \, dx}{\sqrt{-1+x^2}}\\ &=\frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|\frac{5}{2}\right )}{\sqrt{2} \sqrt{-1+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0241583, size = 40, normalized size = 1.25 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right ),\frac{2}{5}\right )}{\sqrt{5} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 29, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}{\it EllipticF} \left ( x,{\frac{\sqrt{10}}{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{-5 \, 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{-5 \, x^{2} + 2}}{5 \, x^{4} - 7 \, x^{2} + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 4.58751, size = 37, normalized size = 1.16 \begin{align*} \begin{cases} - \frac{\sqrt{5} i F\left (\operatorname{asin}{\left (\frac{\sqrt{10} x}{2} \right )}\middle | \frac{2}{5}\right )}{5} & \text{for}\: x > - \frac{\sqrt{10}}{5} \wedge x < \frac{\sqrt{10}}{5} \end{cases} \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{-5 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]