Optimal. Leaf size=53 \[ \frac{\sqrt{3 x^2+2} \text{EllipticF}\left (\tan ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0094579, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {418} \[ \frac{\sqrt{3 x^2+2} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 418
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-x^2} \sqrt{2+3 x^2}} \, dx &=\frac{\sqrt{2+3 x^2} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{-1-x^2} \sqrt{\frac{2+3 x^2}{1+x^2}}}\\ \end{align*}
Mathematica [C] time = 0.0228308, size = 39, normalized size = 0.74 \[ -\frac{i \sqrt{x^2+1} \text{EllipticF}\left (i \sinh ^{-1}(x),\frac{3}{2}\right )}{\sqrt{2} \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.024, size = 36, normalized size = 0.7 \begin{align*}{{\frac{i}{3}}\sqrt{3}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{6},{\frac{\sqrt{6}}{3}} \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{3 \, x^{2} + 2} \sqrt{-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{3 \, x^{2} + 2} \sqrt{-x^{2} - 1}}{3 \, x^{4} + 5 \, 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{- x^{2} - 1} \sqrt{3 x^{2} + 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{3 \, x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]