Optimal. Leaf size=39 \[ -\frac{\text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right ),\frac{1}{42} \left (21+5 \sqrt{21}\right )\right )}{\sqrt [4]{21}} \]
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Rubi [A] time = 0.0725563, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 420} \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )|\frac{1}{42} \left (21+5 \sqrt{21}\right )\right )}{\sqrt [4]{21}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 420
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+5 x^2-x^4}} \, dx &=2 \int \frac{1}{\sqrt{5+\sqrt{21}-2 x^2} \sqrt{-5+\sqrt{21}+2 x^2}} \, dx\\ &=-\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )|\frac{1}{42} \left (21+5 \sqrt{21}\right )\right )}{\sqrt [4]{21}}\\ \end{align*}
Mathematica [B] time = 0.107016, size = 87, normalized size = 2.23 \[ \frac{\sqrt{-2 x^2-\sqrt{21}+5} \sqrt{\left (\sqrt{21}-5\right ) x^2+2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right ),\frac{23}{2}-\frac{5 \sqrt{21}}{2}\right )}{2 \sqrt{-x^4+5 x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.209, size = 82, normalized size = 2.1 \begin{align*}{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{7}}{2}}-{\frac{\sqrt{3}}{2}} \right ) ,{\frac{5}{2}}+{\frac{\sqrt{21}}{2}} \right ) }{{\frac{\sqrt{7}}{2}}-{\frac{\sqrt{3}}{2}}}\sqrt{1- \left ({\frac{5}{2}}-{\frac{\sqrt{21}}{2}} \right ){x}^{2}}\sqrt{1- \left ({\frac{5}{2}}+{\frac{\sqrt{21}}{2}} \right ){x}^{2}}{\frac{1}{\sqrt{-{x}^{4}+5\,{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 5 \, x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 5 \, x^{2} - 1}}{x^{4} - 5 \, x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x^{4} + 5 x^{2} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 5 \, x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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