Optimal. Leaf size=62 \[ -\frac{2^{-m-2} \sqrt{x^2} \left (2-4 x^2\right )^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\left (1-2 x^2\right )^2\right )}{(m+1) x} \]
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Rubi [C] time = 0.010505, antiderivative size = 23, normalized size of antiderivative = 0.37, 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 = {429} \[ x F_1\left (\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right ) \]
Warning: Unable to verify antiderivative.
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Rule 429
Rubi steps
\begin{align*} \int \frac{\left (1-2 x^2\right )^m}{\sqrt{1-x^2}} \, dx &=x F_1\left (\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right )\\ \end{align*}
Mathematica [C] time = 0.123661, size = 122, normalized size = 1.97 \[ \frac{3 x \left (1-2 x^2\right )^m F_1\left (\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right )}{\sqrt{1-x^2} \left (x^2 \left (F_1\left (\frac{3}{2};-m,\frac{3}{2};\frac{5}{2};2 x^2,x^2\right )-4 m F_1\left (\frac{3}{2};1-m,\frac{1}{2};\frac{5}{2};2 x^2,x^2\right )\right )+3 F_1\left (\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -2\,{x}^{2}+1 \right ) ^{m}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \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{{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt{-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^{2} + 1}{\left (-2 \, x^{2} + 1\right )}^{m}}{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{\left (1 - 2 x^{2}\right )^{m}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, 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{{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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