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 = 0.01, 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*}
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Mathematica [C] time = 0.13, 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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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + 1} {\left (-2 \, x^{2} + 1\right )}^{m}}{x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (-2 x^{2}+1\right )^{m}}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (1-2\,x^2\right )}^m}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1 - 2 x^{2}\right )^{m}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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