Optimal. Leaf size=62 \[ -\frac {2^{-2-m} \sqrt {x^2} \left (2-4 x^2\right )^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\left (1-2 x^2\right )^2\right )}{(1+m) x} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
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 = {440}
\begin {gather*} x F_1\left (\frac {1}{2};-m,\frac {1}{2};\frac {3}{2};2 x^2,x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 440
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] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 1.39, size = 122, normalized size = 1.97 \begin {gather*} \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 (3 F_1\left (\frac {1}{2};-m,\frac {1}{2};\frac {3}{2};2 x^2,x^2\right )+x^2 \left (-4 m F_1\left (\frac {3}{2};1-m,\frac {1}{2};\frac {5}{2};2 x^2,x^2\right )+F_1\left (\frac {3}{2};-m,\frac {3}{2};\frac {5}{2};2 x^2,x^2\right )\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (1 - 2 x^{2}\right )^{m}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-2\,x^2\right )}^m}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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