3.298 \(\int \frac{\left (1-2 x^2\right )^m}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2^{-m-2} \sqrt{x^2} \left (2-4 x^2\right )^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\left (1-2 x^2\right )^2\right )}{(m+1) x} \]

[Out]

-((2^(-2 - m)*Sqrt[x^2]*(2 - 4*x^2)^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3
+ m)/2, (1 - 2*x^2)^2])/((1 + m)*x))

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Rubi [C]  time = 0.0345361, 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 \[ x F_1\left (\frac{1}{2};-m,\frac{1}{2};\frac{3}{2};2 x^2,x^2\right ) \]

Warning: Unable to verify antiderivative.

[In]  Int[(1 - 2*x^2)^m/Sqrt[1 - x^2],x]

[Out]

x*AppellF1[1/2, -m, 1/2, 3/2, 2*x^2, x^2]

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Rubi in Sympy [A]  time = 6.13962, size = 15, normalized size = 0.24 \[ x \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{2},- m,\frac{3}{2},x^{2},2 x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*x**2+1)**m/(-x**2+1)**(1/2),x)

[Out]

x*appellf1(1/2, 1/2, -m, 3/2, x**2, 2*x**2)

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Mathematica [C]  time = 0.193788, 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.

[In]  Integrate[(1 - 2*x^2)^m/Sqrt[1 - x^2],x]

[Out]

(3*x*(1 - 2*x^2)^m*AppellF1[1/2, -m, 1/2, 3/2, 2*x^2, x^2])/(Sqrt[1 - x^2]*(3*Ap
pellF1[1/2, -m, 1/2, 3/2, 2*x^2, x^2] + x^2*(-4*m*AppellF1[3/2, 1 - m, 1/2, 5/2,
2*x^2, x^2] + AppellF1[3/2, -m, 3/2, 5/2, 2*x^2, x^2])))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{ \left ( -2\,{x}^{2}+1 \right ) ^{m}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*x^2+1)^m/(-x^2+1)^(1/2),x)

[Out]

int((-2*x^2+1)^m/(-x^2+1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x^2 + 1)^m/sqrt(-x^2 + 1),x, algorithm="maxima")

[Out]

integrate((-2*x^2 + 1)^m/sqrt(-x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt{-x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x^2 + 1)^m/sqrt(-x^2 + 1),x, algorithm="fricas")

[Out]

integral((-2*x^2 + 1)^m/sqrt(-x^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- 2 x^{2} + 1\right )^{m}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x**2+1)**m/(-x**2+1)**(1/2),x)

[Out]

Integral((-2*x**2 + 1)**m/sqrt(-(x - 1)*(x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x^2 + 1)^m/sqrt(-x^2 + 1),x, algorithm="giac")

[Out]

integrate((-2*x^2 + 1)^m/sqrt(-x^2 + 1), x)