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

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 = 0.010505, 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.

[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]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

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.

[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*AppellF1[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 = 0.04, 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.

[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. \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.

[In]

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

[In]

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

[Out]

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

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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.

[In]

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

[Out]

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

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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.

[In]

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

[Out]

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