\(\int \frac {(1-2 x^2)^m}{\sqrt {1-x^2}} \, dx\) [301]

Optimal result

Integrand size = 21, antiderivative size = 62 \[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=-\frac {2^{-2-m} \sqrt {x^2} \left (2-4 x^2\right )^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\left (1-2 x^2\right )^2\right )}{(1+m) x} \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.37, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {440} \[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=x \operatorname {AppellF1}\left (\frac {1}{2},-m,\frac {1}{2},\frac {3}{2},2 x^2,x^2\right ) \]

[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 440

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*} \text {integral}& = x F_1\left (\frac {1}{2};-m,\frac {1}{2};\frac {3}{2};2 x^2,x^2\right ) \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 1.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.97 \[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=\frac {3 x \left (1-2 x^2\right )^m \operatorname {AppellF1}\left (\frac {1}{2},-m,\frac {1}{2},\frac {3}{2},2 x^2,x^2\right )}{\sqrt {1-x^2} \left (3 \operatorname {AppellF1}\left (\frac {1}{2},-m,\frac {1}{2},\frac {3}{2},2 x^2,x^2\right )+x^2 \left (-4 m \operatorname {AppellF1}\left (\frac {3}{2},1-m,\frac {1}{2},\frac {5}{2},2 x^2,x^2\right )+\operatorname {AppellF1}\left (\frac {3}{2},-m,\frac {3}{2},\frac {5}{2},2 x^2,x^2\right )\right )\right )} \]

[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])))

Maple [F]

\[\int \frac {\left (-2 x^{2}+1\right )^{m}}{\sqrt {-x^{2}+1}}d x\]

[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)

Fricas [F]

\[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=\int { \frac {{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt {-x^{2} + 1}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=\int \frac {\left (1 - 2 x^{2}\right )^{m}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=\int { \frac {{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt {-x^{2} + 1}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=\int { \frac {{\left (-2 \, x^{2} + 1\right )}^{m}}{\sqrt {-x^{2} + 1}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-2 x^2\right )^m}{\sqrt {1-x^2}} \, dx=\int \frac {{\left (1-2\,x^2\right )}^m}{\sqrt {1-x^2}} \,d x \]

[In]

int((1 - 2*x^2)^m/(1 - x^2)^(1/2),x)

[Out]

int((1 - 2*x^2)^m/(1 - x^2)^(1/2), x)