\(\int \frac {x-x^2}{\sqrt {1-x^2}} \, dx\) [44]

Optimal result

Integrand size = 19, antiderivative size = 27 \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {\arcsin (x)}{2} \]

[Out]

-1/2*arcsin(x)-1/2*(2-x)*(-x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1607, 794, 222} \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=-\frac {\arcsin (x)}{2}-\frac {1}{2} \sqrt {1-x^2} (2-x) \]

[In]

Int[(x - x^2)/Sqrt[1 - x^2],x]

[Out]

-1/2*((2 - x)*Sqrt[1 - x^2]) - ArcSin[x]/2

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-x) x}{\sqrt {1-x^2}} \, dx \\ & = -\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {1}{2} \sin ^{-1}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} (2-x) \sqrt {1-x^2}+\arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]

[In]

Integrate[(x - x^2)/Sqrt[1 - x^2],x]

[Out]

-1/2*((2 - x)*Sqrt[1 - x^2]) + ArcTan[Sqrt[1 - x^2]/(1 + x)]

Maple [A] (verified)

Time = 3.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {\left (-2+x \right ) \left (x^{2}-1\right )}{2 \sqrt {-x^{2}+1}}-\frac {\arcsin \left (x \right )}{2}\)	\(25\)
default	\(\frac {x \sqrt {-x^{2}+1}}{2}-\frac {\arcsin \left (x \right )}{2}-\sqrt {-x^{2}+1}\)	\(29\)
trager	\(\left (-1+\frac {x}{2}\right ) \sqrt {-x^{2}+1}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{2}\)	\(45\)
meijerg	\(-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}-\frac {i \left (i \sqrt {\pi }\, x \sqrt {-x^{2}+1}-i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{2 \sqrt {\pi }}\)	\(58\)

[In]

int((-x^2+x)/(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-2+x)*(x^2-1)/(-x^2+1)^(1/2)-1/2*arcsin(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} + \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*(x - 2) + arctan((sqrt(-x^2 + 1) - 1)/x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=\frac {x \sqrt {1 - x^{2}}}{2} - \sqrt {1 - x^{2}} - \frac {\operatorname {asin}{\left (x \right )}}{2} \]

[In]

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

[Out]

x*sqrt(1 - x**2)/2 - sqrt(1 - x**2) - asin(x)/2

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1} - \frac {1}{2} \, \arcsin \left (x\right ) \]

[In]

integrate((-x^2+x)/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1) - 1/2*arcsin(x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} - \frac {1}{2} \, \arcsin \left (x\right ) \]

[In]

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

[Out]

1/2*sqrt(-x^2 + 1)*(x - 2) - 1/2*arcsin(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx=\left (\frac {x}{2}-1\right )\,\sqrt {1-x^2}-\frac {\mathrm {asin}\left (x\right )}{2} \]

[In]

int((x - x^2)/(1 - x^2)^(1/2),x)

[Out]

(x/2 - 1)*(1 - x^2)^(1/2) - asin(x)/2