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

Optimal. Leaf size=27 \[ -\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1607, 794, 222} \begin {gather*} -\frac {\text {ArcSin}(x)}{2}-\frac {1}{2} \sqrt {1-x^2} (2-x) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {x-x^2}{\sqrt {1-x^2}} \, dx &=\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*}

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Mathematica [A]
time = 0.00, size = 39, normalized size = 1.44 \begin {gather*} -\frac {1}{2} (2-x) \sqrt {1-x^2}+\tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[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)]

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Maple [A]
time = 0.12, size = 29, normalized size = 1.07

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 28, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1} - \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 4.15, size = 31, normalized size = 1.15 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} + \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.07, size = 24, normalized size = 0.89 \begin {gather*} \frac {x \sqrt {1 - x^{2}}}{2} - \sqrt {1 - x^{2}} - \frac {\operatorname {asin}{\left (x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]
time = 0.71, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} - \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.03, size = 20, normalized size = 0.74 \begin {gather*} \left (\frac {x}{2}-1\right )\,\sqrt {1-x^2}-\frac {\mathrm {asin}\left (x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - x^2)/(1 - x^2)^(1/2),x)

[Out]

(x/2 - 1)*(1 - x^2)^(1/2) - asin(x)/2

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