3.117 \(\int \frac{x \sqrt{1-x^2}}{1+x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0170649, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {785, 780, 216} \[ -\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[1 - x^2])/(1 + x),x]

[Out]

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

Rule 785

Int[(x_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^m*e^m, Int[(x*(a + c*x^2)^(m
+ p))/(a*e + c*d*x)^m, x], x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[
m, 0] && EqQ[m, -1] &&  !ILtQ[p - 1/2, 0]

Rule 780

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 216

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]

Rubi steps

\begin{align*} \int \frac{x \sqrt{1-x^2}}{1+x} \, 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*}

Mathematica [A]  time = 0.0336552, size = 26, normalized size = 0.96 \[ \left (\frac{x}{2}-1\right ) \sqrt{1-x^2}-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 34, normalized size = 1.3 \begin{align*}{\frac{x}{2}\sqrt{-{x}^{2}+1}}-{\frac{\arcsin \left ( x \right ) }{2}}-\sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.48329, size = 38, normalized size = 1.41 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1} - \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+1)^(1/2)/(1+x),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 = 1.61252, size = 82, normalized size = 3.04 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x - 2\right )} + \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+1)^(1/2)/(1+x),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 = 2.86234, size = 29, normalized size = 1.07 \begin{align*} \begin{cases} \frac{x \sqrt{1 - x^{2}}}{2} - \sqrt{1 - x^{2}} - \frac{\operatorname{asin}{\left (x \right )}}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sqrt(1 - x**2)/2 - sqrt(1 - x**2) - asin(x)/2, (x > -1) & (x < 1)))

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Giac [A]  time = 1.28953, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x - 2\right )} - \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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