3.1119 \(\int \frac{\sqrt{1-x}}{(1+x)^{3/2}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 \sqrt{1-x}}{\sqrt{x+1}}-\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0033808, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {47, 41, 216} \[ -\frac{2 \sqrt{1-x}}{\sqrt{x+1}}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/(1 + x)^(3/2),x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

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{\sqrt{1-x}}{(1+x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-x}}{\sqrt{1+x}}-\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 \sqrt{1-x}}{\sqrt{1+x}}-\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 \sqrt{1-x}}{\sqrt{1+x}}-\sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0368303, size = 34, normalized size = 1.48 \[ 2 \left (\frac{x-1}{\sqrt{1-x^2}}+\sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x]/(1 + x)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.011, size = 67, normalized size = 2.9 \begin{align*} 2\,{\frac{ \left ( -1+x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }\sqrt{1-x}\sqrt{1+x}}}-{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(1/2)/(1+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.53654, size = 28, normalized size = 1.22 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{x + 1} - \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(1/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

-2*sqrt(-x^2 + 1)/(x + 1) - arcsin(x)

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Fricas [B]  time = 1.76957, size = 131, normalized size = 5.7 \begin{align*} \frac{2 \,{\left ({\left (x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - x - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(1/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.46917, size = 104, normalized size = 4.52 \begin{align*} \begin{cases} 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{2 i \sqrt{x + 1}}{\sqrt{x - 1}} + \frac{4 i}{\sqrt{x - 1} \sqrt{x + 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{2 \sqrt{x + 1}}{\sqrt{1 - x}} - \frac{4}{\sqrt{1 - x} \sqrt{x + 1}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(1/2)/(1+x)**(3/2),x)

[Out]

Piecewise((2*I*acosh(sqrt(2)*sqrt(x + 1)/2) - 2*I*sqrt(x + 1)/sqrt(x - 1) + 4*I/(sqrt(x - 1)*sqrt(x + 1)), Abs
(x + 1)/2 > 1), (-2*asin(sqrt(2)*sqrt(x + 1)/2) + 2*sqrt(x + 1)/sqrt(1 - x) - 4/(sqrt(1 - x)*sqrt(x + 1)), Tru
e))

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Giac [B]  time = 1.06955, size = 74, normalized size = 3.22 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}} - 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(1/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

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