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

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

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 \sqrt {1-x}}{\sqrt {x+1}}-\sin ^{-1}(x) \end {gather*}

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 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 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 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*}

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Mathematica [A]  time = 0.04, size = 34, normalized size = 1.48 \begin {gather*} 2 \left (\frac {x-1}{\sqrt {1-x^2}}+\sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.04, size = 39, normalized size = 1.70 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right )-\frac {2 \sqrt {1-x}}{\sqrt {x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.77, size = 50, normalized size = 2.17 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.69, size = 55, normalized size = 2.39 \begin {gather*} \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 {gather*}

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)
)

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maple [B]  time = 0.02, size = 67, normalized size = 2.91 \begin {gather*} -\frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}+\frac {2 \left (x -1\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{\sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.86, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{x + 1} - \arcsin \relax (x) \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {1-x}}{{\left (x+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/2)/(x + 1)^(3/2),x)

[Out]

int((1 - x)^(1/2)/(x + 1)^(3/2), x)

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sympy [B]  time = 1.54, size = 104, normalized size = 4.52 \begin {gather*} \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 {gather*}

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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