3.1069 \(\int \frac{\sqrt{1+x}}{(1-x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0037246, 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} \[ \frac{2 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x) \]

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 = 0.0130225, size = 36, normalized size = 1.57 \[ 2 \left (\frac{\sqrt{x+1}}{\sqrt{1-x}}+\sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.022, size = 64, normalized size = 2.8 \begin{align*} 2\,{\frac{\sqrt{1+x}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }\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)^(1/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(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 = 1.56908, 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 = 1.50013, 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 [A]  time = 1.57789, size = 71, normalized size = 3.09 \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}} & \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}} & \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), Abs(x + 1)/2 > 1), (-2*asin(sqrt(2)
*sqrt(x + 1)/2) + 2*sqrt(x + 1)/sqrt(1 - x), True))

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Giac [A]  time = 1.07497, size = 45, normalized size = 1.96 \begin{align*} -\frac{2 \, \sqrt{x + 1} \sqrt{-x + 1}}{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]

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