\(\int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx\) [1119]

Optimal result

Integrand size = 17, antiderivative size = 23 \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=-\frac {2 \sqrt {1-x}}{\sqrt {1+x}}-\arcsin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {49, 41, 222} \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=-\arcsin (x)-\frac {2 \sqrt {1-x}}{\sqrt {x+1}} \]

[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 49

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 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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=-\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+2 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(19)=38\).

Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91

method	result	size
risch	\(\frac {2 \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(67\)

[In]

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (19) = 38\).

Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17 \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=\frac {2 \, {\left ({\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - x - \sqrt {x + 1} \sqrt {-x + 1} - 1\right )}}{x + 1} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.43 \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=\begin {cases} 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {4 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {4}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \]

[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), (-2*asin(sqrt(2)*sqrt(x + 1)/2) + 2*sqrt(x + 1)/sqrt(1 - x) - 4/(sqrt(1 - x)*sqrt(x + 1)), True)
)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{x + 1} - \arcsin \left (x\right ) \]

[In]

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

[Out]

-2*sqrt(-x^2 + 1)/(x + 1) - arcsin(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (19) = 38\).

Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[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)
)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx=\int \frac {\sqrt {1-x}}{{\left (x+1\right )}^{3/2}} \,d x \]

[In]

int((1 - x)^(1/2)/(x + 1)^(3/2),x)

[Out]

int((1 - x)^(1/2)/(x + 1)^(3/2), x)