3.1118 \(\int \frac{(1-x)^{3/2}}{(1+x)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 (1-x)^{3/2}}{\sqrt{x+1}}-3 \sqrt{x+1} \sqrt{1-x}-3 \sin ^{-1}(x) \]

[Out]

(-2*(1 - x)^(3/2))/Sqrt[1 + x] - 3*Sqrt[1 - x]*Sqrt[1 + x] - 3*ArcSin[x]

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Rubi [A]  time = 0.0331637, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{2 (1-x)^{3/2}}{\sqrt{x+1}}-3 \sqrt{x+1} \sqrt{1-x}-3 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(3/2)/(1 + x)^(3/2),x]

[Out]

(-2*(1 - x)^(3/2))/Sqrt[1 + x] - 3*Sqrt[1 - x]*Sqrt[1 + x] - 3*ArcSin[x]

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Rubi in Sympy [A]  time = 5.32194, size = 36, normalized size = 0.88 \[ - \frac{2 \left (- x + 1\right )^{\frac{3}{2}}}{\sqrt{x + 1}} - 3 \sqrt{- x + 1} \sqrt{x + 1} - 3 \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(3/2)/(1+x)**(3/2),x)

[Out]

-2*(-x + 1)**(3/2)/sqrt(x + 1) - 3*sqrt(-x + 1)*sqrt(x + 1) - 3*asin(x)

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Mathematica [A]  time = 0.0284615, size = 38, normalized size = 0.93 \[ -\frac{\sqrt{1-x} (x+5)}{\sqrt{x+1}}-6 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - x]*(5 + x))/Sqrt[1 + x]) - 6*ArcSin[Sqrt[1 + x]/Sqrt[2]]

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Maple [B]  time = 0.025, size = 71, normalized size = 1.7 \[{({x}^{2}+4\,x-5)\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-3\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(3/2)/(1+x)^(3/2),x)

[Out]

(x^2+4*x-5)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-3*
((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.51776, size = 55, normalized size = 1.34 \[ \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{x^{2} + 2 \, x + 1} - \frac{6 \, \sqrt{-x^{2} + 1}}{x + 1} - 3 \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-x^2 + 1)^(3/2)/(x^2 + 2*x + 1) - 6*sqrt(-x^2 + 1)/(x + 1) - 3*arcsin(x)

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Fricas [A]  time = 0.206628, size = 140, normalized size = 3.41 \[ \frac{x^{3} + x^{2} -{\left (x^{2} + 8 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \,{\left (x^{2} +{\left (x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - x - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 8 \, x}{x^{2} +{\left (x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^3 + x^2 - (x^2 + 8*x)*sqrt(x + 1)*sqrt(-x + 1) + 6*(x^2 + (x + 2)*sqrt(x + 1)
*sqrt(-x + 1) - x - 2)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 8*x)/(x^2 + (x
+ 2)*sqrt(x + 1)*sqrt(-x + 1) - x - 2)

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Sympy [A]  time = 11.187, size = 133, normalized size = 3.24 \[ \begin{cases} 6 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x - 1}} - \frac{2 i \sqrt{x + 1}}{\sqrt{x - 1}} + \frac{8 i}{\sqrt{x - 1} \sqrt{x + 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} + \frac{2 \sqrt{x + 1}}{\sqrt{- x + 1}} - \frac{8}{\sqrt{- x + 1} \sqrt{x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((6*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(3/2)/sqrt(x - 1) - 2*I
*sqrt(x + 1)/sqrt(x - 1) + 8*I/(sqrt(x - 1)*sqrt(x + 1)), Abs(x + 1)/2 > 1), (-6
*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(3/2)/sqrt(-x + 1) + 2*sqrt(x + 1)/sqrt(
-x + 1) - 8/(sqrt(-x + 1)*sqrt(x + 1)), True))

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GIAC/XCAS [A]  time = 0.218763, size = 95, normalized size = 2.32 \[ -\sqrt{x + 1} \sqrt{-x + 1} + \frac{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} - \frac{2 \, \sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x + 1)*sqrt(-x + 1) + 2*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 2*sqrt(x +
1)/(sqrt(2) - sqrt(-x + 1)) - 6*arcsin(1/2*sqrt(2)*sqrt(x + 1))