Optimal. Leaf size=41 \[ \frac{2 (x+1)^{3/2}}{\sqrt{1-x}}+3 \sqrt{1-x} \sqrt{x+1}-3 \sin ^{-1}(x) \]
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Rubi [A] time = 0.032995, 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 \[ \frac{2 (x+1)^{3/2}}{\sqrt{1-x}}+3 \sqrt{1-x} \sqrt{x+1}-3 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 + x)^(3/2)/(1 - x)^(3/2),x]
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Rubi in Sympy [A] time = 5.33719, size = 34, normalized size = 0.83 \[ 3 \sqrt{- x + 1} \sqrt{x + 1} - 3 \operatorname{asin}{\left (x \right )} + \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)**(3/2)/(1-x)**(3/2),x)
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Mathematica [A] time = 0.0373503, size = 37, normalized size = 0.9 \[ \frac{(x-5) \sqrt{1-x^2}}{x-1}-6 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x)^(3/2)/(1 - x)^(3/2),x]
[Out]
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Maple [B] time = 0.028, size = 72, normalized size = 1.8 \[ -{({x}^{2}-4\,x-5)\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-3\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)^(3/2)/(1-x)^(3/2),x)
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Maxima [A] time = 1.49155, size = 57, normalized size = 1.39 \[ -\frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{x^{2} - 2 \, x + 1} - \frac{6 \, \sqrt{-x^{2} + 1}}{x - 1} - 3 \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)/(-x + 1)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.216187, size = 139, normalized size = 3.39 \[ \frac{x^{3} - x^{2} +{\left (x^{2} - 8 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \,{\left (x^{2} - \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + x - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 8 \, x}{x^{2} - \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + x - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)/(-x + 1)^(3/2),x, algorithm="fricas")
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Sympy [A] time = 12.1097, size = 100, normalized size = 2.44 \[ \begin{cases} 6 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x - 1}} - \frac{6 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} + \frac{6 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)**(3/2)/(1-x)**(3/2),x)
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GIAC/XCAS [A] time = 0.212893, size = 47, normalized size = 1.15 \[ \frac{\sqrt{x + 1}{\left (x - 5\right )} \sqrt{-x + 1}}{x - 1} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)/(-x + 1)^(3/2),x, algorithm="giac")
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