Optimal. Leaf size=44 \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
[Out]
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Rubi [A] time = 0.0994702, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + x)^(3/2)/(Sqrt[1 - x]*x^2),x]
[Out]
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Rubi in Sympy [A] time = 8.60404, size = 34, normalized size = 0.77 \[ \operatorname{asin}{\left (x \right )} - 2 \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )} - \frac{\sqrt{- x + 1} \sqrt{x + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)**(3/2)/x**2/(1-x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0539123, size = 51, normalized size = 1.16 \[ -\frac{\sqrt{1-x^2}}{x}-2 \log \left (\sqrt{1-x^2}+1\right )+\tan ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )+2 \log (x) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + x)^(3/2)/(Sqrt[1 - x]*x^2),x]
[Out]
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Maple [A] time = 0.017, size = 55, normalized size = 1.3 \[{\frac{1}{x}\sqrt{1-x}\sqrt{1+x} \left ( \arcsin \left ( x \right ) x-2\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x-\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)^(3/2)/x^2/(1-x)^(1/2),x)
[Out]
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Maxima [A] time = 1.47844, size = 57, normalized size = 1.3 \[ -\frac{\sqrt{-x^{2} + 1}}{x} + \arcsin \left (x\right ) - 2 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)/(x^2*sqrt(-x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222333, size = 161, normalized size = 3.66 \[ \frac{x^{2} - 2 \,{\left (\sqrt{x + 1} x \sqrt{-x + 1} - x\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 2 \,{\left (\sqrt{x + 1} x \sqrt{-x + 1} - x\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + \sqrt{x + 1} \sqrt{-x + 1} - 1}{\sqrt{x + 1} x \sqrt{-x + 1} - x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)/(x^2*sqrt(-x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x + 1\right )^{\frac{3}{2}}}{x^{2} \sqrt{- x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)**(3/2)/x**2/(1-x)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)/(x^2*sqrt(-x + 1)),x, algorithm="giac")
[Out]