Optimal. Leaf size=33 \[ -\frac{\sqrt{1-x^2}}{x}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+\sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.119516, antiderivative size = 33, 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 \[ -\frac{\sqrt{1-x^2}}{x}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 + x)^2/(x^2*Sqrt[1 - x^2]),x]
[Out]
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Rubi in Sympy [A] time = 8.45711, size = 24, normalized size = 0.73 \[ \operatorname{asin}{\left (x \right )} - 2 \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )} - \frac{\sqrt{- x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)**2/x**2/(-x**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0317497, size = 39, normalized size = 1.18 \[ -\frac{\sqrt{1-x^2}}{x}-2 \log \left (\sqrt{1-x^2}+1\right )+2 \log (x)+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x)^2/(x^2*Sqrt[1 - x^2]),x]
[Out]
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Maple [A] time = 0.011, size = 30, normalized size = 0.9 \[ \arcsin \left ( x \right ) -{\frac{1}{x}\sqrt{-{x}^{2}+1}}-2\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)^2/x^2/(-x^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 0.786493, size = 57, normalized size = 1.73 \[ -\frac{\sqrt{-x^{2} + 1}}{x} + \arcsin \left (x\right ) - 2 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279256, size = 132, normalized size = 4. \[ \frac{x^{2} - 2 \,{\left (\sqrt{-x^{2} + 1} x - x\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + 2 \,{\left (\sqrt{-x^{2} + 1} x - x\right )} \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + \sqrt{-x^{2} + 1} - 1}{\sqrt{-x^{2} + 1} x - x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.28067, size = 51, normalized size = 1.55 \[ \begin{cases} - \frac{i \sqrt{x^{2} - 1}}{x} & \text{for}\: \left |{x^{2}}\right | > 1 \\- \frac{\sqrt{- x^{2} + 1}}{x} & \text{otherwise} \end{cases} + 2 \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{x} \right )} & \text{for}\: \left |{\frac{1}{x^{2}}}\right | > 1 \\i \operatorname{asin}{\left (\frac{1}{x} \right )} & \text{otherwise} \end{cases}\right ) + \operatorname{asin}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)**2/x**2/(-x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.28351, size = 74, normalized size = 2.24 \[ \frac{x}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{\sqrt{-x^{2} + 1} - 1}{2 \, x} + \arcsin \left (x\right ) + 2 \,{\rm ln}\left (-\frac{\sqrt{-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^2),x, algorithm="giac")
[Out]