3.59 \(\int \frac{(1+x)^2}{x^2 \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\sqrt{1-x^2}}{x}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+\sin ^{-1}(x) \]

[Out]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

-sqrt(-x^2 + 1)/x + arcsin(x) - 2*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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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]

(x^2 - 2*(sqrt(-x^2 + 1)*x - x)*arctan((sqrt(-x^2 + 1) - 1)/x) + 2*(sqrt(-x^2 +
1)*x - x)*log((sqrt(-x^2 + 1) - 1)/x) + sqrt(-x^2 + 1) - 1)/(sqrt(-x^2 + 1)*x -
x)

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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]

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

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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]

1/2*x/(sqrt(-x^2 + 1) - 1) - 1/2*(sqrt(-x^2 + 1) - 1)/x + arcsin(x) + 2*ln(-(sqr
t(-x^2 + 1) - 1)/abs(x))