3.289 \(\int \frac{\left (-\sqrt{1-x}-\sqrt{1+x}\right ) \left (\sqrt{1-x}+\sqrt{1+x}\right )}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.451368, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{1-x^2}}{x^2}+\frac{1}{x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 0.0566216, size = 85, normalized size = 2.58 \[ \frac{\sqrt{1-x^2}}{x^2}+\frac{1}{x^2}+\log \left (1-\sqrt{x+1}\right )-\log \left (\sqrt{1-x}-\sqrt{x+1}+2\right )-\log \left (\sqrt{x+1}+1\right )+\log \left (\sqrt{1-x}+\sqrt{x+1}+2\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.003, size = 57, normalized size = 1.7 \[{x}^{-2}-{\frac{1}{{x}^{2}}\sqrt{1-x}\sqrt{1+x} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}-\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.764547, size = 69, normalized size = 2.09 \[ \sqrt{-x^{2} + 1} + \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{x^{2}} + \frac{1}{x^{2}} - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x^2 + 1) + (-x^2 + 1)^(3/2)/x^2 + 1/x^2 - log(2*sqrt(-x^2 + 1)/abs(x) + 2/
abs(x))

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Fricas [A]  time = 0.271468, size = 104, normalized size = 3.15 \[ \frac{{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + \sqrt{x + 1} \sqrt{-x + 1} - 1}{x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{2}{x^{3}}\, dx - \int \frac{2 \sqrt{- x + 1} \sqrt{x + 1}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(2/x**3, x) - Integral(2*sqrt(-x + 1)*sqrt(x + 1)/x**3, x)

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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(-(sqrt(x + 1) + sqrt(-x + 1))^2/x^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError