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

Optimal. Leaf size=44 \[ -\frac{1}{2} \left (\frac{1}{x^2}-1\right )^{3/2} x^2+\frac{3}{2} \sqrt{\frac{1}{x^2}-1}-\frac{3}{2} \tan ^{-1}\left (\sqrt{\frac{1}{x^2}-1}\right ) \]

[Out]

(3*Sqrt[-1 + x^(-2)])/2 - ((-1 + x^(-2))^(3/2)*x^2)/2 - (3*ArcTan[Sqrt[-1 + x^(-
2)]])/2

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Rubi [A]  time = 0.0409153, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{2} \left (\frac{1}{x^2}-1\right )^{3/2} x^2+\frac{3}{2} \sqrt{\frac{1}{x^2}-1}-\frac{3}{2} \tan ^{-1}\left (\sqrt{\frac{1}{x^2}-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + x^(-2)]*(-1 + x^2))/x,x]

[Out]

(3*Sqrt[-1 + x^(-2)])/2 - ((-1 + x^(-2))^(3/2)*x^2)/2 - (3*ArcTan[Sqrt[-1 + x^(-
2)]])/2

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Rubi in Sympy [A]  time = 2.97041, size = 42, normalized size = 0.95 \[ - \frac{x^{2} \left (-1 + \frac{1}{x^{2}}\right )^{\frac{3}{2}}}{2} + \frac{3 \sqrt{-1 + \frac{1}{x^{2}}}}{2} - \frac{3 \operatorname{atan}{\left (\sqrt{-1 + \frac{1}{x^{2}}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*(-1 + x**(-2))**(3/2)/2 + 3*sqrt(-1 + x**(-2))/2 - 3*atan(sqrt(-1 + x**(-2
)))/2

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Mathematica [A]  time = 0.0261215, size = 53, normalized size = 1.2 \[ \frac{\sqrt{\frac{1}{x^2}-1} \left (\sqrt{x^2-1} \left (x^2+2\right )-3 x \log \left (\sqrt{x^2-1}+x\right )\right )}{2 \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + x^(-2)]*(-1 + x^2))/x,x]

[Out]

(Sqrt[-1 + x^(-2)]*(Sqrt[-1 + x^2]*(2 + x^2) - 3*x*Log[x + Sqrt[-1 + x^2]]))/(2*
Sqrt[-1 + x^2])

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Maple [A]  time = 0.011, size = 55, normalized size = 1.3 \[{\frac{1}{2}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}} \left ( 2\, \left ( -{x}^{2}+1 \right ) ^{3/2}+3\,{x}^{2}\sqrt{-{x}^{2}+1}+3\,\arcsin \left ( x \right ) x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(-(x^2-1)/x^2)^(1/2)*(2*(-x^2+1)^(3/2)+3*x^2*(-x^2+1)^(1/2)+3*arcsin(x)*x)/(
-x^2+1)^(1/2)

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Maxima [A]  time = 0.800166, size = 41, normalized size = 0.93 \[ \frac{1}{2} \, x^{2} \sqrt{\frac{1}{x^{2}} - 1} + \sqrt{\frac{1}{x^{2}} - 1} - \frac{3}{2} \, \arctan \left (\sqrt{\frac{1}{x^{2}} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.26889, size = 58, normalized size = 1.32 \[ \frac{1}{2} \,{\left (x^{2} + 2\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}} - 3 \, \arctan \left (\frac{x \sqrt{-\frac{x^{2} - 1}{x^{2}}} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.4937, size = 112, normalized size = 2.55 \[ - \begin{cases} - \frac{i x}{\sqrt{x^{2} - 1}} + i \operatorname{acosh}{\left (x \right )} + \frac{i}{x \sqrt{x^{2} - 1}} & \text{for}\: \left |{x^{2}}\right | > 1 \\\frac{x}{\sqrt{- x^{2} + 1}} - \operatorname{asin}{\left (x \right )} - \frac{1}{x \sqrt{- x^{2} + 1}} & \text{otherwise} \end{cases} + \begin{cases} \frac{i x^{3}}{2 \sqrt{x^{2} - 1}} - \frac{i x}{2 \sqrt{x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (x \right )}}{2} & \text{for}\: \left |{x^{2}}\right | > 1 \\\frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Piecewise((-I*x/sqrt(x**2 - 1) + I*acosh(x) + I/(x*sqrt(x**2 - 1)), Abs(x**2) >
1), (x/sqrt(-x**2 + 1) - asin(x) - 1/(x*sqrt(-x**2 + 1)), True)) + Piecewise((I
*x**3/(2*sqrt(x**2 - 1)) - I*x/(2*sqrt(x**2 - 1)) - I*acosh(x)/2, Abs(x**2) > 1)
, (x*sqrt(-x**2 + 1)/2 + asin(x)/2, True))

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GIAC/XCAS [A]  time = 0.269938, size = 77, normalized size = 1.75 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x{\rm sign}\left (x\right ) + \frac{3}{2} \, \arcsin \left (x\right ){\rm sign}\left (x\right ) - \frac{x{\rm sign}\left (x\right )}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}{\rm sign}\left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)*sqrt(1/x^2 - 1)/x,x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 + 1)*x*sign(x) + 3/2*arcsin(x)*sign(x) - 1/2*x*sign(x)/(sqrt(-x^2
+ 1) - 1) + 1/2*(sqrt(-x^2 + 1) - 1)*sign(x)/x