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

Optimal. Leaf size=26 \[ \frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 + ArcCosh[x]/2

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Rubi [A]  time = 0.0337403, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 + ArcCosh[x]/2

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Rubi in Sympy [A]  time = 3.00214, size = 20, normalized size = 0.77 \[ \frac{x \sqrt{x - 1} \sqrt{x + 1}}{2} + \frac{\operatorname{acosh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x - 1)*sqrt(x + 1)/2 + acosh(x)/2

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Mathematica [A]  time = 0.0300442, size = 34, normalized size = 1.31 \[ \frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 + ArcSinh[Sqrt[-1 + x]/Sqrt[2]]

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Maple [B]  time = 0.02, size = 40, normalized size = 1.5 \[{\frac{1}{2}\sqrt{-1+x}\sqrt{1+x} \left ( x\sqrt{{x}^{2}-1}+\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35154, size = 36, normalized size = 1.38 \[ \frac{1}{2} \, \sqrt{x^{2} - 1} x + \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229364, size = 127, normalized size = 4.88 \[ \frac{2 \, x^{4} -{\left (2 \, x^{3} - x\right )} \sqrt{x + 1} \sqrt{x - 1} - 2 \, x^{2} -{\left (2 \, \sqrt{x + 1} \sqrt{x - 1} x - 2 \, x^{2} + 1\right )} \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right )}{2 \,{\left (2 \, \sqrt{x + 1} \sqrt{x - 1} x - 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*x^4 - (2*x^3 - x)*sqrt(x + 1)*sqrt(x - 1) - 2*x^2 - (2*sqrt(x + 1)*sqrt(x
- 1)*x - 2*x^2 + 1)*log(sqrt(x + 1)*sqrt(x - 1) - x))/(2*sqrt(x + 1)*sqrt(x - 1
)*x - 2*x^2 + 1)

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Sympy [A]  time = 12.7845, size = 87, normalized size = 3.35 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0), ()),
x**(-2))/(4*pi**(3/2)) - I*meijerg(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4
, -3/4), (-3/2, -1, -1, 0)), exp_polar(2*I*pi)/x**2)/(4*pi**(3/2))

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GIAC/XCAS [A]  time = 0.239845, size = 42, normalized size = 1.62 \[ \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x -{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x + 1)*sqrt(x - 1)*x - ln(abs(-sqrt(x + 1) + sqrt(x - 1)))