3.844 \(\int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\)

Optimal. Leaf size=35 \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + ArcCosh[Sqrt[x]]

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Rubi [A]  time = 0.0747282, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + ArcCosh[Sqrt[x]]

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Rubi in Sympy [A]  time = 10.1108, size = 31, normalized size = 0.89 \[ \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1} + \operatorname{acosh}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0223921, size = 59, normalized size = 1.69 \[ \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + Log[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + S
qrt[x]] + Sqrt[x]]

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Maple [A]  time = 0.013, size = 41, normalized size = 1.2 \[{1\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( \sqrt{x}\sqrt{-1+x}+\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.39534, size = 32, normalized size = 0.91 \[ \sqrt{x - 1} \sqrt{x} + \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220266, size = 153, normalized size = 4.37 \[ -\frac{2 \,{\left (4 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} + 2 \,{\left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) + 6 \, x + 1}{4 \,{\left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*(4*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 8*x^2 + 2*(2*sqr
t(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)*log(2*sqrt(x)*sqrt(sqrt(x) +
1)*sqrt(sqrt(x) - 1) - 2*x + 1) + 6*x + 1)/(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sq
rt(x) - 1) - 2*x + 1)

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Sympy [A]  time = 8.50282, size = 83, normalized size = 2.37 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{x}} \right )}}{2 \pi ^{\frac{3}{2}}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x}} \right )}}{2 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError