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

Optimal. Leaf size=8 \[ 2 \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*ArcCosh[Sqrt[x]]

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Rubi [A]  time = 0.0425341, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ 2 \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*ArcCosh[Sqrt[x]]

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Rubi in Sympy [A]  time = 6.00859, size = 7, normalized size = 0.88 \[ 2 \operatorname{acosh}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*acosh(sqrt(x))

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Mathematica [B]  time = 0.014149, size = 20, normalized size = 2.5 \[ 4 \sinh ^{-1}\left (\frac{\sqrt{\sqrt{x}-1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4*ArcSinh[Sqrt[-1 + Sqrt[x]]/Sqrt[2]]

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Maple [B]  time = 0.006, size = 40, normalized size = 5. \[ 2\,{\frac{\sqrt{ \left ( 1+\sqrt{x} \right ) \left ( -1+\sqrt{x} \right ) }\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) }{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.40053, size = 22, normalized size = 2.75 \[ 2 \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(2*sqrt(x - 1) + 2*sqrt(x))

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Fricas [A]  time = 0.216603, size = 36, normalized size = 4.5 \[ -\log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(1/(sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="giac")

[Out]

Exception raised: TypeError