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