Optimal. Leaf size=89 \[ \tan ^{-1}\left (\frac{3-\sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-2 \tanh ^{-1}\left (\frac{1-2 \sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-\tanh ^{-1}\left (\frac{3 \sqrt{x}+1}{2 \sqrt{x-\sqrt{x}-1}}\right ) \]
[Out]
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Rubi [A] time = 0.470015, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \tan ^{-1}\left (\frac{3-\sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-2 \tanh ^{-1}\left (\frac{1-2 \sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right )-\tanh ^{-1}\left (\frac{3 \sqrt{x}+1}{2 \sqrt{x-\sqrt{x}-1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-1 - Sqrt[x] + x]/((-1 + x)*Sqrt[x]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 \int ^{\sqrt{x}} \left (- \frac{\sqrt{x^{2} - \sqrt{x^{2}} - 1}}{2 \left (x + 1\right )}\right )\, dx + 2 \int ^{\sqrt{x}} \frac{\sqrt{x^{2} - \sqrt{x^{2}} - 1}}{2 x - 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+x-x**(1/2))**(1/2)/(-1+x)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0186521, size = 93, normalized size = 1.04 \[ -\log \left (\sqrt{x}+1\right )+2 \log \left (-2 \sqrt{x}-2 \sqrt{x-\sqrt{x}-1}+1\right )+\log \left (3 \sqrt{x}-2 \sqrt{x-\sqrt{x}-1}+1\right )-\tan ^{-1}\left (\frac{\sqrt{x}-3}{2 \sqrt{x-\sqrt{x}-1}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-1 - Sqrt[x] + x]/((-1 + x)*Sqrt[x]),x]
[Out]
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Maple [A] time = 0.017, size = 130, normalized size = 1.5 \[ -\sqrt{ \left ( 1+\sqrt{x} \right ) ^{2}-2-3\,\sqrt{x}}+{\frac{3}{2}\ln \left ( -{\frac{1}{2}}+\sqrt{x}+\sqrt{ \left ( 1+\sqrt{x} \right ) ^{2}-2-3\,\sqrt{x}} \right ) }+{\it Artanh} \left ({\frac{1}{2} \left ( -1-3\,\sqrt{x} \right ){\frac{1}{\sqrt{ \left ( 1+\sqrt{x} \right ) ^{2}-2-3\,\sqrt{x}}}}} \right ) +\sqrt{ \left ( -1+\sqrt{x} \right ) ^{2}+\sqrt{x}-2}+{\frac{1}{2}\ln \left ( -{\frac{1}{2}}+\sqrt{x}+\sqrt{ \left ( -1+\sqrt{x} \right ) ^{2}+\sqrt{x}-2} \right ) }-\arctan \left ({\frac{1}{2} \left ( \sqrt{x}-3 \right ){\frac{1}{\sqrt{ \left ( -1+\sqrt{x} \right ) ^{2}+\sqrt{x}-2}}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+x-x^(1/2))^(1/2)/(-1+x)/x^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x - \sqrt{x} - 1}}{{\left (x - 1\right )} \sqrt{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - sqrt(x) - 1)/((x - 1)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - sqrt(x) - 1)/((x - 1)*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \sqrt{x} + x - 1}}{\sqrt{x} \left (x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-1+x-x**(1/2))**(1/2)/(-1+x)/x**(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(sqrt(x - sqrt(x) - 1)/((x - 1)*sqrt(x)),x, algorithm="giac")
[Out]