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

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]

ArcTan[(3 - Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - 2*ArcTanh[(1 - 2*Sqrt[x])/(2*
Sqrt[-1 - Sqrt[x] + x])] - ArcTanh[(1 + 3*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])]

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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]

ArcTan[(3 - Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - 2*ArcTanh[(1 - 2*Sqrt[x])/(2*
Sqrt[-1 - Sqrt[x] + x])] - ArcTanh[(1 + 3*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])]

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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]

2*Integral(-sqrt(x**2 - sqrt(x**2) - 1)/(2*(x + 1)), (x, sqrt(x))) + 2*Integral(
sqrt(x**2 - sqrt(x**2) - 1)/(2*x - 2), (x, sqrt(x)))

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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]

-ArcTan[(-3 + Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - Log[1 + Sqrt[x]] + 2*Log[1
- 2*Sqrt[x] - 2*Sqrt[-1 - Sqrt[x] + x]] + Log[1 + 3*Sqrt[x] - 2*Sqrt[-1 - Sqrt[x
] + x]]

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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]

-((1+x^(1/2))^2-2-3*x^(1/2))^(1/2)+3/2*ln(-1/2+x^(1/2)+((1+x^(1/2))^2-2-3*x^(1/2
))^(1/2))+arctanh(1/2*(-1-3*x^(1/2))/((1+x^(1/2))^2-2-3*x^(1/2))^(1/2))+((-1+x^(
1/2))^2+x^(1/2)-2)^(1/2)+1/2*ln(-1/2+x^(1/2)+((-1+x^(1/2))^2+x^(1/2)-2)^(1/2))-a
rctan(1/2*(x^(1/2)-3)/((-1+x^(1/2))^2+x^(1/2)-2)^(1/2))

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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]

integrate(sqrt(x - sqrt(x) - 1)/((x - 1)*sqrt(x)), x)

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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]

Timed 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]

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

[Out]

Exception raised: TypeError