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

Optimal. Leaf size=36 \[ \frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}} \]

[Out]

(Sqrt[1 - x]*ArcSin[x])/(Sqrt[-1 - Sqrt[x]]*Sqrt[-1 + Sqrt[x]])

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Rubi [A]  time = 0.0708708, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ \frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - x]*ArcSin[x])/(Sqrt[-1 - Sqrt[x]]*Sqrt[-1 + Sqrt[x]])

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Rubi in Sympy [A]  time = 8.37875, size = 31, normalized size = 0.86 \[ \frac{\sqrt{- \sqrt{x} - 1} \sqrt{\sqrt{x} - 1} \operatorname{asin}{\left (x \right )}}{\sqrt{- x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-sqrt(x) - 1)*sqrt(sqrt(x) - 1)*asin(x)/sqrt(-x + 1)

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Mathematica [A]  time = 0.947749, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-1-\sqrt{x}} \sqrt{-1+\sqrt{x}} \sqrt{1+x}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{-1-\sqrt{x}}}}{\frac{1}{\sqrt{-1+\sqrt{x}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(1+x)^(1/2)/(-1-x^(1/2))^(1/2)/(-1+x^(1/2))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236123, size = 93, normalized size = 2.58 \[ -i \, \log \left (\frac{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1} + i \, x - 1}{x}\right ) + i \, \log \left (\frac{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1} - i \, x - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*log((sqrt(x + 1)*sqrt(sqrt(x) - 1)*sqrt(-sqrt(x) - 1) + I*x - 1)/x) + I*log((
sqrt(x + 1)*sqrt(sqrt(x) - 1)*sqrt(-sqrt(x) - 1) - I*x - 1)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError