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