Optimal. Leaf size=47 \[ 2 \sqrt{x+\sqrt{x+1}}-\tanh ^{-1}\left (\frac{2 \sqrt{x+1}+1}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0576661, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \sqrt{x+\sqrt{x+1}}-\tanh ^{-1}\left (\frac{2 \sqrt{x+1}+1}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[x + Sqrt[1 + x]],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 2.2801, size = 37, normalized size = 0.79 \[ 2 \sqrt{x + \sqrt{x + 1}} - \operatorname{atanh}{\left (\frac{2 \sqrt{x + 1} + 1}{2 \sqrt{x + \sqrt{x + 1}}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x+(1+x)**(1/2))**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0217176, size = 45, normalized size = 0.96 \[ 2 \sqrt{x+\sqrt{x+1}}-\log \left (2 \sqrt{x+1}+2 \sqrt{x+\sqrt{x+1}}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[x + Sqrt[1 + x]],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 32, normalized size = 0.7 \[ 2\,\sqrt{x+\sqrt{1+x}}-\ln \left ( \sqrt{1+x}+{\frac{1}{2}}+\sqrt{x+\sqrt{1+x}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x+(1+x)^(1/2))^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + \sqrt{x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x + sqrt(x + 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.476033, size = 63, normalized size = 1.34 \[ 2 \, \sqrt{x + \sqrt{x + 1}} + \frac{1}{2} \, \log \left (4 \, \sqrt{x + \sqrt{x + 1}}{\left (2 \, \sqrt{x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt{x + 1} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x + sqrt(x + 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + \sqrt{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x+(1+x)**(1/2))**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x + sqrt(x + 1)),x, algorithm="giac")
[Out]