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

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]

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

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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]

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

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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]

2*sqrt(x + sqrt(x + 1)) - atanh((2*sqrt(x + 1) + 1)/(2*sqrt(x + sqrt(x + 1))))

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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]

2*Sqrt[x + Sqrt[1 + x]] - Log[1 + 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]]

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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]

2*(x+(1+x)^(1/2))^(1/2)-ln((1+x)^(1/2)+1/2+(x+(1+x)^(1/2))^(1/2))

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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]

integrate(1/sqrt(x + sqrt(x + 1)), x)

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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]

2*sqrt(x + sqrt(x + 1)) + 1/2*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) -
8*x - 8*sqrt(x + 1) - 5)

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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]

Integral(1/sqrt(x + sqrt(x + 1)), x)

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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]

Exception raised: NotImplementedError