3.565 \(\int \frac{1}{\sqrt{1+x+\sqrt{-1+2 x}}} \, dx\)

Optimal. Leaf size=44 \[ 2 \sqrt{x+\sqrt{2 x-1}+1}-\sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.070403, antiderivative size = 52, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ 2 \sqrt{x+\sqrt{2 x-1}+1}-\sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + x + Sqrt[-1 + 2*x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.82141, size = 61, normalized size = 1.39 \[ \sqrt{2} \sqrt{2 x + 2 \sqrt{2 x - 1} + 2} - \sqrt{2} \operatorname{atanh}{\left (\frac{2 \sqrt{2 x - 1} + 2}{2 \sqrt{2 x + 2 \sqrt{2 x - 1} + 2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.036288, size = 44, normalized size = 1. \[ 2 \sqrt{x+\sqrt{2 x-1}+1}-\sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 + x + Sqrt[-1 + 2*x]],x]

[Out]

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

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Maple [A]  time = 0.01, size = 38, normalized size = 0.9 \[ \sqrt{4\,x+4+4\,\sqrt{2\,x-1}}-{\it Arcsinh} \left ({\frac{\sqrt{2}}{2} \left ( 1+\sqrt{2\,x-1} \right ) } \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*x+4+4*(2*x-1)^(1/2))^(1/2)-arcsinh(1/2*(1+(2*x-1)^(1/2))*2^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + \sqrt{2 \, x - 1} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x + sqrt(2*x - 1) + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.728963, size = 115, normalized size = 2.61 \[ \frac{1}{4} \, \sqrt{2} \log \left (-8 \, x^{2} - 8 \,{\left (2 \, x + 1\right )} \sqrt{2 \, x - 1} + 2 \,{\left (\sqrt{2}{\left (2 \, x + 3\right )} \sqrt{2 \, x - 1} + \sqrt{2}{\left (6 \, x - 1\right )}\right )} \sqrt{x + \sqrt{2 \, x - 1} + 1} - 24 \, x + 7\right ) + 2 \, \sqrt{x + \sqrt{2 \, x - 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x + sqrt(2*x - 1) + 1),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log(-8*x^2 - 8*(2*x + 1)*sqrt(2*x - 1) + 2*(sqrt(2)*(2*x + 3)*sqrt(2
*x - 1) + sqrt(2)*(6*x - 1))*sqrt(x + sqrt(2*x - 1) + 1) - 24*x + 7) + 2*sqrt(x
+ sqrt(2*x - 1) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + \sqrt{2 x - 1} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273461, size = 92, normalized size = 2.09 \[ -\sqrt{2}{\left (\sqrt{3} +{\rm ln}\left (\sqrt{3} - 1\right )\right )} + \sqrt{2}{\rm ln}\left (\sqrt{2 \, x + 2 \, \sqrt{2 \, x - 1} + 2} - \sqrt{2 \, x - 1} - 1\right ) + \sqrt{2} \sqrt{2 \, x + 2 \, \sqrt{2 \, x - 1} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x + sqrt(2*x - 1) + 1),x, algorithm="giac")

[Out]

-sqrt(2)*(sqrt(3) + ln(sqrt(3) - 1)) + sqrt(2)*ln(sqrt(2*x + 2*sqrt(2*x - 1) + 2
) - sqrt(2*x - 1) - 1) + sqrt(2)*sqrt(2*x + 2*sqrt(2*x - 1) + 2)