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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]