Optimal. Leaf size=68 \[ \frac{2}{3} \left (x+\sqrt{x-1}\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x-1}+1\right ) \sqrt{x+\sqrt{x-1}}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right ) \]
[Out]
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Rubi [A] time = 0.0772413, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{2}{3} \left (x+\sqrt{x-1}\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x-1}+1\right ) \sqrt{x+\sqrt{x-1}}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[Sqrt[-1 + x] + x],x]
[Out]
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Rubi in Sympy [A] time = 2.52094, size = 66, normalized size = 0.97 \[ \frac{2 \left (x + \sqrt{x - 1}\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{x + \sqrt{x - 1}} \left (2 \sqrt{x - 1} + 1\right )}{4} - \frac{3 \operatorname{atanh}{\left (\frac{2 \sqrt{x - 1} + 1}{2 \sqrt{x + \sqrt{x - 1}}} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x+(-1+x)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0365456, size = 54, normalized size = 0.79 \[ \frac{1}{12} \sqrt{x+\sqrt{x-1}} \left (8 x+2 \sqrt{x-1}-3\right )-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[Sqrt[-1 + x] + x],x]
[Out]
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Maple [A] time = 0.008, size = 48, normalized size = 0.7 \[{\frac{2}{3} \left ( x+\sqrt{-1+x} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4} \left ( 1+2\,\sqrt{-1+x} \right ) \sqrt{x+\sqrt{-1+x}}}-{\frac{3}{8}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{-1+x}+{\frac{1}{2}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x+(-1+x)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x + \sqrt{x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.606246, size = 80, normalized size = 1.18 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x - 1} - 3\right )} \sqrt{x + \sqrt{x - 1}} + \frac{3}{16} \, \log \left (-4 \, \sqrt{x + \sqrt{x - 1}}{\left (2 \, \sqrt{x - 1} + 1\right )} + 8 \, x + 8 \, \sqrt{x - 1} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x + \sqrt{x - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x+(-1+x)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.3034, size = 72, normalized size = 1.06 \[ \frac{1}{12} \,{\left (2 \, \sqrt{x - 1}{\left (4 \, \sqrt{x - 1} + 1\right )} + 5\right )} \sqrt{x + \sqrt{x - 1}} + \frac{3}{8} \,{\rm ln}\left (2 \, \sqrt{x + \sqrt{x - 1}} - 2 \, \sqrt{x - 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x - 1)),x, algorithm="giac")
[Out]