Optimal. Leaf size=75 \[ \frac{2}{3} \left (x+\sqrt{x+1}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x+1}+1\right ) \sqrt{x+\sqrt{x+1}+1}+\frac{1}{4} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{x+\sqrt{x+1}+1}}\right ) \]
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Rubi [A] time = 0.0822884, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{2}{3} \left (x+\sqrt{x+1}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x+1}+1\right ) \sqrt{x+\sqrt{x+1}+1}+\frac{1}{4} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{x+\sqrt{x+1}+1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + x + Sqrt[1 + x]],x]
[Out]
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Rubi in Sympy [A] time = 2.68395, size = 65, normalized size = 0.87 \[ - \frac{\left (2 \sqrt{x + 1} + 1\right ) \sqrt{x + \sqrt{x + 1} + 1}}{4} + \frac{2 \left (x + \sqrt{x + 1} + 1\right )^{\frac{3}{2}}}{3} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{x + 1}}{\sqrt{x + \sqrt{x + 1} + 1}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x+(1+x)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0514878, size = 65, normalized size = 0.87 \[ \frac{1}{24} \left (2 \sqrt{x+\sqrt{x+1}+1} \left (8 x+2 \sqrt{x+1}+5\right )+3 \log \left (2 \sqrt{x+1}+2 \sqrt{x+\sqrt{x+1}+1}+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + x + Sqrt[1 + x]],x]
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Maple [A] time = 0.007, size = 55, normalized size = 0.7 \[{\frac{2}{3} \left ( 1+x+\sqrt{1+x} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4} \left ( 1+2\,\sqrt{1+x} \right ) \sqrt{1+x+\sqrt{1+x}}}+{\frac{1}{8}\ln \left ({\frac{1}{2}}+\sqrt{1+x}+\sqrt{1+x+\sqrt{1+x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x+(1+x)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x + \sqrt{x + 1} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x + 1) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.554688, size = 82, normalized size = 1.09 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x + 1} + 5\right )} \sqrt{x + \sqrt{x + 1} + 1} + \frac{1}{16} \, \log \left (-4 \, \sqrt{x + \sqrt{x + 1} + 1}{\left (2 \, \sqrt{x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt{x + 1} - 9\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + sqrt(x + 1) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x + \sqrt{x + 1} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x+(1+x)**(1/2))**(1/2),x)
[Out]
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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(sqrt(x + sqrt(x + 1) + 1),x, algorithm="giac")
[Out]