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

Optimal. Leaf size=75 \[ \frac{2}{3} \left (x+\sqrt{x+1}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x+1}+1\right ) \sqrt{x+\sqrt{x+1}+1}+\frac{1}{4} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{x+\sqrt{x+1}+1}}\right ) \]

[Out]

(2*(1 + x + Sqrt[1 + x])^(3/2))/3 - (Sqrt[1 + x + Sqrt[1 + x]]*(1 + 2*Sqrt[1 + x
]))/4 + ArcTanh[Sqrt[1 + x]/Sqrt[1 + x + Sqrt[1 + x]]]/4

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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]

(2*(1 + x + Sqrt[1 + x])^(3/2))/3 - (Sqrt[1 + x + Sqrt[1 + x]]*(1 + 2*Sqrt[1 + x
]))/4 + ArcTanh[Sqrt[1 + x]/Sqrt[1 + x + Sqrt[1 + x]]]/4

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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]

-(2*sqrt(x + 1) + 1)*sqrt(x + sqrt(x + 1) + 1)/4 + 2*(x + sqrt(x + 1) + 1)**(3/2
)/3 + atanh(sqrt(x + 1)/sqrt(x + sqrt(x + 1) + 1))/4

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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]

[Out]

(2*Sqrt[1 + x + Sqrt[1 + x]]*(5 + 8*x + 2*Sqrt[1 + x]) + 3*Log[1 + 2*Sqrt[1 + x]
+ 2*Sqrt[1 + x + Sqrt[1 + x]]])/24

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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]

2/3*(1+x+(1+x)^(1/2))^(3/2)-1/4*(1+2*(1+x)^(1/2))*(1+x+(1+x)^(1/2))^(1/2)+1/8*ln
(1/2+(1+x)^(1/2)+(1+x+(1+x)^(1/2))^(1/2))

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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]

integrate(sqrt(x + sqrt(x + 1) + 1), x)

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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]

1/12*(8*x + 2*sqrt(x + 1) + 5)*sqrt(x + sqrt(x + 1) + 1) + 1/16*log(-4*sqrt(x +
sqrt(x + 1) + 1)*(2*sqrt(x + 1) + 1) - 8*x - 8*sqrt(x + 1) - 9)

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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]

Integral(sqrt(x + sqrt(x + 1) + 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(sqrt(x + sqrt(x + 1) + 1),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError