3.308 \(\int \sqrt{1+\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 x}{\sqrt{\sqrt{x^2+1}+1}}+\frac{2 x^3}{3 \left (\sqrt{x^2+1}+1\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0194902, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 x}{\sqrt{\sqrt{x^2+1}+1}}+\frac{2 x^3}{3 \left (\sqrt{x^2+1}+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + Sqrt[1 + x^2]],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.15661, size = 36, normalized size = 0.88 \[ \frac{2 x^{3}}{3 \left (\sqrt{x^{2} + 1} + 1\right )^{\frac{3}{2}}} + \frac{2 x}{\sqrt{\sqrt{x^{2} + 1} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**3/(3*(sqrt(x**2 + 1) + 1)**(3/2)) + 2*x/sqrt(sqrt(x**2 + 1) + 1)

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Mathematica [A]  time = 0.080765, size = 44, normalized size = 1.07 \[ \frac{2 \left (\sqrt{x^2+1}-1\right ) \sqrt{\sqrt{x^2+1}+1} \left (\sqrt{x^2+1}+2\right )}{3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + Sqrt[1 + x^2]],x]

[Out]

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

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Maple [C]  time = 0.04, size = 55, normalized size = 1.3 \[ -{\frac{1}{8\,\sqrt{\pi }} \left ( -{\frac{32\,\sqrt{\pi }\sqrt{2}{x}^{3}}{3}\cosh \left ({\frac{3\,{\it Arcsinh} \left ( x \right ) }{2}} \right ) }-8\,{\frac{\sqrt{\pi }\sqrt{2} \left ( -4/3\,{x}^{4}-2/3\,{x}^{2}+2/3 \right ) \sinh \left ( 3/2\,{\it Arcsinh} \left ( x \right ) \right ) }{\sqrt{{x}^{2}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+(x^2+1)^(1/2))^(1/2),x)

[Out]

-1/8/Pi^(1/2)*(-32/3*Pi^(1/2)*2^(1/2)*x^3*cosh(3/2*arcsinh(x))-8*Pi^(1/2)*2^(1/2
)*(-4/3*x^4-2/3*x^2+2/3)*sinh(3/2*arcsinh(x))/(x^2+1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(sqrt(x^2 + 1) + 1), x)

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Fricas [A]  time = 0.291077, size = 38, normalized size = 0.93 \[ \frac{2 \,{\left (x^{2} + \sqrt{x^{2} + 1} - 1\right )} \sqrt{\sqrt{x^{2} + 1} + 1}}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x^2 + sqrt(x^2 + 1) - 1)*sqrt(sqrt(x^2 + 1) + 1)/x

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Sympy [A]  time = 3.8663, size = 197, normalized size = 4.8 \[ - \frac{\sqrt{2} x^{3} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{x^{2} + 1} \sqrt{\sqrt{x^{2} + 1} + 1} + 12 \pi \sqrt{\sqrt{x^{2} + 1} + 1}} - \frac{3 \sqrt{2} x \sqrt{x^{2} + 1} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{x^{2} + 1} \sqrt{\sqrt{x^{2} + 1} + 1} + 12 \pi \sqrt{\sqrt{x^{2} + 1} + 1}} - \frac{3 \sqrt{2} x \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{x^{2} + 1} \sqrt{\sqrt{x^{2} + 1} + 1} + 12 \pi \sqrt{\sqrt{x^{2} + 1} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*x**3*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(x**2 + 1)*sqrt(sqrt(x**2 + 1) +
1) + 12*pi*sqrt(sqrt(x**2 + 1) + 1)) - 3*sqrt(2)*x*sqrt(x**2 + 1)*gamma(-1/4)*g
amma(1/4)/(12*pi*sqrt(x**2 + 1)*sqrt(sqrt(x**2 + 1) + 1) + 12*pi*sqrt(sqrt(x**2
+ 1) + 1)) - 3*sqrt(2)*x*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(x**2 + 1)*sqrt(sqrt(
x**2 + 1) + 1) + 12*pi*sqrt(sqrt(x**2 + 1) + 1))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt{x^{2} + 1} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(sqrt(x^2 + 1) + 1), x)