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

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

[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 = 0.0073905, 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, Rules used = {2129} \[ \frac{2 x^3}{3 \left (\sqrt{x^2+1}+1\right )^{3/2}}+\frac{2 x}{\sqrt{\sqrt{x^2+1}+1}} \]

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]]

Rule 2129

Int[Sqrt[(a_) + (b_.)*Sqrt[(c_) + (d_.)*(x_)^2]], x_Symbol] :> Simp[(2*b^2*d*x^3)/(3*(a + b*Sqrt[c + d*x^2])^(
3/2)), x] + Simp[(2*a*x)/Sqrt[a + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2*c, 0]

Rubi steps

\begin{align*} \int \sqrt{1+\sqrt{1+x^2}} \, dx &=\frac{2 x^3}{3 \left (1+\sqrt{1+x^2}\right )^{3/2}}+\frac{2 x}{\sqrt{1+\sqrt{1+x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0572284, 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 = 0.017, size = 55, normalized size = 1.3 \begin{align*} -{\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 ) } \end{align*}

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. \begin{align*} \int \sqrt{\sqrt{x^{2} + 1} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+1)^(1/2)+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(x^2 + 1) + 1), x)

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Fricas [A]  time = 1.08139, size = 77, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (x^{2} + \sqrt{x^{2} + 1} - 1\right )} \sqrt{\sqrt{x^{2} + 1} + 1}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+1)^(1/2)+1)^(1/2),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 [B]  time = 1.08258, size = 197, normalized size = 4.8 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+1)**(1/2)+1)**(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)*gamma(1/4)/(12*pi*sqrt(x**2 + 1)*sqrt(sqrt(x**2 + 1) + 1) + 1
2*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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{x^{2} + 1} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+1)^(1/2)+1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sqrt(x^2 + 1) + 1), x)