3.5.69 \(\int \sqrt {1+\sqrt {1+x^2}} \, dx\) [469]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {2154} \begin {gather*} \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}+\frac {2 x^3}{3 \left (\sqrt {x^2+1}+1\right )^{3/2}} \end {gather*}

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 2154

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*}

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Mathematica [A]
time = 0.06, size = 31, normalized size = 0.76 \begin {gather*} \frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sqrt[1 + x^2]],x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.02, size = 55, normalized size = 1.34

method	result	size
meijerg	\(-\frac {-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \cosh \left (\frac {3 \arcsinh \left (x \right )}{2}\right )}{3}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {4}{3} x^{4}-\frac {2}{3} x^{2}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (x \right )}{2}\right )}{\sqrt {x^{2}+1}}}{8 \sqrt {\pi }}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+(x^2+1)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(x^2 + 1) + 1), x)

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Fricas [A]
time = 0.38, size = 28, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(x^2+1)^(1/2))^(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] Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (36) = 72\).
time = 0.58, size = 197, normalized size = 4.80 \begin {gather*} - \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 {gather*}

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)*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(x^2 + 1) + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {\sqrt {x^2+1}+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 + 1)^(1/2) + 1)^(1/2),x)

[Out]

int(((x^2 + 1)^(1/2) + 1)^(1/2), x)

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