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

Optimal. Leaf size=49 \[ \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right ) \]

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Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/Sqrt[1 + Sqrt[1 + x^2]],x]

[Out]

Defer[Int][1/Sqrt[1 + Sqrt[1 + x^2]], x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx &=\int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.22, size = 106, normalized size = 2.16 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+1} \left (-2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )+4 \sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )-2\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + Sqrt[1 + x^2]]*(-2 + 4*Sqrt[1 + x^2] - Sqrt[2]*Sqrt[-1 + Sqrt[1 + x^2]]*ArcTan[Sqrt[-1 + Sqrt[1 + x^
2]]/Sqrt[2]] - 2*Hypergeometric2F1[-1/2, 1, 1/2, 1/2 - Sqrt[1 + x^2]/2]))/(2*x)

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IntegrateAlgebraic [A]  time = 0.10, size = 49, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[1 + Sqrt[1 + x^2]],x]

[Out]

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

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fricas [A]  time = 1.36, size = 51, normalized size = 1.04 \begin {gather*} \frac {\sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) + 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} {\left (\sqrt {x^{2} + 1} - 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(sqrt(x^2 + 1) + 1), x)

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maple [C]  time = 0.10, size = 20, normalized size = 0.41

method	result	size
meijerg	\(\frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -x^{2}\right )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*2^(1/2)*x*hypergeom([1/4,1/2,3/4],[3/2,3/2],-x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/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 \frac {1}{\sqrt {\sqrt {x^2+1}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^2 + 1)^(1/2) + 1)^(1/2),x)

[Out]

int(1/((x^2 + 1)^(1/2) + 1)^(1/2), x)

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sympy [C]  time = 0.73, size = 32, normalized size = 0.65 \begin {gather*} \frac {x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{2 \pi } \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+(x**2+1)**(1/2))**(1/2),x)

[Out]

x*gamma(1/4)*gamma(3/4)*hyper((1/4, 1/2, 3/4), (3/2, 3/2), x**2*exp_polar(I*pi))/(2*pi)

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