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

Optimal. Leaf size=50 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right )}{\sqrt {2}}-\frac {\sqrt {\sqrt {x^2+1}+1}}{x} \]

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Rubi [F]  time = 0.08, 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 {\sqrt {1+\sqrt {1+x^2}}}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[1 + Sqrt[1 + x^2]]/x^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 78, normalized size = 1.56 \begin {gather*} \frac {\left (\sqrt {x^2+1}-1\right ) \left (\sqrt {x^2+1}+1\right )^{3/2} \left (\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^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.11, size = 50, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+\sqrt {1+x^2}}}{x}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.60, size = 43, normalized size = 0.86 \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}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(2)*x*arctan(sqrt(2)*sqrt(sqrt(x^2 + 1) + 1)/x) + 2*sqrt(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 {\sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 22, normalized size = 0.44

method	result	size
meijerg	\(-\frac {\sqrt {2}\, \hypergeom \left (\left [-\frac {1}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {1}{2}\right ], -x^{2}\right )}{x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

int(((x^2 + 1)^(1/2) + 1)^(1/2)/x^2, x)

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sympy [C]  time = 0.88, size = 37, normalized size = 0.74 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{4 \pi x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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