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

Optimal. Leaf size=51 \[ -\frac {\sqrt {\sqrt {x^2+1}+1}}{x}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right )}{\sqrt {2}} \]

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Rubi [F]  time = 0.33, 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 \sqrt {1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.11, size = 45, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {\sqrt {x^2+1}+1} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.08, size = 51, 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*Sqrt[1 + 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 = 1.43, size = 43, normalized size = 0.84 \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^2+1)^(1/2),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}}{\sqrt {x^{2} + 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^2+1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x^{2}+1}}}{x^{2} \sqrt {x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}}{\sqrt {x^{2} + 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^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(x^2 + 1) + 1)/(sqrt(x^2 + 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\,\sqrt {x^2+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^2*(x^2 + 1)^(1/2)),x)

[Out]

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

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sympy [C]  time = 1.19, size = 34, normalized size = 0.67 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{\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**2+1)**(1/2),x)

[Out]

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

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