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

Optimal. Leaf size=18 \[ \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}} \]

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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}}}{\sqrt {1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 31, normalized size = 1.72 \begin {gather*} \frac {2 \left (\sqrt {x^2+1}-1\right ) \sqrt {\sqrt {x^2+1}+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {1+\sqrt {1+x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/Sqrt[1 + Sqrt[1 + x^2]]

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fricas [A]  time = 0.49, size = 25, normalized size = 1.39 \begin {gather*} \frac {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+(x^2+1)^(1/2))^(1/2)/(x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {\sqrt {\sqrt {x^2+1}+1}}{\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 + 1)^(1/2),x)

[Out]

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

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sympy [B]  time = 0.73, size = 31, normalized size = 1.72 \begin {gather*} \frac {\sqrt {2} x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{\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**2+1)**(1/2),x)

[Out]

sqrt(2)*x*gamma(1/4)*gamma(3/4)/(pi*sqrt(sqrt(x**2 + 1) + 1))

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