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

Optimal. Leaf size=79 \[ \frac {2 \sqrt {x^2+1} x}{3 \sqrt {\sqrt {x^2+1}+1}}-\frac {2 x}{3 \sqrt {\sqrt {x^2+1}+1}}+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right ) \]

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Rubi [F]  time = 0.10, 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+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.18, size = 111, normalized size = 1.41 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+1} \left (6 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )+4 x^2-8 \sqrt {x^2+1}+3 \sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )+2\right )}{6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + Sqrt[1 + x^2]]*(2 + 4*x^2 - 8*Sqrt[1 + x^2] + 3*Sqrt[2]*Sqrt[-1 + Sqrt[1 + x^2]]*ArcTan[Sqrt[-1 + Sq
rt[1 + x^2]]/Sqrt[2]] + 6*Hypergeometric2F1[-1/2, 1, 1/2, 1/2 - Sqrt[1 + x^2]/2]))/(6*x)

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IntegrateAlgebraic [A]  time = 0.14, size = 79, normalized size = 1.00 \begin {gather*} -\frac {2 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \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[Sqrt[1 + x^2]/Sqrt[1 + Sqrt[1 + x^2]],x]

[Out]

(-2*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*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 = 1.14, size = 58, normalized size = 0.73 \begin {gather*} -\frac {3 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) - 2 \, {\left (x^{2} - 2 \, \sqrt {x^{2} + 1} + 2\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*sqrt(2)*x*arctan(sqrt(2)*sqrt(sqrt(x^2 + 1) + 1)/x) - 2*(x^2 - 2*sqrt(x^2 + 1) + 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 {x^{2} + 1}}{\sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+1}}{\sqrt {1+\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^2+1}}{\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)/((x^2 + 1)^(1/2) + 1)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2 + 1)/sqrt(sqrt(x**2 + 1) + 1), x)

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