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

Optimal. Leaf size=20 \[ 2 \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^2+1}+1}}\right ) \]

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Rubi [F]  time = 0.23, 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}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx &=\int \left (\frac {i \sqrt {1+\sqrt {1+x^2}}}{2 (i-x)}+\frac {i \sqrt {1+\sqrt {1+x^2}}}{2 (i+x)}\right ) \, dx\\ &=\frac {1}{2} i \int \frac {\sqrt {1+\sqrt {1+x^2}}}{i-x} \, dx+\frac {1}{2} i \int \frac {\sqrt {1+\sqrt {1+x^2}}}{i+x} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.06, size = 20, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]]

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fricas [B]  time = 1.77, size = 55, normalized size = 2.75 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) \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),x, algorithm="fricas")

[Out]

-1/2*arctan(4*(x^4 - 12*x^2 + (5*x^2 - 3)*sqrt(x^2 + 1) + 3)*sqrt(sqrt(x^2 + 1) + 1)/(x^5 - 46*x^3 + 17*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} + 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),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {\sqrt {x^2+1}+1}}{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),x)

[Out]

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

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

[Out]

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

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