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

Optimal. Leaf size=71 \[ \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+1}\& \right ] \]

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Rubi [F]  time = 0.55, 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 {x+\sqrt {1+x^2}}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 71, normalized size = 1.00 \begin {gather*} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\&,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+1}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 +
#1^4) & ]

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IntegrateAlgebraic [A]  time = 0.20, size = 71, normalized size = 1.00 \begin {gather*} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{1-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 +
#1^4) & ]

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fricas [B]  time = 0.53, size = 1368, normalized size = 19.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2)*arctan(1/2*sqrt(sqrt((sqrt(2) + 2
)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqrt
(x^2 + 1)) + 2*sqrt(sqrt(2) + 2) + 2)*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) -
2) + sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 1)*sqrt(sqrt(x + sqrt(x^2 + 1))
+ 1) - sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2) - 1) - sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/
4)*(sqrt(2) - 2)*arctan(1/8*sqrt(-16*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) -
2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 32*sqrt(x + sqrt(x^2 + 1)) + 32*sqrt(sqrt(2) + 2) + 32)*sqrt((sqrt(2) +
2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2) + sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt
(2) + 2)^(3/4)*(sqrt(2) - 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2) + 1) + 1/
16*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)*arctan(
1/128*sqrt(sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)
*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 32*sqrt(x + sqrt(x^2 + 1)) + 8*sqrt(-16*sqrt(2) + 32) + 32)*sqrt(-(sqrt(2
) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4) - 1/16*sqrt(-(sqrt(2) -
2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 1)*(-16*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1
)) + 1) - 1/4*sqrt(2)*sqrt(-16*sqrt(2) + 32) - sqrt(2) + 1) + 1/16*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32)
- 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)*arctan(1/128*sqrt(-sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(
2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 32*sqrt(
x + sqrt(x^2 + 1)) + 8*sqrt(-16*sqrt(2) + 32) + 32)*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 1
6)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4) - 1/16*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*
(sqrt(2) + 1)*(-16*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1/4*sqrt(2)*sqrt(-16*sqrt(2) + 32)
+ sqrt(2) - 1) - 1/4*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2)*sqrt(sqrt(2) + 2) - 2*sqrt(2))*(sqrt(2
) + 2)^(1/4)*log(2*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2)*sqrt(sqrt(x + s
qrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4) + 1/4*sqrt((sqrt(2) + 2)^(3/2) + 2*s
qrt(2) + 4)*(sqrt(2)*sqrt(sqrt(2) + 2) - 2*sqrt(2))*(sqrt(2) + 2)^(1/4)*log(-2*sqrt((sqrt(2) + 2)^(3/2) + 2*sq
rt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) + 4
*sqrt(sqrt(2) + 2) + 4) - 1/64*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2)*sqrt(-16*
sqrt(2) + 32) - 8*sqrt(2))*(-16*sqrt(2) + 32)^(1/4)*log(1/8*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqr
t(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)
) + sqrt(-16*sqrt(2) + 32) + 4) + 1/64*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2)*s
qrt(-16*sqrt(2) + 32) - 8*sqrt(2))*(-16*sqrt(2) + 32)^(1/4)*log(-1/8*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32
) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt
(x^2 + 1)) + sqrt(-16*sqrt(2) + 32) + 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x + \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+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

[Out]

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

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