3.30.89 \(\int \frac {(1+x^2)^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2} \, dx\)

Optimal. Leaf size=392 \[ -\frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\& ,\frac {5 \text {$\#$1}^3 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-6 \text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^6-3 \text {$\#$1}^4+4 \text {$\#$1}^2-2}\& \right ]+\frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\& ,\frac {5 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-6 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3}\& \right ]-4 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\frac {\left (8 x^4-7 x^2-5\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+\sqrt {x^2+1} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (8 x^3-11 x\right )}{\left (2 x^2+1\right ) \left (x^2-1\right )+2 x \sqrt {x^2+1} \left (x^2-1\right )} \]

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Rubi [F]  time = 2.35, 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 {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]

[Out]

Defer[Int][((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x)^2, x]/4 + Defer[Int][((1 + x^2)^(3/2)*S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x), x]/4 + Defer[Int][((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]
])/(1 + x)^2, x]/4 + Defer[Int][((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x), x]/4

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx &=\int \left (\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1-x)^2}+\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1+x)^2}+\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \int \left (\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x)}+\frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x)}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.37, size = 440, normalized size = 1.12 \begin {gather*} \frac {\left (x^2+\sqrt {x^2+1} x+1\right ) \left (-\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\&,\frac {5 \text {$\#$1}^3 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-6 \text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^6-3 \text {$\#$1}^4+4 \text {$\#$1}^2-2}\&\right ]+\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\&,\frac {5 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-6 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3}\&\right ]+8 \left (2 \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )-2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\right )+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (8 x^4-7 x^2-11 \sqrt {x^2+1} x+8 \sqrt {x^2+1} x^3-5\right )}{\left (x^2-1\right ) \left (2 x^2+2 \sqrt {x^2+1} x+1\right )}\right )\right )}{8 \sqrt {x^2+1} \left (\sqrt {x^2+1}+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]

[Out]

((1 + x^2 + x*Sqrt[1 + x^2])*(8*(((-5 - 7*x^2 + 8*x^4 - 11*x*Sqrt[1 + x^2] + 8*x^3*Sqrt[1 + x^2])*Sqrt[1 + Sqr
t[x + Sqrt[1 + x^2]]])/((-1 + x^2)*(1 + 2*x^2 + 2*x*Sqrt[1 + x^2])) + 2*Log[1 - Sqrt[1 + Sqrt[x + Sqrt[1 + x^2
]]]] - 2*Log[1 + Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 +
Sqrt[x + Sqrt[1 + x^2]]] - #1] - 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 5*Log[Sqrt[1 + Sqrt[x +
Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-
6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1 + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3)/(-2 + 4
*#1^2 - 3*#1^4 + #1^6) & ]))/(8*Sqrt[1 + x^2]*(x + Sqrt[1 + x^2]))

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IntegrateAlgebraic [A]  time = 0.75, size = 710, normalized size = 1.81 \begin {gather*} \frac {\sqrt {1+x^2} \left (-11 x+8 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-5-7 x^2+8 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 x \left (-1+x^2\right ) \sqrt {1+x^2}+\left (-1+x^2\right ) \left (1+2 x^2\right )}-4 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]

[Out]

(Sqrt[1 + x^2]*(-11*x + 8*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-5 - 7*x^2 + 8*x^4)*Sqrt[1 + Sqrt[x + Sqrt
[1 + x^2]]])/(2*x*(-1 + x^2)*Sqrt[1 + x^2] + (-1 + x^2)*(1 + 2*x^2)) - 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^
2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 +
Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^
7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[
1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5
+ #1^7) & ]/8 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #
1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*
#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x +
Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 +
x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8

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fricas [B]  time = 1.95, size = 6658, normalized size = 16.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="fricas")

[Out]

1/8*(sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*log(1/4*sqrt(1/2)*((295686*s
qrt(1/2)*sqrt(353*sqrt(2) - 497) - 147843*sqrt(2) + 2198386)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) +
16)^2 + 147843*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^3 - 3*(49281*(2*sqrt(1/2)*sqrt(353*sqrt(2)
- 497) - sqrt(2) - 16)^2 + 6307968*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 3153984*sqrt(2) + 6828344)*(2*sqrt(1/2
)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16) + 9461952*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 + 1
040814720*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 520407360*sqrt(2) + 1061465768)*sqrt(2*sqrt(1/2)*sqrt(353*sqrt(2
) - 497) + sqrt(2) + 16) + 1694285845*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2
)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*log(-1/4*sqrt(1/2)*((295686*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 1478
43*sqrt(2) + 2198386)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 + 147843*(2*sqrt(1/2)*sqrt(353*sq
rt(2) - 497) - sqrt(2) - 16)^3 - 3*(49281*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 + 6307968*sqr
t(1/2)*sqrt(353*sqrt(2) - 497) - 3153984*sqrt(2) + 6828344)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 1
6) + 9461952*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 + 1040814720*sqrt(1/2)*sqrt(353*sqrt(2) -
497) - 520407360*sqrt(2) + 1061465768)*sqrt(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16) + 1694285845*s
qrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(1/2)*(x^2 - 1)*sqrt(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)*
log(1/4*sqrt(1/2)*(15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^3 + (272502*sqrt(1/2)*sqrt(5*sqrt(2
) + 7) + 15139*sqrt(2) + 430040)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 + 847784*(18*sqrt(1/2)*sq
rt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - (15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 15260112*sq
rt(1/2)*sqrt(5*sqrt(2) + 7) + 847784*sqrt(2) + 6271112)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 29
8662192*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 16592344*sqrt(2) - 704044536)*sqrt(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) -
sqrt(2) + 14) + 376771509*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(1/2)*(x^2 - 1)*sqrt(18*sqrt(1/2)*sqrt(5*sq
rt(2) + 7) - sqrt(2) + 14)*log(-1/4*sqrt(1/2)*(15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^3 + (27
2502*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + 15139*sqrt(2) + 430040)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)
^2 + 847784*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - (15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + s
qrt(2) - 14)^2 + 15260112*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + 847784*sqrt(2) + 6271112)*(18*sqrt(1/2)*sqrt(5*sqrt(
2) + 7) - sqrt(2) + 14) - 298662192*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 16592344*sqrt(2) - 704044536)*sqrt(18*sqrt
(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) + 376771509*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + (x^2 - 1)*sqrt(sqrt
(2)*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) +
sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqr
t(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284) + 1/2*sqrt(2) + 7)*log(1/8*((272502*sqrt(
1/2)*sqrt(5*sqrt(2) + 7) + 15139*sqrt(2) + 430040)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 + 64198
6*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - (15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 1
4)^2 + 15260112*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + 847784*sqrt(2) + 6271112)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) -
sqrt(2) + 14) + 4*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*s
qrt(2) + 7) + sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt
(2) + 7) - sqrt(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284)*((15139*sqrt(2)*(18*sqrt(1/
2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14) + 641986*sqrt(2))*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 6
41986*sqrt(2)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14) - 17811128*sqrt(2)) + 326521584*sqrt(1/2)*sqrt
(5*sqrt(2) + 7) + 18140088*sqrt(2) - 785943008)*sqrt(sqrt(2)*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sq
rt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqr
t(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2
*sqrt(2) + 284) + 1/2*sqrt(2) + 7) + 376771509*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - (x^2 - 1)*sqrt(sqrt(2)*sqr
t(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2)
+ 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) +
14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284) + 1/2*sqrt(2) + 7)*log(-1/8*((272502*sqrt(1/2)*s
qrt(5*sqrt(2) + 7) + 15139*sqrt(2) + 430040)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 + 641986*(18*
sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - (15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 +
15260112*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + 847784*sqrt(2) + 6271112)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2
) + 14) + 4*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2)
+ 7) + sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) +
7) - sqrt(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284)*((15139*sqrt(2)*(18*sqrt(1/2)*sqr
t(5*sqrt(2) + 7) + sqrt(2) - 14) + 641986*sqrt(2))*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 641986*
sqrt(2)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14) - 17811128*sqrt(2)) + 326521584*sqrt(1/2)*sqrt(5*sqr
t(2) + 7) + 18140088*sqrt(2) - 785943008)*sqrt(sqrt(2)*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2)
- 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) +
14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(
2) + 284) + 1/2*sqrt(2) + 7) + 376771509*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + (x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/
32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) + 42
)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2
- 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284) + 1/2*sqrt(2) + 7)*log(1/8*((272502*sqrt(1/2)*sqrt(5*
sqrt(2) + 7) + 15139*sqrt(2) + 430040)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 + 641986*(18*sqrt(1
/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - (15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 15260
112*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + 847784*sqrt(2) + 6271112)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14
) - 4*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7)
+ sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - s
qrt(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284)*((15139*sqrt(2)*(18*sqrt(1/2)*sqrt(5*sq
rt(2) + 7) + sqrt(2) - 14) + 641986*sqrt(2))*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 641986*sqrt(2
)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14) - 17811128*sqrt(2)) + 326521584*sqrt(1/2)*sqrt(5*sqrt(2) +
7) + 18140088*sqrt(2) - 785943008)*sqrt(-sqrt(2)*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)
^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)
- 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) +
284) + 1/2*sqrt(2) + 7) + 376771509*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - (x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/32*(1
8*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) + 42)*(18
*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 - 63
*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284) + 1/2*sqrt(2) + 7)*log(-1/8*((272502*sqrt(1/2)*sqrt(5*sqrt
(2) + 7) + 15139*sqrt(2) + 430040)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 + 641986*(18*sqrt(1/2)*
sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - (15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 15260112*
sqrt(1/2)*sqrt(5*sqrt(2) + 7) + 847784*sqrt(2) + 6271112)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) -
4*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 + 1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sq
rt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(
2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284)*((15139*sqrt(2)*(18*sqrt(1/2)*sqrt(5*sqrt(2
) + 7) + sqrt(2) - 14) + 641986*sqrt(2))*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 641986*sqrt(2)*(1
8*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14) - 17811128*sqrt(2)) + 326521584*sqrt(1/2)*sqrt(5*sqrt(2) + 7)
+ 18140088*sqrt(2) - 785943008)*sqrt(-sqrt(2)*sqrt(-3/32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 +
1/16*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) + 42)*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14) - 3/
32*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - sqrt(2) + 14)^2 - 63*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 7/2*sqrt(2) + 284)
+ 1/2*sqrt(2) + 7) + 376771509*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 8*(x^2 - 1)*sqrt(-1/64*sqrt(1/2)*sqrt(353
*sqrt(2) - 497) + 1/128*sqrt(2) + 1/8)*log(2*(147843*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^3 +
4898078*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 + 697062192*sqrt(1/2)*sqrt(353*sqrt(2) - 497) -
348531096*sqrt(2) - 2320947704)*sqrt(-1/64*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/128*sqrt(2) + 1/8) + 1694285
845*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 8*(x^2 - 1)*sqrt(-1/64*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/128*sqrt
(2) + 1/8)*log(-2*(147843*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^3 + 4898078*(2*sqrt(1/2)*sqrt(3
53*sqrt(2) - 497) - sqrt(2) - 16)^2 + 697062192*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 348531096*sqrt(2) - 232094
7704)*sqrt(-1/64*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/128*sqrt(2) + 1/8) + 1694285845*sqrt(sqrt(x + sqrt(x^2
+ 1)) + 1)) + 8*(x^2 - 1)*sqrt(-9/64*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 1/128*sqrt(2) + 7/64)*log(2*(15139*(18*sq
rt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^3 + 205798*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - 6
25183776*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 34732432*sqrt(2) + 527573720)*sqrt(-9/64*sqrt(1/2)*sqrt(5*sqrt(2) + 7
) - 1/128*sqrt(2) + 7/64) + 376771509*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 8*(x^2 - 1)*sqrt(-9/64*sqrt(1/2)*sq
rt(5*sqrt(2) + 7) - 1/128*sqrt(2) + 7/64)*log(-2*(15139*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^3 +
205798*(18*sqrt(1/2)*sqrt(5*sqrt(2) + 7) + sqrt(2) - 14)^2 - 625183776*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 3473243
2*sqrt(2) + 527573720)*sqrt(-9/64*sqrt(1/2)*sqrt(5*sqrt(2) + 7) - 1/128*sqrt(2) + 7/64) + 376771509*sqrt(sqrt(
x + sqrt(x^2 + 1)) + 1)) - 2*(x^2 - 1)*sqrt(-1/8*sqrt(2) + sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) +
sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 4
97) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2
) - 497) + 1/2*sqrt(2) - 31) + 2)*log(1/4*((295686*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 147843*sqrt(2) + 219838
6)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 - 3*(49281*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sq
rt(2) - 16)^2 + 6307968*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 3153984*sqrt(2) + 6828344)*(2*sqrt(1/2)*sqrt(353*s
qrt(2) - 497) + sqrt(2) + 16) + 4563874*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 + 16*((295686*s
qrt(1/2)*sqrt(353*sqrt(2) - 497) - 147843*sqrt(2) + 2198386)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) +
16) - 9127748*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 4563874*sqrt(2) - 47189688)*sqrt(-3/256*(2*sqrt(1/2)*sqrt(35
3*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*s
qrt(353*sqrt(2) - 497) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1
/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2) - 31) + 343752528*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 171876264*sqrt
(2) + 2910138672)*sqrt(-1/8*sqrt(2) + sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/1
28*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) + 48) -
3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2
) - 31) + 2) + 1694285845*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 2*(x^2 - 1)*sqrt(-1/8*sqrt(2) + sqrt(-3/256*(2*
sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 1
6)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2)
- 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2) - 31) + 2)*log(-1/4*((295686*sqrt(1/2)*sqrt(353*sqr
t(2) - 497) - 147843*sqrt(2) + 2198386)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 - 3*(49281*(2*s
qrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 + 6307968*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 3153984*sqrt(
2) + 6828344)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16) + 4563874*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 4
97) - sqrt(2) - 16)^2 + 16*((295686*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 147843*sqrt(2) + 2198386)*(2*sqrt(1/2)
*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16) - 9127748*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 4563874*sqrt(2) - 47189
688)*sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2)
- 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt
(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2) - 31) + 343752528*sqrt(1/2)*sqr
t(353*sqrt(2) - 497) - 171876264*sqrt(2) + 2910138672)*sqrt(-1/8*sqrt(2) + sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*s
qrt(2) - 497) + sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt
(353*sqrt(2) - 497) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)
*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2) - 31) + 2) + 1694285845*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 2*(x^2 - 1
)*sqrt(-1/8*sqrt(2) - sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*
sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) + 48) - 3/256*(2*sqrt(1
/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2) - 31) + 2)*log
(1/4*((295686*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 147843*sqrt(2) + 2198386)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 49
7) + sqrt(2) + 16)^2 - 3*(49281*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 + 6307968*sqrt(1/2)*sqr
t(353*sqrt(2) - 497) - 3153984*sqrt(2) + 6828344)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16) + 45638
74*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - 16*((295686*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 14
7843*sqrt(2) + 2198386)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16) - 9127748*sqrt(1/2)*sqrt(353*sqrt
(2) - 497) + 4563874*sqrt(2) - 47189688)*sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 +
1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) + 48
) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqr
t(2) - 31) + 343752528*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 171876264*sqrt(2) + 2910138672)*sqrt(-1/8*sqrt(2) -
sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 49
7) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2)
- 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2) - 31) + 2) + 1694285845*sqrt(sqrt(x
+ sqrt(x^2 + 1)) + 1)) + 2*(x^2 - 1)*sqrt(-1/8*sqrt(2) - sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + s
qrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 49
7) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2)
- 497) + 1/2*sqrt(2) - 31) + 2)*log(-1/4*((295686*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 147843*sqrt(2) + 219838
6)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 - 3*(49281*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sq
rt(2) - 16)^2 + 6307968*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 3153984*sqrt(2) + 6828344)*(2*sqrt(1/2)*sqrt(353*s
qrt(2) - 497) + sqrt(2) + 16) + 4563874*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - 16*((295686*s
qrt(1/2)*sqrt(353*sqrt(2) - 497) - 147843*sqrt(2) + 2198386)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) +
16) - 9127748*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 4563874*sqrt(2) - 47189688)*sqrt(-3/256*(2*sqrt(1/2)*sqrt(35
3*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/128*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*s
qrt(353*sqrt(2) - 497) - sqrt(2) + 48) - 3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1
/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2) - 31) + 343752528*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - 171876264*sqrt
(2) + 2910138672)*sqrt(-1/8*sqrt(2) - sqrt(-3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)^2 + 1/1
28*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) + sqrt(2) + 16)*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) + 48) -
3/256*(2*sqrt(1/2)*sqrt(353*sqrt(2) - 497) - sqrt(2) - 16)^2 - sqrt(1/2)*sqrt(353*sqrt(2) - 497) + 1/2*sqrt(2
) - 31) + 2) + 1694285845*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 16*(x^2 - 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) +
1) + 1) + 16*(x^2 - 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1) + 8*(5*x^2 - sqrt(x^2 + 1)*x - 5)*sqrt(sqrt
(x + sqrt(x^2 + 1)) + 1))/(x^2 - 1)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x)

[Out]

int((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} - 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="maxima")

[Out]

integrate((x^2 + 1)^(3/2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 - 1)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,{\left (x^2+1\right )}^{3/2}}{{\left (x^2-1\right )}^2} \,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)^(3/2))/(x^2 - 1)^2,x)

[Out]

int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(3/2))/(x^2 - 1)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 1\right )^{\frac {3}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)**(3/2)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**2+1)**2,x)

[Out]

Integral((x**2 + 1)**(3/2)*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/((x - 1)**2*(x + 1)**2), x)

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