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

Optimal. Leaf size=639 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\& ,\frac {7 \text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-15 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+13 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3}\& \right ]+\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\& ,\frac {7 \text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-15 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+5 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+4 \text {$\#$1}^3-2 \text {$\#$1}}\& \right ]-\tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (24 x^5+32 x^4-6 x^3-16 x^2-18 x-16\right )+\sqrt {x^2+1} \left (\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (24 x^4+32 x^3-18 x^2-32 x-6\right )+\left (240 x^5-32 x^4-1170 x^3+24 x^2-120 x+8\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (240 x^6-32 x^5-1050 x^4+8 x^3-735 x^2+24 x+75\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{105 \left (x^2-1\right ) \left (\sqrt {x^2+1}+x\right )^{5/2}} \]

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Rubi [F]  time = 1.87, 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 )^2 \sqrt {x+\sqrt {1+x^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)^2*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 7.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^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]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.30, size = 914, normalized size = 1.43 \begin {gather*} \frac {\left (75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-16-18 x-16 x^2-6 x^3+32 x^4+24 x^5\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (8-120 x+24 x^2-1170 x^3-32 x^4+240 x^5\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-6-32 x-18 x^2+32 x^3+24 x^4\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{105 \left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{5/2}}-\tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]-\frac {1}{4} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-9 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((75 + 24*x - 735*x^2 + 8*x^3 - 1050*x^4 - 32*x^5 + 240*x^6)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-16 - 18*x -
16*x^2 - 6*x^3 + 32*x^4 + 24*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*(
(8 - 120*x + 24*x^2 - 1170*x^3 - 32*x^4 + 240*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-6 - 32*x - 18*x^2 + 3
2*x^3 + 24*x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(105*(-1 + x^2)*(x + Sqrt[1 + x^2]
)^(5/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (5*Log[Sqrt[1
+ Sqrt[x + Sqrt[1 + x^2]]] - #1] - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 2*Log[Sqrt[1 + Sqrt[x
+ Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1
+ Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x +
Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4
+ RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) - 4*Log[
Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^
2 - 3*#1^4 + #1^6) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]
] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^
4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4

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fricas [B]  time = 2.13, size = 7099, normalized size = 11.11

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/840*(105*sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) +
132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) +
80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(
1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45361) - 101/2*sqrt(2) + 66)*log(1/8*(5*(1087899451*sqrt(2)*
(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132) - 11039605670*sqrt(2))*(2*sqrt(1/2)*sqrt(56941*s
qrt(2) + 80521) + 101*sqrt(2) + 132)^2 - 55198028350*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sq
rt(2) - 132)^2 - (5439497255*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 + 2872054
550640*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132) - 28437296565606*sqrt(2))*(2*sqrt
(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 8*(5*(2175798902*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521
) - 109877844551*sqrt(2) - 154642333202)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 11039
6056700*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 5575000863350*sqrt(2) - 6578877339006)*sqrt(-3/32*(2*sqrt(1/2)
*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt
(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2
) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45361) - 2843729
6565606*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132) + 402764487053168*sqrt(2))*sqrt(
sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56
941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32
*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) +
3333*sqrt(2) + 45361) - 101/2*sqrt(2) + 66) + 3462064407728593*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 105*sqrt(2
)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*
sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt
(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sq
rt(2) + 80521) + 3333*sqrt(2) + 45361) - 101/2*sqrt(2) + 66)*log(-1/8*(5*(1087899451*sqrt(2)*(2*sqrt(1/2)*sqrt
(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132) - 11039605670*sqrt(2))*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) +
101*sqrt(2) + 132)^2 - 55198028350*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 -
(5439497255*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 + 2872054550640*sqrt(2)*(2
*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132) - 28437296565606*sqrt(2))*(2*sqrt(1/2)*sqrt(56941*
sqrt(2) + 80521) + 101*sqrt(2) + 132) + 8*(5*(2175798902*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 109877844551*
sqrt(2) - 154642333202)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 110396056700*sqrt(1/2)
*sqrt(56941*sqrt(2) + 80521) - 5575000863350*sqrt(2) - 6578877339006)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(
2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqr
t(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*
sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45361) - 28437296565606*sqrt(2)*(
2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132) + 402764487053168*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/3
2*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 805
21) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqr
t(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45
361) - 101/2*sqrt(2) + 66) + 3462064407728593*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 105*sqrt(2)*(x^2 - 1)*sqrt(
-sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(5
6941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/3
2*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) +
3333*sqrt(2) + 45361) - 101/2*sqrt(2) + 66)*log(1/8*(5*(1087899451*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) +
80521) - 101*sqrt(2) - 132) - 11039605670*sqrt(2))*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 13
2)^2 - 55198028350*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - (5439497255*sqrt(
2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 + 2872054550640*sqrt(2)*(2*sqrt(1/2)*sqrt(5
6941*sqrt(2) + 80521) - 101*sqrt(2) - 132) - 28437296565606*sqrt(2))*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521)
+ 101*sqrt(2) + 132) - 8*(5*(2175798902*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 109877844551*sqrt(2) - 1546423
33202)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 110396056700*sqrt(1/2)*sqrt(56941*sqrt(
2) + 80521) - 5575000863350*sqrt(2) - 6578877339006)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101
*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941
*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2
- 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45361) - 28437296565606*sqrt(2)*(2*sqrt(1/2)*sqrt(
56941*sqrt(2) + 80521) - 101*sqrt(2) - 132) + 402764487053168*sqrt(2))*sqrt(-sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*s
qrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2
) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2)
+ 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45361) - 101/2*sqr
t(2) + 66) + 3462064407728593*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 105*sqrt(2)*(x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3
/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 8
0521) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*s
qrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) +
45361) - 101/2*sqrt(2) + 66)*log(-1/8*(5*(1087899451*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sq
rt(2) - 132) - 11039605670*sqrt(2))*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 - 55198028
350*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - (5439497255*sqrt(2)*(2*sqrt(1/2)
*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 + 2872054550640*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) +
80521) - 101*sqrt(2) - 132) - 28437296565606*sqrt(2))*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) +
132) - 8*(5*(2175798902*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 109877844551*sqrt(2) - 154642333202)*(2*sqrt(
1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 110396056700*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 5
575000863350*sqrt(2) - 6578877339006)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)
^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 8052
1) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 66*sqrt(1/2)*
sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45361) - 28437296565606*sqrt(2)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) +
80521) - 101*sqrt(2) - 132) + 402764487053168*sqrt(2))*sqrt(-sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(
2) + 80521) + 101*sqrt(2) + 132)^2 + 1/16*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)*(2*sqr
t(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) + 396) - 3/32*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*
sqrt(2) - 132)^2 - 66*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 3333*sqrt(2) + 45361) - 101/2*sqrt(2) + 66) + 34
62064407728593*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 105*(x^2 - 1)*sqrt(4*sqrt(2)*sqrt(-3/128*(2*sqrt(1/2)*sqrt
(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2)
+ 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2)
- 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 33899/4) - 103*s
qrt(2) - 134)*log(1/4*(3*(3123031798*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 160836137597*sqrt(2) + 2281421395
95)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - (4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt
(2) - 78407) - 103*sqrt(2) + 134)^2 - 5021835131184*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 258624509255976*sq
rt(2) - 369498919235591)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) + 56697027387*(2*sqrt(1
/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 8*(3*(1561515899*sqrt(2)*(2*sqrt(1/2)*sqrt(55445*sqrt
(2) - 78407) - 103*sqrt(2) + 134) + 18899009129*sqrt(2))*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2
) - 134) - 56697027387*sqrt(2)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134) - 2646358766831*s
qrt(2))*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(
55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) -
134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407
) - 6901/8*sqrt(2) - 33899/4) - 66071930892526*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 3402704440965089*sqrt(2
) - 4761501573839354)*sqrt(4*sqrt(2)*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)
^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt
(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/
2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 33899/4) - 103*sqrt(2) - 134) + 9925267848380161*sqrt(sqrt(x
+ sqrt(x^2 + 1)) + 1)) - 105*(x^2 - 1)*sqrt(4*sqrt(2)*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) +
103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/
2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) - 4
02) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 33899/4) - 103*sqrt(2) - 134)*log(-1/4*(3*
(3123031798*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 160836137597*sqrt(2) + 228142139595)*(2*sqrt(1/2)*sqrt(554
45*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - (4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2
) + 134)^2 - 5021835131184*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 258624509255976*sqrt(2) - 369498919235591)*
(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) + 56697027387*(2*sqrt(1/2)*sqrt(55445*sqrt(2) -
78407) - 103*sqrt(2) + 134)^2 + 8*(3*(1561515899*sqrt(2)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2
) + 134) + 18899009129*sqrt(2))*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) - 56697027387*sq
rt(2)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134) - 2646358766831*sqrt(2))*sqrt(-3/128*(2*sq
rt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) -
103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55
445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 3389
9/4) - 66071930892526*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 3402704440965089*sqrt(2) - 4761501573839354)*sqr
t(4*sqrt(2)*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*s
qrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2
) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 7
8407) - 6901/8*sqrt(2) - 33899/4) - 103*sqrt(2) - 134) + 9925267848380161*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) +
210*(x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/1
28*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78
407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(
55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 33899/4) - 103/4*sqrt(2) - 67/2)*log(1/2*(3*(3123031798*sqrt(1/2)*sq
rt(55445*sqrt(2) - 78407) - 160836137597*sqrt(2) + 228142139595)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 10
3*sqrt(2) - 134)^2 - (4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 - 50218351311
84*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 258624509255976*sqrt(2) - 369498919235591)*(2*sqrt(1/2)*sqrt(55445*
sqrt(2) - 78407) + 103*sqrt(2) - 134) + 56697027387*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 1
34)^2 - 8*(3*(1561515899*sqrt(2)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134) + 18899009129*s
qrt(2))*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) - 56697027387*sqrt(2)*(2*sqrt(1/2)*sqrt(
55445*sqrt(2) - 78407) - 103*sqrt(2) + 134) - 2646358766831*sqrt(2))*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(
2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1
/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 1
03*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 33899/4) - 66071930892526*sq
rt(1/2)*sqrt(55445*sqrt(2) - 78407) + 3402704440965089*sqrt(2) - 4761501573839354)*sqrt(-sqrt(2)*sqrt(-3/128*(
2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407
) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqr
t(55445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) -
33899/4) - 103/4*sqrt(2) - 67/2) + 9925267848380161*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 210*(x^2 - 1)*sqrt(-s
qrt(2)*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(5
5445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 1
34)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407)
- 6901/8*sqrt(2) - 33899/4) - 103/4*sqrt(2) - 67/2)*log(-1/2*(3*(3123031798*sqrt(1/2)*sqrt(55445*sqrt(2) - 78
407) - 160836137597*sqrt(2) + 228142139595)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 -
(4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 - 5021835131184*sqrt(1/2)*sqrt(554
45*sqrt(2) - 78407) + 258624509255976*sqrt(2) - 369498919235591)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 10
3*sqrt(2) - 134) + 56697027387*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 - 8*(3*(1561515
899*sqrt(2)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134) + 18899009129*sqrt(2))*(2*sqrt(1/2)*
sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) - 56697027387*sqrt(2)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407
) - 103*sqrt(2) + 134) - 2646358766831*sqrt(2))*sqrt(-3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqr
t(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 1/64*(2*sqrt(1/2)*sqrt
(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) - 402) + 6
7/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 33899/4) - 66071930892526*sqrt(1/2)*sqrt(55445*sq
rt(2) - 78407) + 3402704440965089*sqrt(2) - 4761501573839354)*sqrt(-sqrt(2)*sqrt(-3/128*(2*sqrt(1/2)*sqrt(5544
5*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 - 3/128*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134
)^2 + 1/64*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 784
07) - 103*sqrt(2) - 402) + 67/4*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 6901/8*sqrt(2) - 33899/4) - 103/4*sqrt
(2) - 67/2) + 9925267848380161*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 105*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(56941*
sqrt(2) + 80521) + 101*sqrt(2) + 132)*log(1/4*(5*(2175798902*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 109877844
551*sqrt(2) - 154642333202)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 5439497255*(2*sq
rt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^3 - (5439497255*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 805
21) - 101*sqrt(2) - 132)^2 + 5744109101280*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 290077509614640*sqrt(2) - 4
07548497250086)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 2872054550640*(2*sqrt(1/2)*sqr
t(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 2810522957691440*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 141
931409363417720*sqrt(2) + 199739929674604072)*sqrt(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132
) + 3462064407728593*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 105*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(56941*sqrt(2) +
80521) + 101*sqrt(2) + 132)*log(-1/4*(5*(2175798902*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 109877844551*sqrt(
2) - 154642333202)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132)^2 + 5439497255*(2*sqrt(1/2)*s
qrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^3 - (5439497255*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101
*sqrt(2) - 132)^2 + 5744109101280*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 290077509614640*sqrt(2) - 4075484972
50086)*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 2872054550640*(2*sqrt(1/2)*sqrt(56941*s
qrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 2810522957691440*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 141931409363
417720*sqrt(2) + 199739929674604072)*sqrt(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101*sqrt(2) + 132) + 34620
64407728593*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 840*(x^2 - 1)*sqrt(-1/32*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521
) + 101/64*sqrt(2) + 33/16)*log(2*(5439497255*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^3
+ 2927252578990*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^2 - 2753648364560228*sqrt(1/2)*s
qrt(56941*sqrt(2) + 80521) + 139059242410291514*sqrt(2) + 196945337924312496)*sqrt(-1/32*sqrt(1/2)*sqrt(56941*
sqrt(2) + 80521) + 101/64*sqrt(2) + 33/16) + 3462064407728593*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 840*(x^2 -
1)*sqrt(-1/32*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101/64*sqrt(2) + 33/16)*log(-2*(5439497255*(2*sqrt(1/2)*
sqrt(56941*sqrt(2) + 80521) - 101*sqrt(2) - 132)^3 + 2927252578990*(2*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) -
101*sqrt(2) - 132)^2 - 2753648364560228*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 139059242410291514*sqrt(2) + 1
96945337924312496)*sqrt(-1/32*sqrt(1/2)*sqrt(56941*sqrt(2) + 80521) + 101/64*sqrt(2) + 33/16) + 34620644077285
93*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 105*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(
2) - 134)*log(1/2*(3*(3123031798*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 160836137597*sqrt(2) + 228142139595)*
(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 + 4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) -
78407) - 103*sqrt(2) + 134)^3 - (4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 -
5021835131184*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 258624509255976*sqrt(2) - 369498919235591)*(2*sqrt(1/2)
*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) - 2510917565592*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 1
03*sqrt(2) + 134)^2 + 3550212579457632*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 182835947842068048*sqrt(2) + 26
0224899268893244)*sqrt(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) + 9925267848380161*sqrt(sq
rt(x + sqrt(x^2 + 1)) + 1)) + 105*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)*
log(-1/2*(3*(3123031798*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 160836137597*sqrt(2) + 228142139595)*(2*sqrt(1
/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134)^2 + 4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) -
103*sqrt(2) + 134)^3 - (4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 - 50218351
31184*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 258624509255976*sqrt(2) - 369498919235591)*(2*sqrt(1/2)*sqrt(554
45*sqrt(2) - 78407) + 103*sqrt(2) - 134) - 2510917565592*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2
) + 134)^2 + 3550212579457632*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 182835947842068048*sqrt(2) + 26022489926
8893244)*sqrt(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103*sqrt(2) - 134) + 9925267848380161*sqrt(sqrt(x + sq
rt(x^2 + 1)) + 1)) + 840*(x^2 - 1)*sqrt(-1/32*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103/64*sqrt(2) - 67/32)*
log(4*(4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^3 - 2567614592979*(2*sqrt(1/2)
*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 3616284510350158*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) -
186238652283033137*sqrt(2) + 262416854871597822)*sqrt(-1/32*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103/64*sq
rt(2) - 67/32) + 9925267848380161*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 840*(x^2 - 1)*sqrt(-1/32*sqrt(1/2)*sqrt
(55445*sqrt(2) - 78407) + 103/64*sqrt(2) - 67/32)*log(-4*(4684547697*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407)
- 103*sqrt(2) + 134)^3 - 2567614592979*(2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 103*sqrt(2) + 134)^2 + 36162
84510350158*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) - 186238652283033137*sqrt(2) + 262416854871597822)*sqrt(-1/3
2*sqrt(1/2)*sqrt(55445*sqrt(2) - 78407) + 103/64*sqrt(2) - 67/32) + 9925267848380161*sqrt(sqrt(x + sqrt(x^2 +
1)) + 1)) + 420*(x^2 - 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) - 420*(x^2 - 1)*log(sqrt(sqrt(x + sqrt(x^
2 + 1)) + 1) - 1) - 8*(6*x^3 + 16*x^2 + 6*sqrt(x^2 + 1)*(x^2 - 1) + (135*x^3 - 8*x^2 - 75*sqrt(x^2 + 1)*(x^2 -
1) - 345*x + 8)*sqrt(x + sqrt(x^2 + 1)) - 6*x - 16)*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)^2*(x+(x^2+1)^(1/2))^(1/2)*(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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((x^2 + 1)^2*sqrt(x + sqrt(x^2 + 1))*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 )}^2\,\sqrt {x+\sqrt {x^2+1}}}{{\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)^2*(x + (x^2 + 1)^(1/2))^(1/2))/(x^2 - 1)^2,x)

[Out]

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

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

[Out]

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

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