3.32.14 \(\int \frac {(1+x^2)^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2} \, dx\) [3114]

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

[Out]

Unintegrable

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Rubi [F]
time = 1.30, 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 [A]
time = 0.00, size = 849, normalized size = 1.33 \begin {gather*} \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6+2 \left (-8-9 x-8 x^2-3 x^3+16 x^4+12 x^5\right ) \sqrt {x+\sqrt {1+x^2}}+2 \sqrt {1+x^2} \left (4-60 x+12 x^2-585 x^3-16 x^4+120 x^5+\left (-3-16 x-9 x^2+16 x^3+12 x^4\right ) \sqrt {x+\sqrt {1+x^2}}\right )\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]

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]

(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(75 + 24*x - 735*x^2 + 8*x^3 - 1050*x^4 - 32*x^5 + 240*x^6 + 2*(-8 - 9*x -
8*x^2 - 3*x^3 + 16*x^4 + 12*x^5)*Sqrt[x + Sqrt[1 + x^2]] + 2*Sqrt[1 + x^2]*(4 - 60*x + 12*x^2 - 585*x^3 - 16*x
^4 + 120*x^5 + (-3 - 16*x - 9*x^2 + 16*x^3 + 12*x^4)*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[Sqr
t[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 + Sqr
t[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[Sq
rt[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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Maple [F]
time = 0.00, 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*} \text {Failed to integrate} \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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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.64, size = 7099, normalized size = 11.11 \begin {gather*} \text {Too large to display} \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="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) - 28...

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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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Giac [F(-1)] Timed out
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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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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