3.30.6 \(\int \frac {(1+x^2) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2} \, dx\) [2906]

Optimal. Leaf size=323 \[ -\frac {x \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{-1+x^2}+\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\& \right ]+\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^5}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 2.29, 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 ) \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)*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]]])/(1 - x)^2, x]/2 + Defer[Int][(Sqrt[x +
Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x)^2, x]/2

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx &=\int \left (\frac {2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2}-\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2}\right ) \, dx\\ &=2 \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx-\int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx\\ &=2 \int \left (\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1-x)^2}+\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1+x)^2}+\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right )}\right ) \, dx-\int \left (\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x)}+\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x)}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x} \, dx+\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx-\frac {1}{2} \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\\ &=\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x} \, dx+\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx-\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x} \, dx+\int \left (\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x)}+\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x)}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{2} \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 [B] Leaf count is larger than twice the leaf count of optimal. \(682\) vs. \(2(323)=646\).
time = 0.01, size = 682, normalized size = 2.11 \begin {gather*} \frac {1}{8} \left (-\frac {8 x \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{-1+x^2}+4 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+4 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]-\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-9 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*x*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(-1 + x^2) + 4*RootSum[-2 + 4*#1^4 - 4*#1^6
+ #1^8 & , (3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*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*#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) - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + 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 + Sq
rt[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) &
])/8

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)*(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)*(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)*(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)*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 = 1.53, size = 6976, normalized size = 21.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(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]

1/16*(sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 + 1/1
6*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) + 1
2) - 3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*
sqrt(2) + 673) - 5/2*sqrt(2) + 2)*log(1/8*(5*(300981*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2)
- 4) - 1111618*sqrt(2))*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 - 5558090*sqrt(2)*(2*sqrt(1/
2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 - (1504905*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sq
rt(2) - 4)^2 + 24078480*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4) - 213263242*sqrt(2))*(2
*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4) + 8*(5*(601962*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 15049
05*sqrt(2) - 2315542)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4) + 11116180*sqrt(1/2)*sqrt(941*sqr
t(2) + 1321) - 27790450*sqrt(2) - 146566162)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)
^2 + 1/16*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqr
t(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(941*sqrt(2) + 13
21) + 5*sqrt(2) + 673) - 213263242*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4) - 5544442608
*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)
*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) + 12) - 3/32*(2*s
qrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 673)
- 5/2*sqrt(2) + 2) + 18101760817*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/
32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*s
qrt(2) + 4)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 132
1) - 5*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 673) - 5/2*sqrt(2) + 2)*log(-1/8*(5
*(300981*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4) - 1111618*sqrt(2))*(2*sqrt(1/2)*sqrt(9
41*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 - 5558090*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)
^2 - (1504905*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 + 24078480*sqrt(2)*(2*sqrt(1/2)
*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4) - 213263242*sqrt(2))*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt
(2) + 4) + 8*(5*(601962*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 1504905*sqrt(2) - 2315542)*(2*sqrt(1/2)*sqrt(941*
sqrt(2) + 1321) + 5*sqrt(2) + 4) + 11116180*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 27790450*sqrt(2) - 146566162)
*sqrt(-3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 13
21) + 5*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(941*sqrt
(2) + 1321) - 5*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 673) - 213263242*sqrt(2)*(
2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4) - 5544442608*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/
2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)*(
2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2
) - 4)^2 - 2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 673) - 5/2*sqrt(2) + 2) + 18101760817*sqrt(sqrt(
x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) +
5*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(941*sqrt(2) +
1321) - 5*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(94
1*sqrt(2) + 1321) + 5*sqrt(2) + 673) - 5/2*sqrt(2) + 2)*log(1/8*(5*(300981*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(
2) + 1321) - 5*sqrt(2) - 4) - 1111618*sqrt(2))*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)^2 - 5558
090*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 - (1504905*sqrt(2)*(2*sqrt(1/2)*sqrt(941*
sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 + 24078480*sqrt(2)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4) -
213263242*sqrt(2))*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4) - 8*(5*(601962*sqrt(1/2)*sqrt(941*s
qrt(2) + 1321) - 1504905*sqrt(2) - 2315542)*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4) + 11116180*
sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 27790450*sqrt(2) - 146566162)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) +
1321) + 5*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(941*s
qrt(2) + 1321) - 5*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(941*sqrt(2) + 1321) - 5*sqrt(2) - 4)^2 - 2*sqrt(1/2)
*sqrt(941*sqrt(2) + 1321) + 5*sqrt(2) + 673) - ...

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

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