3.31.1 \(\int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2} \, dx\) [3001]

Optimal. Leaf size=401 \[ -\frac {x \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (-1+x^2\right )}-\frac {1}{16} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ]-\frac {1}{16} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {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[(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]/4 + Defer[Int][(Sqrt[x +
Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x), x]/4 + Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 +
Sqrt[x + Sqrt[1 + x^2]]])/(1 + x)^2, x]/4 + Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^
2]]])/(1 + x), x]/4

Rubi steps

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

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Mathematica [A]
time = 0.48, size = 546, normalized size = 1.36 \begin {gather*} \frac {1}{16} \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 {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-4 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]-\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-9 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(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 + 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) & ])/16

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+(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+(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+(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(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 = 1.59, size = 6966, normalized size = 17.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

-1/32*(sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 + 1/
16*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) +
12) - 3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37
*sqrt(2) + 913) - 37/2*sqrt(2) + 2)*log(1/8*((7252423*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2
) - 4) - 86307886*sqrt(2))*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 - 86307886*sqrt(2)*(2*sqrt
(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - (7252423*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37
*sqrt(2) - 4)^2 + 116038768*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4) - 10516321158*sqrt(
2))*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4) + 8*((14504846*sqrt(1/2)*sqrt(877*sqrt(2) + 457) -
268339651*sqrt(2) - 115317578)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4) + 172615772*sqrt(1/2)*sq
rt(877*sqrt(2) + 457) - 3193391782*sqrt(2) - 9480626526)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*
sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 4
57) - 37*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(877*
sqrt(2) + 457) + 37*sqrt(2) + 913) - 10516321158*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4
) - 128487844240*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 + 1
/16*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) +
12) - 3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 3
7*sqrt(2) + 913) - 37/2*sqrt(2) + 2) + 1238984819345*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 - 1)*sq
rt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(877*sq
rt(2) + 457) + 37*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqr
t(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 913) - 37/2*sqrt
(2) + 2)*log(-1/8*((7252423*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4) - 86307886*sqrt(2))
*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 - 86307886*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 4
57) - 37*sqrt(2) - 4)^2 - (7252423*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 + 11603876
8*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4) - 10516321158*sqrt(2))*(2*sqrt(1/2)*sqrt(877*
sqrt(2) + 457) + 37*sqrt(2) + 4) + 8*((14504846*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 268339651*sqrt(2) - 115317
578)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4) + 172615772*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 31
93391782*sqrt(2) - 9480626526)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 + 1/16*(2*s
qrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) + 12) - 3
/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2
) + 913) - 10516321158*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4) - 128487844240*sqrt(2))*
sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(877*
sqrt(2) + 457) + 37*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*s
qrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 913) - 37/2*sq
rt(2) + 2) + 1238984819345*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/32*(2*
sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2)
+ 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37
*sqrt(2) - 4)^2 - 2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 913) - 37/2*sqrt(2) + 2)*log(1/8*((725242
3*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4) - 86307886*sqrt(2))*(2*sqrt(1/2)*sqrt(877*sqr
t(2) + 457) + 37*sqrt(2) + 4)^2 - 86307886*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 -
(7252423*sqrt(2)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) - 4)^2 + 116038768*sqrt(2)*(2*sqrt(1/2)*sqr
t(877*sqrt(2) + 457) - 37*sqrt(2) - 4) - 10516321158*sqrt(2))*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2
) + 4) - 8*((14504846*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 268339651*sqrt(2) - 115317578)*(2*sqrt(1/2)*sqrt(877
*sqrt(2) + 457) + 37*sqrt(2) + 4) + 172615772*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 3193391782*sqrt(2) - 9480626
526)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) + 37*sqrt(2) + 4)^2 + 1/16*(2*sqrt(1/2)*sqrt(877*sqrt(2)
+ 457) + 37*sqrt(2) + 4)*(2*sqrt(1/2)*sqrt(877*sqrt(2) + 457) - 37*sqrt(2) + 12) - 3/32*(2*sqrt(1/2)*sqrt(877*
sqrt(2) + 457) - 37*sqrt(2) - 4)^2 - 2*sqrt(1/2...

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

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