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3.31.6 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\) [3006]

Optimal. Leaf size=404 \[ -\frac {x \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}-\frac {1}{16} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ]+\frac {1}{16} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.61, 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 {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.46, size = 552, normalized size = 1.37 \begin {gather*} \frac {1}{16} \left (-\frac {8 x \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}-4 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+4 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]-\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*x*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((-1 + x^2)*Sqrt[x + Sqrt[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*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] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2
)/(2*#1^3 - 3*#1^5 + #1^7) & ] + 4*RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (3*Log[Sqrt[1 + Sqrt[x + Sq
rt[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[Sqr
t[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-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 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2/(x+(x^2+1)^(1/2))^(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((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.47, size = 6878, normalized size = 17.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)^2 - 3
/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 3
9*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(1345*sqrt(2) - 223)
 - 78*sqrt(2) + 649) - 39/2*sqrt(2) - 4)*log(1/4*((8674646*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*
sqrt(2) + 8) + 295619989*sqrt(2))*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)^2 + 295619989*sqrt(2
)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 - (8674646*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2)
- 223) - 39*sqrt(2) + 8)^2 - 277588672*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8) + 34946
05993*sqrt(2))*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8) + 8*((17349292*sqrt(1/2)*sqrt(1345*sqrt
(2) - 223) - 338311194*sqrt(2) + 365017157)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8) - 59123997
8*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 11529179571*sqrt(2) + 10589485729)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1345*sq
rt(2) - 223) + 39*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16*(2*sq
rt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) - 24) +
4*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 78*sqrt(2) + 649) + 3494605993*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) -
 223) - 39*sqrt(2) + 8) - 4975202382104*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223)
 + 39*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt
(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) - 24) + 4*sqrt(1/2)*
sqrt(1345*sqrt(2) - 223) - 78*sqrt(2) + 649) - 39/2*sqrt(2) - 4) + 14527409494457*sqrt(sqrt(x + sqrt(x^2 + 1))
 + 1)) - sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)^2 -
 3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) +
 39*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(1345*sqrt(2) - 22
3) - 78*sqrt(2) + 649) - 39/2*sqrt(2) - 4)*log(-1/4*((8674646*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) -
39*sqrt(2) + 8) + 295619989*sqrt(2))*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)^2 + 295619989*sqr
t(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 - (8674646*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(
2) - 223) - 39*sqrt(2) + 8)^2 - 277588672*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8) + 34
94605993*sqrt(2))*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8) + 8*((17349292*sqrt(1/2)*sqrt(1345*s
qrt(2) - 223) - 338311194*sqrt(2) + 365017157)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8) - 59123
9978*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 11529179571*sqrt(2) + 10589485729)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1345
*sqrt(2) - 223) + 39*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16*(2
*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) - 24)
 + 4*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 78*sqrt(2) + 649) + 3494605993*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2
) - 223) - 39*sqrt(2) + 8) - 4975202382104*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 2
23) + 39*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*s
qrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) - 24) + 4*sqrt(1/
2)*sqrt(1345*sqrt(2) - 223) - 78*sqrt(2) + 649) - 39/2*sqrt(2) - 4) + 14527409494457*sqrt(sqrt(x + sqrt(x^2 +
1)) + 1)) + sqrt(2)*(x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)
^2 - 3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 22
3) + 39*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(1345*sqrt(2)
- 223) - 78*sqrt(2) + 649) - 39/2*sqrt(2) - 4)*log(1/4*((8674646*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223)
 - 39*sqrt(2) + 8) + 295619989*sqrt(2))*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8)^2 + 295619989*
sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 - (8674646*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sq
rt(2) - 223) - 39*sqrt(2) + 8)^2 - 277588672*sqrt(2)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8) +
 3494605993*sqrt(2))*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8) - 8*((17349292*sqrt(1/2)*sqrt(134
5*sqrt(2) - 223) - 338311194*sqrt(2) + 365017157)*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqrt(2) - 8) - 59
1239978*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 11529179571*sqrt(2) + 10589485729)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(1
345*sqrt(2) - 223) + 39*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) - 39*sqrt(2) + 8)^2 + 1/16
*(2*sqrt(1/2)*sqrt(1345*sqrt(2) - 223) + 39*sqr...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[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)

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