3.32.12 \(\int \frac {(1+x^2)^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\) [3112]

Optimal. Leaf size=639 \[ \frac {\left (-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (2-48 x+2 x^2-16 x^3-4 x^4+64 x^5\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-16+4 x-48 x^2-4 x^3+64 x^4\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{24 \left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{7/2}}-\frac {1}{8} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ]-\frac {1}{4} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+9 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.48, 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 {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[((1 + x^2)^2*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]]]/Sqrt[x + Sqrt[1 + x^2]], x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1
+ x^2]]]/((-1 - x)*Sqrt[x + Sqrt[1 + x^2]]), x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((-1 + x)^2*Sq
rt[x + Sqrt[1 + x^2]]), x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((-1 + x)*Sqrt[x + Sqrt[1 + x^2]]),
x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x)^2*Sqrt[x + Sqrt[1 + x^2]]), x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 636, normalized size = 1.00 \begin {gather*} \frac {1}{24} \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6+2 \left (1-24 x+x^2-8 x^3-2 x^4+32 x^5\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5+4 \left (-4+x-12 x^2-x^3+16 x^4\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{7/2}}-3 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+24 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]-6 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-24 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]-6 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-8 - 9*x - 248*x^2 - 3*x^3 - 208*x^4 + 12*x^5 + 128*x^6 + 2*(1 - 24*x + x
^2 - 8*x^3 - 2*x^4 + 32*x^5)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-3 - 96*x - 9*x^2 - 272*x^3 + 12*x^4 + 1
28*x^5 + 4*(-4 + x - 12*x^2 - x^3 + 16*x^4)*Sqrt[x + Sqrt[1 + x^2]])))/((-1 + x^2)*(x + Sqrt[1 + x^2])^(7/2))
- 3*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + 24*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqr
t[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + #1^3) & ] - 6*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) & ] - 24*RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log
[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3)/(2 - 2*#1^2 +
#1^4) & ] - 6*RootSum[2 - 8*#1^2 + 8*#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 + 4*#1^3 - 3*#1^5 + #1^7) & ])/24

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)^2*(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((x^2+1)^2*(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((x^2+1)^2*(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((x^2 + 1)^2*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 = 1.55, size = 6983, normalized size = 10.93 \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*(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]

1/48*(6*sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2
- 3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136)^2 + 1/16*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487)
+ 103*sqrt(2) - 136)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) - 408) + 204*sqrt(1/2)*sqrt(377*sqrt(
2) - 487) - 3502*sqrt(2) + 3113) - 103/2*sqrt(2) - 68)*log(1/4*((87218678*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2
) - 487) - 103*sqrt(2) + 136) + 2664194769*sqrt(2))*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^
2 + 2664194769*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136)^2 - (87218678*sqrt(2)*(6*sqrt
(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136)^2 - 47446960832*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487
) - 103*sqrt(2) + 136) - 980021093843*sqrt(2))*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136) + 8*(
(523312068*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 8983523834*sqrt(2) + 14525934977)*(6*sqrt(1/2)*sqrt(377*sqrt(2)
- 487) + 103*sqrt(2) - 136) - 15985168614*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 274412061207*sqrt(2) + 10697037
1909)*sqrt(-3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 - 3/32*(6*sqrt(1/2)*sqrt(377*sqrt
(2) - 487) - 103*sqrt(2) + 136)^2 + 1/16*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)*(6*sqrt(1/2
)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) - 408) + 204*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 3502*sqrt(2) + 3113)
- 980021093843*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136) - 50300006241720*sqrt(2))*sqr
t(sqrt(2)*sqrt(-3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 - 3/32*(6*sqrt(1/2)*sqrt(377*
sqrt(2) - 487) - 103*sqrt(2) + 136)^2 + 1/16*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)*(6*sqrt
(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) - 408) + 204*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 3502*sqrt(2) + 31
13) - 103/2*sqrt(2) - 68) + 643948190735955*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 6*sqrt(2)*(x^2 - 1)*sqrt(sqrt
(2)*sqrt(-3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 - 3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2
) - 487) - 103*sqrt(2) + 136)^2 + 1/16*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)*(6*sqrt(1/2)*
sqrt(377*sqrt(2) - 487) - 103*sqrt(2) - 408) + 204*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 3502*sqrt(2) + 3113) -
103/2*sqrt(2) - 68)*log(-1/4*((87218678*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136) + 26
64194769*sqrt(2))*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 + 2664194769*sqrt(2)*(6*sqrt(1/2
)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136)^2 - (87218678*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 10
3*sqrt(2) + 136)^2 - 47446960832*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136) - 980021093
843*sqrt(2))*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136) + 8*((523312068*sqrt(1/2)*sqrt(377*sqrt
(2) - 487) - 8983523834*sqrt(2) + 14525934977)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136) - 159
85168614*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 274412061207*sqrt(2) + 106970371909)*sqrt(-3/32*(6*sqrt(1/2)*sqrt
(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 - 3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136)^2
+ 1/16*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sq
rt(2) - 408) + 204*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 3502*sqrt(2) + 3113) - 980021093843*sqrt(2)*(6*sqrt(1/2
)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136) - 50300006241720*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(6*sqrt(1/2)*
sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 - 3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136
)^2 + 1/16*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 10
3*sqrt(2) - 408) + 204*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 3502*sqrt(2) + 3113) - 103/2*sqrt(2) - 68) + 643948
190735955*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 6*sqrt(2)*(x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/32*(6*sqrt(1/2)*sqrt(
377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 - 3/32*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136)^2 +
1/16*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqr
t(2) - 408) + 204*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 3502*sqrt(2) + 3113) - 103/2*sqrt(2) - 68)*log(1/4*((872
18678*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136) + 2664194769*sqrt(2))*(6*sqrt(1/2)*sqr
t(377*sqrt(2) - 487) + 103*sqrt(2) - 136)^2 + 2664194769*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sq
rt(2) + 136)^2 - (87218678*sqrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136)^2 - 47446960832*s
qrt(2)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 103*sqrt(2) + 136) - 980021093843*sqrt(2))*(6*sqrt(1/2)*sqrt(377
*sqrt(2) - 487) + 103*sqrt(2) - 136) - 8*((523312068*sqrt(1/2)*sqrt(377*sqrt(2) - 487) - 8983523834*sqrt(2) +
14525934977)*(6*sqrt(1/2)*sqrt(377*sqrt(2) - 487) + 103*sqrt(2) - 136) - 15985168614*sqrt(1/2)*sqrt(377*sqrt(2
) - 487) + 274412061207*sqrt(2) + 106970371909)...

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

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