3.32.5 \(\int \frac {(1+x^4) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx\) [3105]

Optimal. Leaf size=590 \[ \frac {\left (6+16 x-48 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+(15-8 x) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left ((16-48 x) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-8 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{60 x+60 \sqrt {1+x^2}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ]+\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{1-2 \text {$\#$1}^2+\text {$\#$1}^4}\& \right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.61, 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 {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 551, normalized size = 0.93 \begin {gather*} \frac {1}{4} \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-6+48 x^2-16 \sqrt {1+x^2}-15 \sqrt {x+\sqrt {1+x^2}}+8 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}+8 x \left (-2+6 \sqrt {1+x^2}+\sqrt {x+\sqrt {1+x^2}}\right )\right )}{15 \left (x+\sqrt {1+x^2}\right )}-\tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-2 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+4 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{1-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]+2 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]

[Out]

(-1/15*(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-6 + 48*x^2 - 16*Sqrt[1 + x^2] - 15*Sqrt[x + Sqrt[1 + x^2]] + 8*Sqr
t[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]] + 8*x*(-2 + 6*Sqrt[1 + x^2] + Sqrt[x + Sqrt[1 + x^2]])))/(x + Sqrt[1 + x^2]
) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqr
t[x + Sqrt[1 + x^2]]] - #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 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 + #1^4) & ] + 2*RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 +
#1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2
- 4*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 \[\int \frac {\left (x^{4}+1\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{-x^{4}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x)

[Out]

int((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),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^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="maxima")

[Out]

-integrate((x^4 + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^4 - 1), x)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.76, size = 3408, normalized size = 5.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="fricas")

[Out]

1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4)*sqrt(sqrt(2) + 1)*(sqrt(2) - 2)*a
rctan(1/2*sqrt(sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(sqr
t(2) + 2)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqrt(x^2 + 1)) + 2*sqrt(sqrt(2) + 2) + 2)*((2*s
qrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)
*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4) + 1/2*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (
3*sqrt(2) - 4)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4)*sqr
t(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(2) - 1) - sqrt(sqrt(2) + 1)*(sqrt(2
) - 1)) + 1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4)*sqrt(sqrt(2) + 1)*(sqrt
(2) - 2)*arctan(1/2*sqrt(-sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(sqrt(2) + 2)*(sqrt(2) - 2
) - 2)*(sqrt(2) + 2)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqrt(x^2 + 1)) + 2*sqrt(sqrt(2) + 2)
+ 2)*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(s
qrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4) + 1/2*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(sqrt(
2) + 1) + (3*sqrt(2) - 4)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2
)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(2) - 1) + sqrt(sqrt(2) +
1)*(sqrt(2) - 1)) - sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2)*arctan(1/2*sq
rt(sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2)*sqrt(sqrt(x + sqrt(x^2 + 1)) +
1) + 2*sqrt(x + sqrt(x^2 + 1)) + 2*sqrt(sqrt(2) + 2) + 2)*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) +
2)^(3/4)*(sqrt(2) - 2) + sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 1)*sqrt(sqr
t(x + sqrt(x^2 + 1)) + 1) - sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2) - 1) - sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) +
4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2)*arctan(1/8*sqrt(-16*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) +
2)^(3/4)*(sqrt(2) - 2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 32*sqrt(x + sqrt(x^2 + 1)) + 32*sqrt(sqrt(2) + 2) +
32)*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 2) + sqrt((sqrt(2) + 2)^(3/2) +
2*sqrt(2) + 4)*(sqrt(2) + 2)^(3/4)*(sqrt(2) - 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(2)*sqrt(sqrt(2) + 2)
+ sqrt(2) + 1) + 1/16*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2)
+ 32)^(3/4)*arctan(1/128*sqrt(sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16
*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 32*sqrt(x + sqrt(x^2 + 1)) + 8*sqrt(-16*sqrt(2) + 32)
+ 32)*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4) - 1
/16*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 1)*(-16*sqrt(2) + 32)^(3/4)*sqrt(s
qrt(x + sqrt(x^2 + 1)) + 1) - 1/4*sqrt(2)*sqrt(-16*sqrt(2) + 32) - sqrt(2) + 1) + 1/16*sqrt(-(sqrt(2) - 2)*sqr
t(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)*arctan(1/128*sqrt(-sqrt(-(sqrt(2)
- 2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 +
1)) + 1) + 32*sqrt(x + sqrt(x^2 + 1)) + 8*sqrt(-16*sqrt(2) + 32) + 32)*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) +
32) - 8*sqrt(2) + 16)*(sqrt(2) + 2)*(-16*sqrt(2) + 32)^(3/4) - 1/16*sqrt(-(sqrt(2) - 2)*sqrt(-16*sqrt(2) + 32
) - 8*sqrt(2) + 16)*(sqrt(2) + 1)*(-16*sqrt(2) + 32)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1/4*sqrt(2)*sqr
t(-16*sqrt(2) + 32) + sqrt(2) - 1) - 1/2*2^(7/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(2) -
1)*arctan(1/2*2^(3/8)*sqrt(2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*sqrt(sqrt(x +
sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqrt(x^2 + 1)) + 2*2^(1/4) + 2)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (
sqrt(2) + 1)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4) - 1/2*2^(3/8)*(2^(1/4)*(sqrt(2)
+ 2)*sqrt(sqrt(2) - 1) + (sqrt(2) + 2)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt
(sqrt(x + sqrt(x^2 + 1)) + 1) - 2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) - (sqrt(2) + 1)*sqrt(sqrt(2) - 1)) - 1
/2*2^(7/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(2) - 1)*arctan(1/2*2^(3/8)*sqrt(-2^(1/8)*s
qrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqr
t(x^2 + 1)) + 2*2^(1/4) + 2)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1))*sqrt(
-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4) - 1/2*2^(3/8)*(2^(1/4)*(sqrt(2) + 2)*sqrt(sqrt(2) - 1) + (sqrt(2) +
2)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2^(1/
4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) +...

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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}}{x^{4} - 1}\, dx - \int \frac {x^{4} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**4+1),x)

[Out]

-Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4 - 1), x) - Integral(x**4*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)
/(x**4 - 1), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="giac")

[Out]

integrate(-(x^4 + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^4 - 1), x)

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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^4+1\right )}{x^4-1} \,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^4 + 1))/(x^4 - 1),x)

[Out]

int(-(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^4 + 1))/(x^4 - 1), x)

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