3.32.5 \(\int \frac {(1+x^4) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx\)
Optimal. Leaf size=590 \[ \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+1}\& \right ]-\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\& ,\frac {\text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-2 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3}\& \right ]+\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-4 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+6 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+4 \text {$\#$1}^3-2 \text {$\#$1}}\& \right ]+\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (15-8 x)+\sqrt {x^2+1} \left ((16-48 x) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-8 \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (-48 x^2+16 x+6\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{60 \sqrt {x^2+1}+60 x}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 0.90, 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*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 571, normalized size = 0.97 \begin {gather*} \frac {1}{120} \left (120 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\&,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+1}\&\right ]-60 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\&,\frac {\text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-2 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+2 \text {$\#$1}^3}\&\right ]+60 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\&,\frac {\text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-4 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+6 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+4 \text {$\#$1}^3-2 \text {$\#$1}}\&\right ]-48 \left (\sqrt {\sqrt {x^2+1}+x}+1\right )^{5/2}+80 \left (\sqrt {\sqrt {x^2+1}+x}+1\right )^{3/2}+\frac {30 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {\sqrt {x^2+1}+x}}+\frac {60 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {x^2+1}+x}+15 \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )-15 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\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]
((60*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(x + Sqrt[1 + x^2]) + (30*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[x +
Sqrt[1 + x^2]] + 80*(1 + Sqrt[x + Sqrt[1 + x^2]])^(3/2) - 48*(1 + Sqrt[x + Sqrt[1 + x^2]])^(5/2) + 15*Log[1 -
Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 15*Log[1 + Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 60*RootSum[-2 + 4*#1^4 -
4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[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) & ] + 120*RootSum[2 - 4*#
1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 + #1^4) & ] + 60
*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 + Sq
rt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ])/120
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 590, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]
[Out]
((6 + 16*x - 48*x^2)*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]*((16 - 48*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 8*Sqrt[x + Sqrt[1 + x^2]]*Sqr
t[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(60*x + 60*Sqrt[1 + x^2]) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/4 - Ro
otSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[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) & ]/2
+ RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^
2 + #1^4) & ] + 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) & ]/2
________________________________________________________________________________________
fricas [B] time = 1.61, size = 3408, normalized size = 5.78
result too large to display
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) + 1)*sqrt(sqrt(2) - 1)) - 1/4*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(
2) + 4)*(sqrt(2)*sqrt(sqrt(2) + 2) - 2*sqrt(2))*(sqrt(2) + 2)^(1/4)*log(2*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) + 4*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt
(sqrt(2) + 2) + 4) + 1/4*sqrt((sqrt(2) + 2)^(3/2) + 2*sqrt(2) + 4)*(sqrt(2)*sqrt(sqrt(2) + 2) - 2*sqrt(2))*(sq
rt(2) + 2)^(1/4)*log(-2*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) + 4*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4) - 1/64*sqrt(-(sqrt(2) - 2)*sqrt
(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2))*(-16*sqrt(2) + 32)^(1/4)*log
(1/8*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) + 4*sqrt(x + sqrt(x^2 + 1)) + sqrt(-16*sqrt(2) + 32) + 4) + 1/64*sqrt(-(sqrt(2) -
2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2) + 16)*(sqrt(2)*sqrt(-16*sqrt(2) + 32) - 8*sqrt(2))*(-16*sqrt(2) + 32)^(
1/4)*log(-1/8*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) + 4*sqrt(x + sqrt(x^2 + 1)) + sqrt(-16*sqrt(2) + 32) + 4) - 1/8*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)*log(1/2
*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) + sqrt(x + sqrt(x^2 + 1)) + sqrt(sqrt(2) + 2) + 1) + 1/8*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)*log(-1/2*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(s
qrt(x + sqrt(x^2 + 1)) + 1) + sqrt(x + sqrt(x^2 + 1)) + sqrt(sqrt(2) + 2) + 1) + 1/8*2^(1/8)*sqrt(-2*2^(1/4)*(
sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*log(1/2*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) + sqrt(x + sqrt(x^2 + 1)) + 2^(1/4) + 1) - 1/8*2^(1/8)*sqrt(-2*2^
(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*log(-1/2*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) + sqrt(x + sqrt(x^2 + 1)) + 2^(1/4) + 1) - 1/60*((15*x -
15*sqrt(x^2 + 1) + 8)*sqrt(x + sqrt(x^2 + 1)) + 54*x - 6*sqrt(x^2 + 1) - 16)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)
- 2*sqrt(sqrt(sqrt(2) + 1) - 1)*arctan(1/2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*sqrt(sqrt(x + sqrt(x^2 + 1))
+ sqrt(sqrt(2) + 1))*sqrt(sqrt(sqrt(2) + 1) - 1) - 1/2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*sqrt(sqrt(x + sq
rt(x^2 + 1)) + 1)*sqrt(sqrt(sqrt(2) + 1) - 1)) + 1/2*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 +
1)) + 1) + sqrt(sqrt(sqrt(2) + 1) + 1)) - 1/2*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1
) - sqrt(sqrt(sqrt(2) + 1) + 1)) - 1/2*sqrt(sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqr
t(sqrt(sqrt(2) - 1) + 1)) + 1/2*sqrt(sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(
sqrt(2) - 1) + 1)) - 1/2*sqrt(-sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(-sqrt(sqrt(
2) - 1) + 1)) + 1/2*sqrt(-sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(-sqrt(sqrt(2) -
1) + 1)) - 1/8*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) + 1/8*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{4} + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1}\,{d x} \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)
________________________________________________________________________________________
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)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{4} + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1}\,{d x} \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)
________________________________________________________________________________________
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)
________________________________________________________________________________________
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)
________________________________________________________________________________________