3.15.56 \(\int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx\)
Optimal. Leaf size=102 \[ 2 \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3-1\& ,\frac {\text {$\#$1}^3 \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )-\log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+3 \text {$\#$1}^2}\& \right ]-2 \sqrt {\sqrt {x^2+1}+x}+x \]
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Rubi [F] time = 1.54, 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 {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[x/(x + Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
x + Log[-1 - 2*x^3 + x^4]/4 + Defer[Int][(-1 - 2*x^3 + x^4)^(-1), x] + (3*Defer[Int][x^2/(-1 - 2*x^3 + x^4), x
])/2 + Defer[Int][(x^2*Sqrt[1 + x^2])/(-1 - 2*x^3 + x^4), x] + Defer[Int][(x^2*Sqrt[x + Sqrt[1 + x^2]])/(-1 -
2*x^3 + x^4), x] - Defer[Int][(x^3*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^3 + x^4), x] - Defer[Int][(x*Sqrt[1 + x^
2]*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^3 + x^4), x]
Rubi steps
\begin {align*} \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (1+\frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4}+\frac {1+x^3}{-1-2 x^3+x^4}+\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}\right ) \, dx\\ &=x+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {1+x^3}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{4} \int \frac {4+6 x^2}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{4} \int \left (\frac {4}{-1-2 x^3+x^4}+\frac {6 x^2}{-1-2 x^3+x^4}\right ) \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {3}{2} \int \frac {x^2}{-1-2 x^3+x^4} \, dx+\int \frac {1}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ \end {align*}
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Mathematica [B] time = 6.96, size = 3760, normalized size = 36.86 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[x/(x + Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
x + ArcSinh[x] + (Log[x - Root[-1 - 2*#1^3 + #1^4 & , 1, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^2] - L
og[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^2]]/
Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^2])/((Root[-1 - 2*#1^3 + #1^4 & , 1, 0] - Root[-1 - 2*#1^3 + #1^4 &
, 2, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 1, 0] - Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(Root[-1 - 2*#1^3 + #1^4 & ,
1, 0] - Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + (Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^2*(Log[x - Root[-1 - 2*#1^3
+ #1^4 & , 1, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 1, 0
] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^
2]))/((Root[-1 - 2*#1^3 + #1^4 & , 1, 0] - Root[-1 - 2*#1^3 + #1^4 & , 2, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 1,
0] - Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 1, 0] - Root[-1 - 2*#1^3 + #1^4 & , 4, 0]
)) + (2*Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^3*(Log[x - Root[-1 - 2*#1^3 + #1^4 & , 1, 0]]/Sqrt[1 + Root[-1 - 2*#
1^3 + #1^4 & , 1, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2
*#1^3 + #1^4 & , 1, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 1, 0]^2]))/((Root[-1 - 2*#1^3 + #1^4 & , 1, 0]
- Root[-1 - 2*#1^3 + #1^4 & , 2, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 1, 0] - Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*
(Root[-1 - 2*#1^3 + #1^4 & , 1, 0] - Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + (Log[x - Root[-1 - 2*#1^3 + #1^4 &
, 2, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 2, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 2, 0] + 2*Sqr
t[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 2, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 2, 0]^2])/((-Ro
ot[-1 - 2*#1^3 + #1^4 & , 1, 0] + Root[-1 - 2*#1^3 + #1^4 & , 2, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 2, 0] - Root
[-1 - 2*#1^3 + #1^4 & , 3, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 2, 0] - Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + (Roo
t[-1 - 2*#1^3 + #1^4 & , 2, 0]^2*(Log[x - Root[-1 - 2*#1^3 + #1^4 & , 2, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4
& , 2, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 2, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^
4 & , 2, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 2, 0]^2]))/((-Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + Root[-1
- 2*#1^3 + #1^4 & , 2, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 2, 0] - Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(Root[-1 -
2*#1^3 + #1^4 & , 2, 0] - Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + (2*Root[-1 - 2*#1^3 + #1^4 & , 2, 0]^3*(Log[x
- Root[-1 - 2*#1^3 + #1^4 & , 2, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 2, 0]^2] - Log[2 + 2*x*Root[-1 - 2*
#1^3 + #1^4 & , 2, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 2, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1
^3 + #1^4 & , 2, 0]^2]))/((-Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + Root[-1 - 2*#1^3 + #1^4 & , 2, 0])*(Root[-1 -
2*#1^3 + #1^4 & , 2, 0] - Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 2, 0] - Root[-1 - 2*
#1^3 + #1^4 & , 4, 0])) + (Log[x - Root[-1 - 2*#1^3 + #1^4 & , 3, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 3,
0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 3, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 3
, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^2])/((-Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + Root[-1 - 2*#1^
3 + #1^4 & , 3, 0])*(-Root[-1 - 2*#1^3 + #1^4 & , 2, 0] + Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(Root[-1 - 2*#1^3
+ #1^4 & , 3, 0] - Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + (Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^2*(Log[x - Root[-
1 - 2*#1^3 + #1^4 & , 3, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1
^4 & , 3, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4
& , 3, 0]^2]))/((-Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(-Root[-1 - 2*#1^3 +
#1^4 & , 2, 0] + Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 3, 0] - Root[-1 - 2*#1^3 + #
1^4 & , 4, 0])) + (2*Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^3*(Log[x - Root[-1 - 2*#1^3 + #1^4 & , 3, 0]]/Sqrt[1 +
Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 3, 0] + 2*Sqrt[1 + x^2]*Sqrt[1
+ Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 3, 0]^2]))/((-Root[-1 - 2*#1^3 +
#1^4 & , 1, 0] + Root[-1 - 2*#1^3 + #1^4 & , 3, 0])*(-Root[-1 - 2*#1^3 + #1^4 & , 2, 0] + Root[-1 - 2*#1^3 + #
1^4 & , 3, 0])*(Root[-1 - 2*#1^3 + #1^4 & , 3, 0] - Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + (Log[x - Root[-1 - 2
*#1^3 + #1^4 & , 4, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 4, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 &
, 4, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 4, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & ,
4, 0]^2])/((-Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + Root[-1 - 2*#1^3 + #1^4 & , 4, 0])*(-Root[-1 - 2*#1^3 + #1^4
& , 2, 0] + Root[-1 - 2*#1^3 + #1^4 & , 4, 0])*(-Root[-1 - 2*#1^3 + #1^4 & , 3, 0] + Root[-1 - 2*#1^3 + #1^4 &
, 4, 0])) + (Root[-1 - 2*#1^3 + #1^4 & , 4, 0]^2*(Log[x - Root[-1 - 2*#1^3 + #1^4 & , 4, 0]]/Sqrt[1 + Root[-1
- 2*#1^3 + #1^4 & , 4, 0]^2] - Log[2 + 2*x*Root[-1 - 2*#1^3 + #1^4 & , 4, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[
-1 - 2*#1^3 + #1^4 & , 4, 0]^2]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 4, 0]^2]))/((-Root[-1 - 2*#1^3 + #1^4 &
, 1, 0] + Root[-1 - 2*#1^3 + #1^4 & , 4, 0])*(-Root[-1 - 2*#1^3 + #1^4 & , 2, 0] + Root[-1 - 2*#1^3 + #1^4 & ,
4, 0])*(-Root[-1 - 2*#1^3 + #1^4 & , 3, 0] + Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + (2*Root[-1 - 2*#1^3 + #1^4
& , 4, 0]^3*(Log[x - Root[-1 - 2*#1^3 + #1^4 & , 4, 0]]/Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 4, 0]^2] - Log[2
+ 2*x*Root[-1 - 2*#1^3 + #1^4 & , 4, 0] + 2*Sqrt[1 + x^2]*Sqrt[1 + Root[-1 - 2*#1^3 + #1^4 & , 4, 0]^2]]/Sqrt
[1 + Root[-1 - 2*#1^3 + #1^4 & , 4, 0]^2]))/((-Root[-1 - 2*#1^3 + #1^4 & , 1, 0] + Root[-1 - 2*#1^3 + #1^4 & ,
4, 0])*(-Root[-1 - 2*#1^3 + #1^4 & , 2, 0] + Root[-1 - 2*#1^3 + #1^4 & , 4, 0])*(-Root[-1 - 2*#1^3 + #1^4 & ,
3, 0] + Root[-1 - 2*#1^3 + #1^4 & , 4, 0])) + RootSum[-1 - 2*#1^3 + #1^4 & , (Log[x - #1] + Log[x - #1]*#1^3)
/(-3*#1^2 + 2*#1^3) & ]/2 - ((x + Sqrt[1 + x^2])*Sqrt[2 + (x + Sqrt[1 + x^2])^(-2) + (x + Sqrt[1 + x^2])^2]*(2
*RootSum[-1 - 2*#1^3 + #1^4 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3)/(
-3*#1^2 + 2*#1^3) & ] - 2*RootSum[-1 + 2*#1^3 + #1^4 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sq
rt[1 + x^2]] - #1]*#1^3)/(3*#1^2 + 2*#1^3) & ] - RootSum[1 + 4*#1^2 - 2*#1^4 + #1^8 & , (-Log[Sqrt[x + Sqrt[1
+ x^2]] - #1] - 2*Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^2 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^4)/(#1 - #1^3
+ #1^7) & ]))/(4*(1 + (x + Sqrt[1 + x^2])^2)) - 2*(Sqrt[x + Sqrt[1 + x^2]] + RootSum[-1 - 2*#1^3 + #1^4 & , (L
og[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3)/(-3*#1^2 + 2*#1^3) & ]/4 - RootSum[
-1 + 2*#1^3 + #1^4 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3)/(3*#1^2 +
2*#1^3) & ]/4 + RootSum[1 + 4*#1^2 - 2*#1^4 + #1^8 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] - 2*Log[Sqrt[x + S
qrt[1 + x^2]] - #1]*#1^2 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^4)/(#1 - #1^3 + #1^7) & ]/8)
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IntegrateAlgebraic [A] time = 0.61, size = 102, normalized size = 1.00 \begin {gather*} x-2 \sqrt {x+\sqrt {1+x^2}}+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x/(x + Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
x - 2*Sqrt[x + Sqrt[1 + x^2]] + 2*RootSum[-1 + 2*#1^3 + #1^4 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqr
t[x + Sqrt[1 + x^2]] - #1]*#1^3)/(3*#1^2 + 2*#1^3) & ]
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fricas [F(-1)] 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+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="fricas")
[Out]
Timed out
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="giac")
[Out]
integrate(x/(x + sqrt(x + sqrt(x^2 + 1))), x)
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x}{x +\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(x+(x+(x^2+1)^(1/2))^(1/2)),x)
[Out]
int(x/(x+(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*} \int \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="maxima")
[Out]
integrate(x/(x + sqrt(x + sqrt(x^2 + 1))), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{x+\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(x + (x + (x^2 + 1)^(1/2))^(1/2)),x)
[Out]
int(x/(x + (x + (x^2 + 1)^(1/2))^(1/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x+(x+(x**2+1)**(1/2))**(1/2)),x)
[Out]
Integral(x/(x + sqrt(x + sqrt(x**2 + 1))), x)
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