3.22.36 \(\int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx\)

Optimal. Leaf size=155 \[ 4 \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3-5 \text {$\#$1}^2+3 \text {$\#$1}+1\& ,\frac {\text {$\#$1}^3 \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )+\text {$\#$1} \log \left (\text {$\#$1}+\sqrt {x+1}-\sqrt {x+\sqrt {x+1}}\right )}{4 \text {$\#$1}^3+6 \text {$\#$1}^2-10 \text {$\#$1}+3}\& \right ]+x-2 \sqrt {x+\sqrt {x+1}}-\log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]

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Rubi [F]  time = 0.94, 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}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x/(x + Sqrt[x + Sqrt[1 + x]]),x]

[Out]

x + Log[2 - Sqrt[1 + x] - 3*(1 + x) + (1 + x)^2]/2 + Defer[Subst][Defer[Int][(2 - x - 3*x^2 + x^4)^(-1), x], x
, Sqrt[1 + x]]/2 + Defer[Subst][Defer[Int][x/(2 - x - 3*x^2 + x^4), x], x, Sqrt[1 + x]] + 2*Defer[Subst][Defer
[Int][x^2/(2 - x - 3*x^2 + x^4), x], x, Sqrt[1 + x]] + 2*Defer[Subst][Defer[Int][(x*Sqrt[-1 + x + x^2])/(2 - x
- 3*x^2 + x^4), x], x, Sqrt[1 + x]] - 2*Defer[Subst][Defer[Int][(x^3*Sqrt[-1 + x + x^2])/(2 - x - 3*x^2 + x^4
), x], x, Sqrt[1 + x]]

Rubi steps

\begin {align*} \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {x}{-1+x^2+\sqrt {-1+x+x^2}}+\frac {x^3}{-1+x^2+\sqrt {-1+x+x^2}}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {x}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\right )+2 \operatorname {Subst}\left (\int \frac {x^3}{-1+x^2+\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \left (\frac {x \left (-1+x^2\right )}{2-x-3 x^2+x^4}-\frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )\right )+2 \operatorname {Subst}\left (\int \left (x-\frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4}+\frac {x \left (-2+x+2 x^2\right )}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=x-2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \left (-2+x+2 x^2\right )}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+2 x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+4 x+4 x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{2-x-3 x^2+x^4}+\frac {2 x}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {2}{2-x-3 x^2+x^4}+\frac {4 x}{2-x-3 x^2+x^4}+\frac {4 x^2}{2-x-3 x^2+x^4}\right ) \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=x+\frac {1}{2} \log \left (2-\sqrt {1+x}-3 (1+x)+(1+x)^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x^2}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {x^3 \sqrt {-1+x+x^2}}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {1}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )-\operatorname {Subst}\left (\int \frac {x}{2-x-3 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ \end {align*}

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Mathematica [B]  time = 6.41, size = 4996, normalized size = 32.23 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(x + Sqrt[x + Sqrt[1 + x]]),x]

[Out]

x - 2*Sqrt[x + Sqrt[1 + x]] - Log[1 + 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] - (2*(1 + x)*Sqrt[(1 + (-1 + Sq
rt[1 + x])/(1 + x))/(1 + x)]*((2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]*(-1 + 2/Sqrt[1 + x])*(-(Log[-1 + 2/Sqr
t[1 + x]]/Sqrt[5]) + Log[5 + 2*Sqrt[5]*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]]/Sqrt[5]))/((-2 + Sqrt[1 + x])*S
qrt[(1 + (-1 + Sqrt[1 + x])/(1 + x))/(1 + x)]) - ((1 + 2/(1 + x)^2 - (1 + x)^(-3/2) - 3/(1 + x))*Sqrt[1 - (1 +
x)^(-1) + 1/Sqrt[1 + x]]*(-1 + 2/Sqrt[1 + x])*((Log[-1 + 2/Sqrt[1 + x]]/Sqrt[5] - Log[5 + 2*Sqrt[5]*Sqrt[1 -
(1 + x)^(-1) + 1/Sqrt[1 + x]]]/Sqrt[5])/(2*(1/2 - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0])*(1/2 - Root[1 - 3
*#1^2 - #1^3 + 2*#1^4 & , 2, 0])*(1/2 - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(1/2 - Root[1 - 3*#1^2 - #1
^3 + 2*#1^4 & , 4, 0])) + (Log[1/Sqrt[1 + x] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0]]/Sqrt[1 + Root[1 - 3*
#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0]^2] - Log[2 + 1/Sqrt[1 + x] + Root[1
- 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - (2*Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0])/Sqrt[1 + x] + 2*Sqrt[1 - (
1 + x)^(-1) + 1/Sqrt[1 + x]]*Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#
1^4 & , 1, 0]^2]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1,
0]^2])/(4*(-1/2 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root
[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2 - #1^3 + 2
*#1^4 & , 3, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])) + (L
og[1/Sqrt[1 + x] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2,
0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]^2] - Log[2 + 1/Sqrt[1 + x] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 &
, 2, 0] - (2*Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0])/Sqrt[1 + x] + 2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]
*Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]^2]]/Sqrt[1 + R
oot[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]^2])/(4*(-1/2 + Root[1 - 3
*#1^2 - #1^3 + 2*#1^4 & , 2, 0])*(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4
& , 2, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(Root[1 -
3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])) + (Log[1/Sqrt[1 + x] - Root[1 -
3*#1^2 - #1^3 + 2*#1^4 & , 3, 0]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0] - Root[1 - 3*#1^2 - #1^3
+ 2*#1^4 & , 3, 0]^2] - Log[2 + 1/Sqrt[1 + x] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0] - (2*Root[1 - 3*#1^
2 - #1^3 + 2*#1^4 & , 3, 0])/Sqrt[1 + x] + 2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]*Sqrt[1 + Root[1 - 3*#1^2 -
#1^3 + 2*#1^4 & , 3, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0]^2]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#
1^4 & , 3, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0]^2])/(4*(-1/2 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3
, 0])*(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(-Root[1 - 3*#1
^2 - #1^3 + 2*#1^4 & , 2, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & ,
3, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])) + (Log[1/Sqrt[1 + x] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 &
, 4, 0]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]^2] - L
og[2 + 1/Sqrt[1 + x] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0] - (2*Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0
])/Sqrt[1 + x] + 2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]*Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0] -
Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]^2]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0] - Root[1 - 3
*#1^2 - #1^3 + 2*#1^4 & , 4, 0]^2])/(4*(-1/2 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])*(-Root[1 - 3*#1^2 -
#1^3 + 2*#1^4 & , 1, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])*(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2,
0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])*(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0] + Root[1 - 3*#1^2
- #1^3 + 2*#1^4 & , 4, 0]))))/((-1 - 2/(1 + x)^2 + (1 + x)^(-3/2) + 3/(1 + x))*(-2 + Sqrt[1 + x])*Sqrt[(1 + (-
1 + Sqrt[1 + x])/(1 + x))/(1 + x)]) + ((1 + 2/(1 + x)^2 - (1 + x)^(-3/2) - 3/(1 + x))*Sqrt[1 - (1 + x)^(-1) +
1/Sqrt[1 + x]]*(-1 + 2/Sqrt[1 + x])*((Log[-1 + 2/Sqrt[1 + x]]/Sqrt[5] - Log[5 + 2*Sqrt[5]*Sqrt[1 - (1 + x)^(-1
) + 1/Sqrt[1 + x]]]/Sqrt[5])/(4*(1/2 - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0])*(1/2 - Root[1 - 3*#1^2 - #1^
3 + 2*#1^4 & , 2, 0])*(1/2 - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(1/2 - Root[1 - 3*#1^2 - #1^3 + 2*#1^4
& , 4, 0])) + (Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0]*(Log[1/Sqrt[1 + x] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4
& , 1, 0]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0]^2]
- Log[2 + 1/Sqrt[1 + x] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - (2*Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1
, 0])/Sqrt[1 + x] + 2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]*Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0
] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0]^2]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1
- 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0]^2]))/(4*(-1/2 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0])*(Root[1 - 3*#1^2
- #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1
, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] - Root[1 - 3*#1^2
- #1^3 + 2*#1^4 & , 4, 0])) + (Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]*(Log[1/Sqrt[1 + x] - Root[1 - 3*#1^2
- #1^3 + 2*#1^4 & , 2, 0]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1
^4 & , 2, 0]^2] - Log[2 + 1/Sqrt[1 + x] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] - (2*Root[1 - 3*#1^2 - #1^
3 + 2*#1^4 & , 2, 0])/Sqrt[1 + x] + 2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]*Sqrt[1 + Root[1 - 3*#1^2 - #1^3 +
2*#1^4 & , 2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]^2]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & ,
2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]^2]))/(4*(-1/2 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0])*
(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0])*(Root[1 - 3*#1^2 - #1
^3 + 2*#1^4 & , 2, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0]
- Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])) + (Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0]*(Log[1/Sqrt[1 + x]
- Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0] - Root[1 - 3*#
1^2 - #1^3 + 2*#1^4 & , 3, 0]^2] - Log[2 + 1/Sqrt[1 + x] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0] - (2*Root
[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])/Sqrt[1 + x] + 2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]]*Sqrt[1 + Root[1
- 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0]^2]]/Sqrt[1 + Root[1 - 3*#1^2 -
#1^3 + 2*#1^4 & , 3, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0]^2]))/(4*(-1/2 + Root[1 - 3*#1^2 - #1^3 + 2
*#1^4 & , 3, 0])*(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(-Ro
ot[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0])*(Root[1 - 3*#1^2 - #1^3 +
2*#1^4 & , 3, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])) + (Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]*(
Log[1/Sqrt[1 + x] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]]/Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4,
0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]^2] - Log[2 + 1/Sqrt[1 + x] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 &
, 4, 0] - (2*Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])/Sqrt[1 + x] + 2*Sqrt[1 - (1 + x)^(-1) + 1/Sqrt[1 + x]
]*Sqrt[1 + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]^2]]/Sqrt[1 +
Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0] - Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]^2]))/(4*(-1/2 + Root[1 -
3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])*(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 1, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1
^4 & , 4, 0])*(-Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 2, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0])*(-Root[
1 - 3*#1^2 - #1^3 + 2*#1^4 & , 3, 0] + Root[1 - 3*#1^2 - #1^3 + 2*#1^4 & , 4, 0]))))/((-1 - 2/(1 + x)^2 + (1 +
x)^(-3/2) + 3/(1 + x))*(-2 + Sqrt[1 + x])*Sqrt[(1 + (-1 + Sqrt[1 + x])/(1 + x))/(1 + x)])))/Sqrt[(1 + x)*(1 +
(-1 + Sqrt[1 + x])/(1 + x))] + 2*RootSum[2 - #1 - 3*#1^2 + #1^4 & , (-(Log[Sqrt[1 + x] - #1]*#1) + Log[Sqrt[1
+ x] - #1]*#1^2 + Log[Sqrt[1 + x] - #1]*#1^3)/(-1 - 6*#1 + 4*#1^3) & ]

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IntegrateAlgebraic [A]  time = 0.00, size = 160, normalized size = 1.03 \begin {gather*} 1+x-2 \sqrt {x+\sqrt {1+x}}-\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/(x + Sqrt[x + Sqrt[1 + x]]),x]

[Out]

1 + x - 2*Sqrt[x + Sqrt[1 + x]] - Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[1 + 3*#1 - 5*#
1^2 + 2*#1^3 + #1^4 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[
1 + x]] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#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+(1+x)^(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 + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(x + sqrt(x + sqrt(x + 1))), x)

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maple [B]  time = 0.04, size = 502, normalized size = 3.24

method	result	size
derivativedivides	\(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )\)	\(502\)
default	\(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )-2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )\)	\(502\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x+(x+(1+x)^(1/2))^(1/2)),x,method=_RETURNVERBOSE)

[Out]

-(x+(1+x)^(1/2))^(1/2)+(1+x)^(1/2)-2*sum((2*_R^3-3*_R^2-_R+6)/(4*_R^3-6*_R^2+2*_R+5)*ln((x+(1+x)^(1/2))^(1/2)-
(1+x)^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^3+_Z^2+5*_Z-1))+5/2/(-1-2*(1+x)^(1/2)+2*(x+(1+x)^(1/2))^(1/2))+ln(-1-2*(1+
x)^(1/2)+2*(x+(1+x)^(1/2))^(1/2))+2*sum((2*_R^3+_R^2-_R)/(4*_R^3+6*_R^2-10*_R+3)*ln((x+(1+x)^(1/2))^(1/2)-(1+x
)^(1/2)-_R),_R=RootOf(_Z^4+2*_Z^3-5*_Z^2+3*_Z+1))+2*sum(_R/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3
*_Z^2-_Z+2))-4*sum(_R^3/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3*_Z^2-_Z+2))+1+x+2*sum((3*_R^3+_R^2
-2*_R)/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3*_Z^2-_Z+2))+2*sum((_R^3-_R^2+2)/(4*_R^3-6*_R^2+2*_R
+5)*ln((x+(1+x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^3+_Z^2+5*_Z-1))-2*sum((_R^3+_R^2-2*_R)/(4*_R^
3+6*_R^2-10*_R+3)*ln((x+(1+x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4+2*_Z^3-5*_Z^2+3*_Z+1))

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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 + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(x + sqrt(x + sqrt(x + 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+1}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/(x + (x + (x + 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 + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(x + sqrt(x + sqrt(x + 1))), x)

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