\(\int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx\) [1457]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=x-2 \sqrt {x+\sqrt {1+x^2}}+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=x-2 \sqrt {x+\sqrt {1+x^2}}+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[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) & ]

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.17

\[\int \frac {x}{x +\sqrt {x +\sqrt {x^{2}+1}}}d x\]

[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)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.03 (sec) , antiderivative size = 7672, normalized size of antiderivative = 75.22 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.17 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.19 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.19 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.19 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x}{x+\sqrt {x+\sqrt {x^2+1}}} \,d x \]

[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)