\(\int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx\) [1385]

Optimal result

Integrand size = 31, antiderivative size = 99 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\log \left (x+\sqrt {1+x^2}\right )-2 \text {RootSum}\left [1-2 \text {$\#$1}^4+4 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{-2+5 \text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

Defer[Int][x^2/(-1 - 2*x^5 + x^8), x] + Defer[Int][x^4/(-1 - 2*x^5 + x^8), x] - Defer[Int][(x^3*Sqrt[1 + x^2])
/(-1 - 2*x^5 + x^8), x] + Defer[Int][(x^6*Sqrt[1 + x^2])/(-1 - 2*x^5 + x^8), x] - Defer[Int][Sqrt[x + Sqrt[1 +
x^2]]/(-1 - 2*x^5 + x^8), x] - Defer[Int][(x^2*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^5 + x^8), x] + Defer[Int][(
x*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^5 + x^8), x] - Defer[Int][(x^4*Sqrt[1 + x^2]*Sqrt[x + Sqrt[
1 + x^2]])/(-1 - 2*x^5 + x^8), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^2 \left (1+x^2\right )}{-1-2 x^5+x^8}+\frac {x^3 \sqrt {1+x^2} \left (-1+x^3\right )}{-1-2 x^5+x^8}+\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}\right ) \, dx \\ & = \int \frac {x^2 \left (1+x^2\right )}{-1-2 x^5+x^8} \, dx+\int \frac {x^3 \sqrt {1+x^2} \left (-1+x^3\right )}{-1-2 x^5+x^8} \, dx+\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx \\ & = \int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \left (\frac {x^2}{-1-2 x^5+x^8}+\frac {x^4}{-1-2 x^5+x^8}\right ) \, dx+\int \left (-\frac {x^3 \sqrt {1+x^2}}{-1-2 x^5+x^8}+\frac {x^6 \sqrt {1+x^2}}{-1-2 x^5+x^8}\right ) \, dx-\int \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}+\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}\right ) \, dx \\ & = \int \frac {x^2}{-1-2 x^5+x^8} \, dx+\int \frac {x^4}{-1-2 x^5+x^8} \, dx-\int \frac {x^3 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx-\int \frac {\sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\log \left (x+\sqrt {1+x^2}\right )-2 \text {RootSum}\left [1-2 \text {$\#$1}^4+4 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{-2+5 \text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]

[In]

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

[Out]

Log[x + Sqrt[1 + x^2]] - 2*RootSum[1 - 2*#1^4 + 4*#1^5 + #1^8 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sq
rt[x + Sqrt[1 + x^2]] - #1]*#1)/(-2 + 5*#1 + 2*#1^4) & ]

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.25

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {x^{2} + 1}}{x^{2} + \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + 1}}{x^{2} + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^2 + 1)/(x^2 + sqrt(x + sqrt(x^2 + 1))), x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + 1}}{x^{2} + \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^2 + 1)/(x^2 + sqrt(x + sqrt(x^2 + 1))), x)

Mupad [N/A]

Not integrable

Time = 6.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {x^2+1}}{\sqrt {x+\sqrt {x^2+1}}+x^2} \,d x \]

[In]

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

[Out]

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