\(\int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx\) [2754]

Optimal result

Integrand size = 46, antiderivative size = 257 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x), x]/2 + Defer[Int][(Sqrt[x + Sq
rt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x), x]/2

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]

[In]

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

[Out]

-1/2*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 + S
qrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5)
& ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + L
og[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 1.13 (sec) , antiderivative size = 5201, normalized size of antiderivative = 20.24 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 2.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=- \int \frac {\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} - 1}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=\int { -\frac {\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} - 1} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=-\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x+\sqrt {x^2+1}}}{x^2-1} \,d x \]

[In]

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

[Out]

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