\(\int \frac {(1+x^2)^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2} \, dx\) [2990]

Optimal result

Integrand size = 40, antiderivative size = 392 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {\sqrt {1+x^2} \left (-11 x+8 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-5-7 x^2+8 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 x \left (-1+x^2\right ) \sqrt {1+x^2}+\left (-1+x^2\right ) \left (1+2 x^2\right )}-4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.76 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {\left (-5-7 x^2+8 x^4-11 x \sqrt {1+x^2}+8 x^3 \sqrt {1+x^2}\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (-1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}-4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\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 )-\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}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {14 \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+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\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 )-\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}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-16 \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+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]

[In]

Integrate[((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]

[Out]

((-5 - 7*x^2 + 8*x^4 - 11*x*Sqrt[1 + x^2] + 8*x^3*Sqrt[1 + x^2])*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((-1 + x^2
)*(1 + 2*x^2 + 2*x*Sqrt[1 + x^2])) - 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1
^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^
2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#
1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]
*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 - RootSum[2 - 8*#1^
2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1
+ x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] +
RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt
[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3
- 3*#1^5 + #1^7) & ]/8

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08

\[\int \frac {\left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}d x\]

[In]

int((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x)

[Out]

int((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x)

Fricas [C] (verification not implemented)

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

Time = 1.10 (sec) , antiderivative size = 6658, normalized size of antiderivative = 16.98 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 47.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\left (x^{2} + 1\right )^{\frac {3}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx \]

[In]

integrate((x**2+1)**(3/2)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**2+1)**2,x)

[Out]

Integral((x**2 + 1)**(3/2)*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/((x - 1)**2*(x + 1)**2), x)

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} - 1\right )}^{2}} \,d x } \]

[In]

integrate((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="maxima")

[Out]

integrate((x^2 + 1)^(3/2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 - 1)^2, x)

Giac [F(-1)]

Timed out. \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((x^2+1)^(3/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2\right )^{3/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,{\left (x^2+1\right )}^{3/2}}{{\left (x^2-1\right )}^2} \,d x \]

[In]

int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(3/2))/(x^2 - 1)^2,x)

[Out]

int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(3/2))/(x^2 - 1)^2, x)