\(\int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx\) [190]

Optimal result

Integrand size = 23, antiderivative size = 20 \[ \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=2 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \]

[Out]

2*arctan(x/(1+(x^2+1)^(1/2))^(1/2))

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=2 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \]

[In]

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

[Out]

2*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]]

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (16) = 32\).

Time = 0.96 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=-\frac {1}{2} \, \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) \]

[In]

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

[Out]

-1/2*arctan(4*(x^4 - 12*x^2 + (5*x^2 - 3)*sqrt(x^2 + 1) + 3)*sqrt(sqrt(x^2 + 1) + 1)/(x^5 - 46*x^3 + 17*x))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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