\(\int \frac {(-1+x^2) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx\) [928]

Optimal result

Integrand size = 28, antiderivative size = 70 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}-4 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \]

[Out]

4/3*x/(1+(x^2+1)^(1/2))^(1/2)+2/3*x*(x^2+1)^(1/2)/(1+(x^2+1)^(1/2))^(1/2)-4*arctan(x/(1+(x^2+1)^(1/2))^(1/2))

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}}-4 \arctan \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \]

[In]

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

[Out]

(2*x*(2 + Sqrt[1 + x^2]))/(3*Sqrt[1 + Sqrt[1 + x^2]]) - 4*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]]

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.89 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx=\frac {3 \, x \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) + 2 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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