\(\int \frac {-1+x^2}{(1+x^2) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [2679]

Optimal result

Integrand size = 30, antiderivative size = 242 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-2 \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+2 \sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-2 \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]

[In]

Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

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

Maple [F]

\[\int \frac {x^{2}-1}{\left (x^{2}+1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}d x\]

[In]

int((x^2-1)/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2),x)

[Out]

int((x^2-1)/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (189) = 378\).

Time = 2.76 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.37 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {1}{2} \, {\left (x^{3} - \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + \frac {1}{4} \, \sqrt {-4 \, \sqrt {2} - 4} \log \left (-\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )} \sqrt {-4 \, \sqrt {2} - 4} + {\left (x^{3} - \sqrt {2} {\left (x^{3} + 2 \, x\right )} + 3 \, x\right )} \sqrt {-4 \, \sqrt {2} - 4}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {-4 \, \sqrt {2} - 4} \log \left (-\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )} \sqrt {-4 \, \sqrt {2} - 4} + {\left (x^{3} - \sqrt {2} {\left (x^{3} + 2 \, x\right )} + 3 \, x\right )} \sqrt {-4 \, \sqrt {2} - 4}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) \]

[In]

integrate((x^2-1)/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/2*(x^3 - sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) - sqrt(2)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 +
1))*sqrt(x^2 + sqrt(x^4 + 1))/x) + 1/8*sqrt(2)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqr
t(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1) + 1/4*sqrt(-4*sqrt(2) - 4)*log(-(2*sqrt(2)*x^2 - 4*x^2 + sqrt(x^2
+ sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(sqrt(2)*x - x)*sqrt(-4*sqrt(2) - 4) + (x^3 - sqrt(2)*(x^3 + 2*x) + 3*x)*sqrt
(-4*sqrt(2) - 4)) + 2*sqrt(x^4 + 1)*(sqrt(2) - 1))/(x^2 + 1)) - 1/4*sqrt(-4*sqrt(2) - 4)*log(-(2*sqrt(2)*x^2 -
4*x^2 - sqrt(x^2 + sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(sqrt(2)*x - x)*sqrt(-4*sqrt(2) - 4) + (x^3 - sqrt(2)*(x^3 +
2*x) + 3*x)*sqrt(-4*sqrt(2) - 4)) + 2*sqrt(x^4 + 1)*(sqrt(2) - 1))/(x^2 + 1)) - 1/2*sqrt(sqrt(2) - 1)*log(2*(
sqrt(2)*x^2 + 2*x^2 + (x^3 + sqrt(2)*(x^3 + 2*x) - sqrt(x^4 + 1)*(sqrt(2)*x + x) + 3*x)*sqrt(x^2 + sqrt(x^4 +
1))*sqrt(sqrt(2) - 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1)) + 1/2*sqrt(sqrt(2) - 1)*log(2*(sqrt(2)*x^2 + 2
*x^2 - (x^3 + sqrt(2)*(x^3 + 2*x) - sqrt(x^4 + 1)*(sqrt(2)*x + x) + 3*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2
) - 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1))

Sympy [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\, dx \]

[In]

integrate((x**2-1)/(x**2+1)/(x**2+(x**4+1)**(1/2))**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)/((x**2 + 1)*sqrt(x**2 + sqrt(x**4 + 1))), x)

Maxima [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}} \,d x } \]

[In]

integrate((x^2-1)/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((x^2 - 1)/(sqrt(x^2 + sqrt(x^4 + 1))*(x^2 + 1)), x)

Giac [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}} \,d x } \]

[In]

integrate((x^2-1)/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate((x^2 - 1)/(sqrt(x^2 + sqrt(x^4 + 1))*(x^2 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {\sqrt {x^4+1}+x^2}} \,d x \]

[In]

int((x^2 - 1)/((x^2 + 1)*((x^4 + 1)^(1/2) + x^2)^(1/2)),x)

[Out]

int((x^2 - 1)/((x^2 + 1)*((x^4 + 1)^(1/2) + x^2)^(1/2)), x)