\(\int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx\) [2883]

Optimal result

Integrand size = 34, antiderivative size = 311 \[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx=\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

1/2*(-2+2*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*(-
2+2*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)))+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)-1/2*(2+2*2^(1/2))^(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)))-1/2*(2+2*2^(1/2))^(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]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx=-\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {-1+\sqrt {2}} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{4}-1}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (238) = 476\).

Time = 3.20 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx=\frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} + 1} \log \left (-\frac {2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} - x^{2}\right )} \sqrt {-\sqrt {2} + 1} + 2 \, {\left (\sqrt {2} x^{3} - 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, \sqrt {2} x^{4} - 3 \, x^{4} - 1\right )} \sqrt {-\sqrt {2} + 1}}{x^{4} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} + 1} \log \left (\frac {2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} - x^{2}\right )} \sqrt {-\sqrt {2} + 1} - 2 \, {\left (\sqrt {2} x^{3} - 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, \sqrt {2} x^{4} - 3 \, x^{4} - 1\right )} \sqrt {-\sqrt {2} + 1}}{x^{4} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) \]

[In]

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

[Out]

1/8*sqrt(2)*sqrt(-sqrt(2) + 1)*log(-(2*sqrt(x^4 + 1)*(sqrt(2)*x^2 - x^2)*sqrt(-sqrt(2) + 1) + 2*(sqrt(2)*x^3 -
2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x - x))*sqrt(x^2 + sqrt(x^4 + 1)) + (2*sqrt(2)*x^4 - 3*x^4 - 1)*sqrt(-sqrt(2)
+ 1))/(x^4 - 1)) - 1/8*sqrt(2)*sqrt(-sqrt(2) + 1)*log((2*sqrt(x^4 + 1)*(sqrt(2)*x^2 - x^2)*sqrt(-sqrt(2) + 1)
- 2*(sqrt(2)*x^3 - 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x - x))*sqrt(x^2 + sqrt(x^4 + 1)) + (2*sqrt(2)*x^4 - 3*x^4 -
1)*sqrt(-sqrt(2) + 1))/(x^4 - 1)) - 1/8*sqrt(2)*sqrt(sqrt(2) + 1)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1)
*(sqrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) + (2*sqrt(2)*x^4 + 3*x^4 + 2*sqrt(x^4 + 1)*(sqrt(2)*x^2 + x^2) + 1
)*sqrt(sqrt(2) + 1))/(x^4 - 1)) + 1/8*sqrt(2)*sqrt(sqrt(2) + 1)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1)*(s
qrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) - (2*sqrt(2)*x^4 + 3*x^4 + 2*sqrt(x^4 + 1)*(sqrt(2)*x^2 + x^2) + 1)*s
qrt(sqrt(2) + 1))/(x^4 - 1)) + 1/4*sqrt(2)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4
+ 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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