\(\int \frac {\sqrt {1+2 x^6} (-1+4 x^6)}{2+x^4+8 x^6+8 x^{12}} \, dx\) [1431]

Optimal result

Integrand size = 36, antiderivative size = 101 \[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=-\frac {\arctan \left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}-x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}+x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}} \]

[Out]

-1/8*arctan(2^(3/4)*x*(2*x^6+1)^(1/2)/(2^(1/2)-x^2+2*2^(1/2)*x^6))*2^(3/4)-1/8*arctanh(2^(3/4)*x*(2*x^6+1)^(1/
2)/(2^(1/2)+x^2+2*2^(1/2)*x^6))*2^(3/4)

Rubi [F]

\[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=\int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \]

[In]

Int[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12),x]

[Out]

Defer[Int][Sqrt[1 + 2*x^6]/(-2 - x^4 - 8*x^6 - 8*x^12), x] + 4*Defer[Int][(x^6*Sqrt[1 + 2*x^6])/(2 + x^4 + 8*x
^6 + 8*x^12), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}}+\frac {4 x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}}\right ) \, dx \\ & = 4 \int \frac {x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}} \, dx+\int \frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=-\frac {\arctan \left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}-x^2+2 \sqrt {2} x^6}\right )+\text {arctanh}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}+x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}} \]

[In]

Integrate[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12),x]

[Out]

-1/4*(ArcTan[(2^(3/4)*x*Sqrt[1 + 2*x^6])/(Sqrt[2] - x^2 + 2*Sqrt[2]*x^6)] + ArcTanh[(2^(3/4)*x*Sqrt[1 + 2*x^6]
)/(Sqrt[2] + x^2 + 2*Sqrt[2]*x^6)])/2^(1/4)

Maple [A] (verified)

Time = 2.67 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {4 x^{6}-\sqrt {4 x^{6}+2}\, 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+2}{4 x^{6}+\sqrt {4 x^{6}+2}\, 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+2}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {4 x^{6}+2}+x}{x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {4 x^{6}+2}-x}{x}\right )\right )}{16}\)	\(114\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} x^{2}-4 \sqrt {2 x^{6}+1}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )}{4 x^{6}+x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}+2}\right )}{8}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+4 \sqrt {2 x^{6}+1}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right )}{-4 x^{6}+x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}-2}\right )}{8}\)	\(187\)

[In]

int((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x,method=_RETURNVERBOSE)

[Out]

1/16*2^(3/4)*(ln((4*x^6-(4*x^6+2)^(1/2)*2^(3/4)*x+2^(1/2)*x^2+2)/(4*x^6+(4*x^6+2)^(1/2)*2^(3/4)*x+2^(1/2)*x^2+
2))+2*arctan((2^(1/4)*(4*x^6+2)^(1/2)+x)/x)+2*arctan((2^(1/4)*(4*x^6+2)^(1/2)-x)/x))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 397, normalized size of antiderivative = 3.93 \[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=-\left (\frac {1}{128} i + \frac {1}{128}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (\left (8 i + 8\right ) \, x^{12} + \left (8 i + 8\right ) \, x^{6} - \left (i + 1\right ) \, x^{4} + 2 i + 2\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i - 2\right ) \, x^{8} + \left (i - 1\right ) \, x^{2}\right )} + 16 \, {\left (4 \, x^{7} - i \, \sqrt {2} x^{3} + 2 \, x\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\right ) + \left (\frac {1}{128} i - \frac {1}{128}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (-\left (8 i - 8\right ) \, x^{12} - \left (8 i - 8\right ) \, x^{6} + \left (i - 1\right ) \, x^{4} - 2 i + 2\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i + 2\right ) \, x^{8} - \left (i + 1\right ) \, x^{2}\right )} + 16 \, {\left (4 \, x^{7} + i \, \sqrt {2} x^{3} + 2 \, x\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\right ) - \left (\frac {1}{128} i - \frac {1}{128}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (\left (8 i - 8\right ) \, x^{12} + \left (8 i - 8\right ) \, x^{6} - \left (i - 1\right ) \, x^{4} + 2 i - 2\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i + 2\right ) \, x^{8} + \left (i + 1\right ) \, x^{2}\right )} + 16 \, {\left (4 \, x^{7} + i \, \sqrt {2} x^{3} + 2 \, x\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\right ) + \left (\frac {1}{128} i + \frac {1}{128}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (-\left (8 i + 8\right ) \, x^{12} - \left (8 i + 8\right ) \, x^{6} + \left (i + 1\right ) \, x^{4} - 2 i - 2\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i - 2\right ) \, x^{8} - \left (i - 1\right ) \, x^{2}\right )} + 16 \, {\left (4 \, x^{7} - i \, \sqrt {2} x^{3} + 2 \, x\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\right ) \]

[In]

integrate((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x, algorithm="fricas")

[Out]

-(1/128*I + 1/128)*8^(3/4)*sqrt(2)*log((8^(3/4)*sqrt(2)*((8*I + 8)*x^12 + (8*I + 8)*x^6 - (I + 1)*x^4 + 2*I +
2) - 8*8^(1/4)*sqrt(2)*((2*I - 2)*x^8 + (I - 1)*x^2) + 16*(4*x^7 - I*sqrt(2)*x^3 + 2*x)*sqrt(2*x^6 + 1))/(8*x^
12 + 8*x^6 + x^4 + 2)) + (1/128*I - 1/128)*8^(3/4)*sqrt(2)*log((8^(3/4)*sqrt(2)*(-(8*I - 8)*x^12 - (8*I - 8)*x
^6 + (I - 1)*x^4 - 2*I + 2) - 8*8^(1/4)*sqrt(2)*(-(2*I + 2)*x^8 - (I + 1)*x^2) + 16*(4*x^7 + I*sqrt(2)*x^3 + 2
*x)*sqrt(2*x^6 + 1))/(8*x^12 + 8*x^6 + x^4 + 2)) - (1/128*I - 1/128)*8^(3/4)*sqrt(2)*log((8^(3/4)*sqrt(2)*((8*
I - 8)*x^12 + (8*I - 8)*x^6 - (I - 1)*x^4 + 2*I - 2) - 8*8^(1/4)*sqrt(2)*((2*I + 2)*x^8 + (I + 1)*x^2) + 16*(4
*x^7 + I*sqrt(2)*x^3 + 2*x)*sqrt(2*x^6 + 1))/(8*x^12 + 8*x^6 + x^4 + 2)) + (1/128*I + 1/128)*8^(3/4)*sqrt(2)*l
og((8^(3/4)*sqrt(2)*(-(8*I + 8)*x^12 - (8*I + 8)*x^6 + (I + 1)*x^4 - 2*I - 2) - 8*8^(1/4)*sqrt(2)*(-(2*I - 2)*
x^8 - (I - 1)*x^2) + 16*(4*x^7 - I*sqrt(2)*x^3 + 2*x)*sqrt(2*x^6 + 1))/(8*x^12 + 8*x^6 + x^4 + 2))

Sympy [F]

\[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=\int \frac {\left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right ) \sqrt {2 x^{6} + 1}}{8 x^{12} + 8 x^{6} + x^{4} + 2}\, dx \]

[In]

integrate((2*x**6+1)**(1/2)*(4*x**6-1)/(8*x**12+8*x**6+x**4+2),x)

[Out]

Integral((2*x**3 - 1)*(2*x**3 + 1)*sqrt(2*x**6 + 1)/(8*x**12 + 8*x**6 + x**4 + 2), x)

Maxima [F]

\[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=\int { \frac {{\left (4 \, x^{6} - 1\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2} \,d x } \]

[In]

integrate((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x, algorithm="maxima")

[Out]

integrate((4*x^6 - 1)*sqrt(2*x^6 + 1)/(8*x^12 + 8*x^6 + x^4 + 2), x)

Giac [F]

\[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=\int { \frac {{\left (4 \, x^{6} - 1\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2} \,d x } \]

[In]

integrate((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x, algorithm="giac")

[Out]

integrate((4*x^6 - 1)*sqrt(2*x^6 + 1)/(8*x^12 + 8*x^6 + x^4 + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx=\int \frac {\sqrt {2\,x^6+1}\,\left (4\,x^6-1\right )}{8\,x^{12}+8\,x^6+x^4+2} \,d x \]

[In]

int(((2*x^6 + 1)^(1/2)*(4*x^6 - 1))/(x^4 + 8*x^6 + 8*x^12 + 2),x)

[Out]

int(((2*x^6 + 1)^(1/2)*(4*x^6 - 1))/(x^4 + 8*x^6 + 8*x^12 + 2), x)