\(\int \frac {\sqrt {1-x^6} (1+2 x^6)}{1+x^4-2 x^6+x^{12}} \, dx\) [1055]

Optimal result

Integrand size = 34, antiderivative size = 79 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {1-x^6}}{-1+x^2+x^6}\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {1-x^6}}{-1-x^2+x^6}\right )}{2 \sqrt {2}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 3.59 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=-\frac {\arctan \left (\frac {x \sqrt {2-2 x^6}}{-1+x^2+x^6}\right )+\text {arctanh}\left (\frac {-1-x^2+x^6}{x \sqrt {2-2 x^6}}\right )}{2 \sqrt {2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 7.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.34

method	result	size
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{6}+\sqrt {-x^{6}+1}\, \sqrt {2}\, x -x^{2}-1}{x^{6}-\sqrt {-x^{6}+1}\, \sqrt {2}\, x -x^{2}-1}\right )+2 \arctan \left (\frac {\sqrt {-x^{6}+1}\, \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\sqrt {-x^{6}+1}\, \sqrt {2}-x}{x}\right )\right )}{8}\)	\(106\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {-x^{6}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{-x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {-x^{6}+1}\, x}{x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1}\right )}{4}\)	\(155\)

[In]

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

[Out]

-1/8*2^(1/2)*(ln((x^6+(-x^6+1)^(1/2)*2^(1/2)*x-x^2-1)/(x^6-(-x^6+1)^(1/2)*2^(1/2)*x-x^2-1))+2*arctan(((-x^6+1)
^(1/2)*2^(1/2)+x)/x)+2*arctan(((-x^6+1)^(1/2)*2^(1/2)-x)/x))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.91 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x^{12} + \left (2 i - 2\right ) \, x^{8} - \left (2 i + 2\right ) \, x^{6} - \left (i + 1\right ) \, x^{4} - \left (2 i - 2\right ) \, x^{2} + i + 1\right )} - 4 \, {\left (x^{7} + i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x^{12} - \left (2 i + 2\right ) \, x^{8} + \left (2 i - 2\right ) \, x^{6} + \left (i - 1\right ) \, x^{4} + \left (2 i + 2\right ) \, x^{2} - i + 1\right )} - 4 \, {\left (x^{7} - i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x^{12} + \left (2 i + 2\right ) \, x^{8} - \left (2 i - 2\right ) \, x^{6} - \left (i - 1\right ) \, x^{4} - \left (2 i + 2\right ) \, x^{2} + i - 1\right )} - 4 \, {\left (x^{7} - i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x^{12} - \left (2 i - 2\right ) \, x^{8} + \left (2 i + 2\right ) \, x^{6} + \left (i + 1\right ) \, x^{4} + \left (2 i - 2\right ) \, x^{2} - i - 1\right )} - 4 \, {\left (x^{7} + i \, x^{3} - x\right )} \sqrt {-x^{6} + 1}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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