\(\int \frac {\sqrt {-1+x^6} (1+2 x^6)}{x^2 (-1+x^2+x^6)} \, dx\) [293]

Optimal result

Integrand size = 30, antiderivative size = 26 \[ \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \] Output:

(x^6-1)^(1/2)/x+arctan(x/(x^6-1)^(1/2))
 

Mathematica [A] (verified)

Time = 3.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \] Input:

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

Output:

Sqrt[-1 + x^6]/x + ArcTan[x/Sqrt[-1 + x^6]]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{x^2 \left (x^6+x^2-1\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (3 x^4+1\right ) \sqrt {x^6-1}}{x^6+x^2-1}-\frac {\sqrt {x^6-1}}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {\sqrt {x^6-1}}{x^6+x^2-1}dx+3 \int \frac {x^4 \sqrt {x^6-1}}{x^6+x^2-1}dx+\frac {3^{3/4} \left (1-\sqrt {3}\right ) \left (1-x^2\right ) \sqrt {\frac {x^4+x^2+1}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} x \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {x^6-1}}+\frac {3 \sqrt [4]{3} \left (1-x^2\right ) \sqrt {\frac {x^4+x^2+1}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} x E\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {x^6-1}}+\frac {\sqrt {x^6-1}}{x}+\frac {3 \left (1+\sqrt {3}\right ) \sqrt {x^6-1} x}{2 \left (1-\left (1+\sqrt {3}\right ) x^2\right )}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {-\arctan \left (\frac {\sqrt {x^{6}-1}}{x}\right ) x +\sqrt {x^{6}-1}}{x}\)	\(28\)
trager	\(\frac {\sqrt {x^{6}-1}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{6}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(73\)
risch	\(\frac {\sqrt {x^{6}-1}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{6}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(74\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx=\frac {x \arctan \left (\frac {2 \, \sqrt {x^{6} - 1} x}{x^{6} - x^{2} - 1}\right ) + 2 \, \sqrt {x^{6} - 1}}{2 \, x} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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