\(\int \frac {\sqrt {1+x^2+x^6} (-1+2 x^6)}{(1+x^6) (2-x^2+2 x^6)} \, dx\) [600]

Optimal result

Integrand size = 41, antiderivative size = 47 \[ \int \frac {\sqrt {1+x^2+x^6} \left (-1+2 x^6\right )}{\left (1+x^6\right ) \left (2-x^2+2 x^6\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {1+x^2+x^6}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.45 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^2+x^6} \left (-1+2 x^6\right )}{\left (1+x^6\right ) \left (2-x^2+2 x^6\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {1+x^2+x^6}}\right ) \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.40

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {x +\sqrt {x^{6}+x^{2}+1}}{x}\right )}{2}-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{6}+x^{2}+1}\, \sqrt {6}}{3 x}\right )}{2}-\frac {\ln \left (\frac {\sqrt {x^{6}+x^{2}+1}-x}{x}\right )}{2}\)	\(66\)
trager	\(\frac {\ln \left (-\frac {x^{6}+2 \sqrt {x^{6}+x^{2}+1}\, x +2 x^{2}+1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{6}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{2}-12 \sqrt {x^{6}+x^{2}+1}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{2 x^{6}-x^{2}+2}\right )}{4}\)	\(117\)

Input:

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

Output:

1/2*ln((x+(x^6+x^2+1)^(1/2))/x)-1/2*6^(1/2)*arctanh(1/3*(x^6+x^2+1)^(1/2)/ 
x*6^(1/2))-1/2*ln(((x^6+x^2+1)^(1/2)-x)/x)
 

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   catd 
ef: division by zero
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

 - int(sqrt(x**6 + x**2 + 1)/(2*x**12 - x**8 + 4*x**6 - x**2 + 2),x) + 2*i 
nt((sqrt(x**6 + x**2 + 1)*x**6)/(2*x**12 - x**8 + 4*x**6 - x**2 + 2),x)