3.14.21 \(\int \frac {(-2+x^6) (1-x^4+x^6)}{x^4 \sqrt [4]{1+x^6} (1+x^4+x^6)} \, dx\) [1321]

3.14.21.1 Optimal result

Integrand size = 38, antiderivative size = 95 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right ) \]

output

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

3.14.21.2 Mathematica [A] (verified)

Time = 4.55 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right ) \]

input

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

output

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

3.14.21.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle 6 \int \frac {1}{\sqrt [4]{x^6+1} \left (x^6+x^4+1\right )}dx+2 \int \frac {x^4}{\sqrt [4]{x^6+1} \left (x^6+x^4+1\right )}dx-2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )+\frac {2 \left (x^6+1\right )^{3/4}}{3 x^3}\)

input

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

output

$Aborted
 

3.14.21.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.14.21.4 Maple [A] (verified)

Time = 26.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36

method	result	size
pseudoelliptic	\(\frac {-6 \arctan \left (\frac {\left (x^{6}+1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}\, x^{3}-6 \arctan \left (\frac {\left (x^{6}+1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}\, x^{3}-3 \ln \left (\frac {-\left (x^{6}+1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{6}+1}}{\left (x^{6}+1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{6}+1}}\right ) \sqrt {2}\, x^{3}+4 \left (x^{6}+1\right )^{\frac {3}{4}}}{6 x^{3}}\)	\(129\)
trager	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (x^{6}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )\)	\(219\)
risch	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (x^{6}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )\)	\(220\)

input

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

output

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

3.14.21.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 112.85 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.27 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx=\frac {-\left (3 i + 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {4 i \, {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + 1} x^{2} - 4 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} - \left (i + 1\right ) \, x^{4} + i + 1\right )}}{x^{6} + x^{4} + 1}\right ) + \left (3 i + 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {4 i \, {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + 1} x^{2} - 4 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} + \left (i + 1\right ) \, x^{4} - i - 1\right )}}{x^{6} + x^{4} + 1}\right ) + \left (3 i - 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {-4 i \, {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + 1} x^{2} - 4 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} + \left (i - 1\right ) \, x^{4} - i + 1\right )}}{x^{6} + x^{4} + 1}\right ) - \left (3 i - 3\right ) \, \sqrt {2} x^{3} \log \left (\frac {-4 i \, {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + 1} x^{2} - 4 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} - \left (i - 1\right ) \, x^{4} + i - 1\right )}}{x^{6} + x^{4} + 1}\right ) + 8 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \]

input

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

output

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

3.14.21.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.14.21.7 Maxima [F]

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

input

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

output

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

3.14.21.8 Giac [F]

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

input

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

output

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

3.14.21.9 Mupad [F(-1)]

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

input

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

output

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