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

Optimal result

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

[Out]

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)))

Rubi [F]

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

[In]

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

[Out]

(2*(1 + x^6)^(3/4))/(3*x^3) - 2*x*Hypergeometric2F1[1/6, 1/4, 7/6, -x^6] + 6*Defer[Int][1/((1 + x^6)^(1/4)*(1
+ x^4 + x^6)), x] + 2*Defer[Int][x^4/((1 + x^6)^(1/4)*(1 + x^4 + x^6)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2}{\sqrt [4]{1+x^6}}-\frac {2}{x^4 \sqrt [4]{1+x^6}}+\frac {x^2}{\sqrt [4]{1+x^6}}+\frac {2 \left (3+x^4\right )}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\sqrt [4]{1+x^6}} \, dx\right )-2 \int \frac {1}{x^4 \sqrt [4]{1+x^6}} \, dx+2 \int \frac {3+x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+\int \frac {x^2}{\sqrt [4]{1+x^6}} \, dx \\ & = -2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2}} \, dx,x,x^3\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x^2}} \, dx,x,x^3\right )+2 \int \left (\frac {3}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )}+\frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )}\right ) \, dx \\ & = \frac {2 x^3}{3 \sqrt [4]{1+x^6}}+\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}-2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx,x,x^3\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2}} \, dx,x,x^3\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\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}-\frac {2}{3} E\left (\left .\frac {\arctan \left (x^3\right )}{2}\right |2\right )-2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx,x,x^3\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\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}-2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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]*Arc
Tanh[(Sqrt[2]*x*(1 + x^6)^(1/4))/(x^2 + Sqrt[1 + x^6])]

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	\(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (x^{6}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )\)	\(220\)

[In]

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

[Out]

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

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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