\(\int \frac {(-1+x^6) (1+x^3+x^6)^{2/3}}{1+x^6+x^{12}} \, dx\) [1866]

Optimal result

Integrand size = 28, antiderivative size = 128 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3+x^6}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3+x^6}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3+x^6}+\sqrt [3]{2} \left (1+x^3+x^6\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]

[Out]

-1/6*arctan(3^(1/2)*x/(x+2^(2/3)*(x^6+x^3+1)^(1/3)))*2^(2/3)*3^(1/2)+1/6*ln(-2*x+2^(2/3)*(x^6+x^3+1)^(1/3))*2^
(2/3)-1/12*ln(2*x^2+2^(2/3)*x*(x^6+x^3+1)^(1/3)+2^(1/3)*(x^6+x^3+1)^(2/3))*2^(2/3)

Rubi [F]

\[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx \]

[In]

Int[((-1 + x^6)*(1 + x^3 + x^6)^(2/3))/(1 + x^6 + x^12),x]

[Out]

(x*(1 + (2*x^3)/(1 - I*Sqrt[3]))^(1/3)*(1 + (2*x^3)/(1 + I*Sqrt[3]))^(1/3)*AppellF1[1/3, 1/3, 1/3, 4/3, (-2*x^
3)/(1 - I*Sqrt[3]), (-2*x^3)/(1 + I*Sqrt[3])])/(1 + x^3 + x^6)^(1/3) + (1 + I*Sqrt[3])*Defer[Int][1/((-1 - I*S
qrt[3] + 2*x^3)*(1 + x^3 + x^6)^(1/3)), x] + (1 - I*Sqrt[3])*Defer[Int][1/((-1 + I*Sqrt[3] + 2*x^3)*(1 + x^3 +
x^6)^(1/3)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+x^6}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}} \, dx \\ & = \int \left (\frac {1}{\sqrt [3]{1+x^3+x^6}}-\frac {2-x^3}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}}\right ) \, dx \\ & = \int \frac {1}{\sqrt [3]{1+x^3+x^6}} \, dx-\int \frac {2-x^3}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}} \, dx \\ & = \frac {\left (\sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}}} \, dx}{\sqrt [3]{1+x^3+x^6}}-\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}}\right ) \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{1-i \sqrt {3}},-\frac {2 x^3}{1+i \sqrt {3}}\right )}{\sqrt [3]{1+x^3+x^6}}-\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}} \, dx-\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3+x^6}}\right )-2 \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3+x^6}\right )+\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3+x^6}+\sqrt [3]{2} \left (1+x^3+x^6\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]

[In]

Integrate[((-1 + x^6)*(1 + x^3 + x^6)^(2/3))/(1 + x^6 + x^12),x]

[Out]

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

Maple [A] (verified)

Time = 6.50 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{6}+x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{6}+x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{12}\)	\(101\)
trager	\(\text {Expression too large to display}\)	\(699\)

[In]

int((x^6-1)*(x^6+x^3+1)^(2/3)/(x^12+x^6+1),x,method=_RETURNVERBOSE)

[Out]

1/12*2^(2/3)*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^6+x^3+1)^(1/3)))+2*ln((-2^(1/3)*x+(x^6+x^3+1)^(1/3)
)/x)-ln((2^(2/3)*x^2+2^(1/3)*(x^6+x^3+1)^(1/3)*x+(x^6+x^3+1)^(2/3))/x^2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (98) = 196\).

Time = 6.30 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=-\frac {1}{18} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (x^{13} + 16 \, x^{10} + 21 \, x^{7} + 16 \, x^{4} + x\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{18} - 21 \, x^{15} - 102 \, x^{12} - 133 \, x^{9} - 102 \, x^{6} - 21 \, x^{3} + 1\right )} - 24 \, \sqrt {6} {\left (x^{14} + x^{11} + x^{5} + x^{2}\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{18} + 51 \, x^{15} + 114 \, x^{12} + 155 \, x^{9} + 114 \, x^{6} + 51 \, x^{3} + 1\right )}}\right ) + \frac {1}{18} \cdot 2^{\frac {2}{3}} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} x - 6 \, {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2^{\frac {1}{3}} {\left (x^{6} - x^{3} + 1\right )}}{x^{6} - x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{12} + 16 \, x^{9} + 21 \, x^{6} + 16 \, x^{3} + 1\right )} + 12 \cdot 2^{\frac {1}{3}} {\left (x^{8} + 2 \, x^{5} + x^{2}\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x^{7} + 5 \, x^{4} + x\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}}}{x^{12} - 2 \, x^{9} + 3 \, x^{6} - 2 \, x^{3} + 1}\right ) \]

[In]

integrate((x^6-1)*(x^6+x^3+1)^(2/3)/(x^12+x^6+1),x, algorithm="fricas")

[Out]

-1/18*sqrt(6)*2^(1/6)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(x^13 + 16*x^10 + 21*x^7 + 16*x^4 + x)*(x^6 + x^3
+ 1)^(2/3) - sqrt(6)*2^(1/3)*(x^18 - 21*x^15 - 102*x^12 - 133*x^9 - 102*x^6 - 21*x^3 + 1) - 24*sqrt(6)*(x^14 +
x^11 + x^5 + x^2)*(x^6 + x^3 + 1)^(1/3))/(x^18 + 51*x^15 + 114*x^12 + 155*x^9 + 114*x^6 + 51*x^3 + 1)) + 1/18
*2^(2/3)*log(-(3*2^(2/3)*(x^6 + x^3 + 1)^(2/3)*x - 6*(x^6 + x^3 + 1)^(1/3)*x^2 - 2^(1/3)*(x^6 - x^3 + 1))/(x^6
- x^3 + 1)) - 1/36*2^(2/3)*log((2^(2/3)*(x^12 + 16*x^9 + 21*x^6 + 16*x^3 + 1) + 12*2^(1/3)*(x^8 + 2*x^5 + x^2
)*(x^6 + x^3 + 1)^(1/3) + 6*(x^7 + 5*x^4 + x)*(x^6 + x^3 + 1)^(2/3))/(x^12 - 2*x^9 + 3*x^6 - 2*x^3 + 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\text {Timed out} \]

[In]

integrate((x**6-1)*(x**6+x**3+1)**(2/3)/(x**12+x**6+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{x^{12} + x^{6} + 1} \,d x } \]

[In]

integrate((x^6-1)*(x^6+x^3+1)^(2/3)/(x^12+x^6+1),x, algorithm="maxima")

[Out]

integrate((x^6 + x^3 + 1)^(2/3)*(x^6 - 1)/(x^12 + x^6 + 1), x)

Giac [F]

\[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{x^{12} + x^{6} + 1} \,d x } \]

[In]

integrate((x^6-1)*(x^6+x^3+1)^(2/3)/(x^12+x^6+1),x, algorithm="giac")

[Out]

integrate((x^6 + x^3 + 1)^(2/3)*(x^6 - 1)/(x^12 + x^6 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\int \frac {\left (x^6-1\right )\,{\left (x^6+x^3+1\right )}^{2/3}}{x^{12}+x^6+1} \,d x \]

[In]

int(((x^6 - 1)*(x^3 + x^6 + 1)^(2/3))/(x^6 + x^12 + 1),x)

[Out]

int(((x^6 - 1)*(x^3 + x^6 + 1)^(2/3))/(x^6 + x^12 + 1), x)