\(\int \frac {\sqrt {-1-x^2+x^6} (1+2 x^6)}{8-x^4-16 x^6+8 x^{12}} \, dx\) [1522]

Optimal result

Integrand size = 41, antiderivative size = 105 \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=\frac {1}{8} \sqrt {\frac {1}{2} \left (4-\sqrt {2}\right )} \arctan \left (\frac {\sqrt {4-\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (4+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right ) \]

[Out]

1/16*(8-2*2^(1/2))^(1/2)*arctan(1/2*(4-2^(1/2))^(1/2)*x/(x^6-x^2-1)^(1/2))-1/16*(8+2*2^(1/2))^(1/2)*arctan(1/2
*(4+2^(1/2))^(1/2)*x/(x^6-x^2-1)^(1/2))

Rubi [F]

\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=\int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx \]

[In]

Int[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12),x]

[Out]

Defer[Int][Sqrt[-1 - x^2 + x^6]/(8 - x^4 - 16*x^6 + 8*x^12), x] + 2*Defer[Int][(x^6*Sqrt[-1 - x^2 + x^6])/(8 -
x^4 - 16*x^6 + 8*x^12), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}}+\frac {2 x^6 \sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}}\right ) \, dx \\ & = 2 \int \frac {x^6 \sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}} \, dx+\int \frac {\sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.45 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=\frac {1}{16} \left (\sqrt {8-2 \sqrt {2}} \arctan \left (\frac {x}{2 \sqrt {\frac {1+x^2-x^6}{-4+\sqrt {2}}}}\right )-\sqrt {2 \left (4+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right )\right ) \]

[In]

Integrate[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12),x]

[Out]

(Sqrt[8 - 2*Sqrt[2]]*ArcTan[x/(2*Sqrt[(1 + x^2 - x^6)/(-4 + Sqrt[2])])] - Sqrt[2*(4 + Sqrt[2])]*ArcTan[(Sqrt[4
+ Sqrt[2]]*x)/(2*Sqrt[-1 - x^2 + x^6])])/16

Maple [A] (verified)

Time = 3.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\left (\sqrt {2}-4\right ) \arctan \left (\frac {2 \sqrt {x^{6}-x^{2}-1}}{\sqrt {4-\sqrt {2}}\, x}\right )+\sqrt {4+\sqrt {2}}\, \arctan \left (\frac {2 \sqrt {x^{6}-x^{2}-1}}{\sqrt {4+\sqrt {2}}\, x}\right ) \sqrt {4-\sqrt {2}}\right )}{16 \sqrt {4-\sqrt {2}}}\)	\(91\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \ln \left (\frac {-2048 \operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x^{6}+262144 \operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{4} x^{2}-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{6}+8448 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{2}+2048 \sqrt {x^{6}-x^{2}-1}\, \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x +2048 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right )+54 \operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{2}+25 \sqrt {x^{6}-x^{2}-1}\, x +18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+64 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right )}{x^{6}+128 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x^{2}+x^{2}-1}\right )}{8}-\operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) \ln \left (-\frac {16384 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3} x^{6}+2097152 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{5} x^{2}+112 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) x^{6}-2048 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3} x^{2}+2048 \sqrt {x^{6}-x^{2}-1}\, \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x -16384 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3}-112 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) x^{2}+7 \sqrt {x^{6}-x^{2}-1}\, x -112 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )}{-x^{6}+128 \operatorname {RootOf}\left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x^{2}+x^{2}+1}\right )\)	\(579\)

[In]

int((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x,method=_RETURNVERBOSE)

[Out]

1/16*2^(1/2)*((2^(1/2)-4)*arctan(2/(4-2^(1/2))^(1/2)/x*(x^6-x^2-1)^(1/2))+(4+2^(1/2))^(1/2)*arctan(2/(4+2^(1/2
))^(1/2)/x*(x^6-x^2-1)^(1/2))*(4-2^(1/2))^(1/2))/(4-2^(1/2))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (71) = 142\).

Time = 0.39 (sec) , antiderivative size = 599, normalized size of antiderivative = 5.70 \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=-\frac {1}{64} \, \sqrt {2} \sqrt {-\sqrt {2} - 4} \log \left (\frac {4 \, {\left (36 \, x^{7} - 8 \, x^{3} + \sqrt {2} {\left (16 \, x^{7} - 9 \, x^{3} - 16 \, x\right )} - 36 \, x\right )} \sqrt {x^{6} - x^{2} - 1} + {\left (32 \, x^{12} - 72 \, x^{8} - 64 \, x^{6} + 12 \, x^{4} + 72 \, x^{2} + \sqrt {2} {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 17 \, x^{4} + 32 \, x^{2} + 8\right )} + 32\right )} \sqrt {-\sqrt {2} - 4}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8}\right ) + \frac {1}{64} \, \sqrt {2} \sqrt {-\sqrt {2} - 4} \log \left (\frac {4 \, {\left (36 \, x^{7} - 8 \, x^{3} + \sqrt {2} {\left (16 \, x^{7} - 9 \, x^{3} - 16 \, x\right )} - 36 \, x\right )} \sqrt {x^{6} - x^{2} - 1} - {\left (32 \, x^{12} - 72 \, x^{8} - 64 \, x^{6} + 12 \, x^{4} + 72 \, x^{2} + \sqrt {2} {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 17 \, x^{4} + 32 \, x^{2} + 8\right )} + 32\right )} \sqrt {-\sqrt {2} - 4}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8}\right ) + \frac {1}{64} \, \sqrt {2} \sqrt {\sqrt {2} - 4} \log \left (\frac {4 \, {\left (36 \, x^{7} - 8 \, x^{3} - \sqrt {2} {\left (16 \, x^{7} - 9 \, x^{3} - 16 \, x\right )} - 36 \, x\right )} \sqrt {x^{6} - x^{2} - 1} + {\left (32 \, x^{12} - 72 \, x^{8} - 64 \, x^{6} + 12 \, x^{4} + 72 \, x^{2} - \sqrt {2} {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 17 \, x^{4} + 32 \, x^{2} + 8\right )} + 32\right )} \sqrt {\sqrt {2} - 4}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8}\right ) - \frac {1}{64} \, \sqrt {2} \sqrt {\sqrt {2} - 4} \log \left (\frac {4 \, {\left (36 \, x^{7} - 8 \, x^{3} - \sqrt {2} {\left (16 \, x^{7} - 9 \, x^{3} - 16 \, x\right )} - 36 \, x\right )} \sqrt {x^{6} - x^{2} - 1} - {\left (32 \, x^{12} - 72 \, x^{8} - 64 \, x^{6} + 12 \, x^{4} + 72 \, x^{2} - \sqrt {2} {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 17 \, x^{4} + 32 \, x^{2} + 8\right )} + 32\right )} \sqrt {\sqrt {2} - 4}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8}\right ) \]

[In]

integrate((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x, algorithm="fricas")

[Out]

-1/64*sqrt(2)*sqrt(-sqrt(2) - 4)*log((4*(36*x^7 - 8*x^3 + sqrt(2)*(16*x^7 - 9*x^3 - 16*x) - 36*x)*sqrt(x^6 - x
^2 - 1) + (32*x^12 - 72*x^8 - 64*x^6 + 12*x^4 + 72*x^2 + sqrt(2)*(8*x^12 - 32*x^8 - 16*x^6 + 17*x^4 + 32*x^2 +
8) + 32)*sqrt(-sqrt(2) - 4))/(8*x^12 - 16*x^6 - x^4 + 8)) + 1/64*sqrt(2)*sqrt(-sqrt(2) - 4)*log((4*(36*x^7 -
8*x^3 + sqrt(2)*(16*x^7 - 9*x^3 - 16*x) - 36*x)*sqrt(x^6 - x^2 - 1) - (32*x^12 - 72*x^8 - 64*x^6 + 12*x^4 + 72
*x^2 + sqrt(2)*(8*x^12 - 32*x^8 - 16*x^6 + 17*x^4 + 32*x^2 + 8) + 32)*sqrt(-sqrt(2) - 4))/(8*x^12 - 16*x^6 - x
^4 + 8)) + 1/64*sqrt(2)*sqrt(sqrt(2) - 4)*log((4*(36*x^7 - 8*x^3 - sqrt(2)*(16*x^7 - 9*x^3 - 16*x) - 36*x)*sqr
t(x^6 - x^2 - 1) + (32*x^12 - 72*x^8 - 64*x^6 + 12*x^4 + 72*x^2 - sqrt(2)*(8*x^12 - 32*x^8 - 16*x^6 + 17*x^4 +
32*x^2 + 8) + 32)*sqrt(sqrt(2) - 4))/(8*x^12 - 16*x^6 - x^4 + 8)) - 1/64*sqrt(2)*sqrt(sqrt(2) - 4)*log((4*(36
*x^7 - 8*x^3 - sqrt(2)*(16*x^7 - 9*x^3 - 16*x) - 36*x)*sqrt(x^6 - x^2 - 1) - (32*x^12 - 72*x^8 - 64*x^6 + 12*x
^4 + 72*x^2 - sqrt(2)*(8*x^12 - 32*x^8 - 16*x^6 + 17*x^4 + 32*x^2 + 8) + 32)*sqrt(sqrt(2) - 4))/(8*x^12 - 16*x
^6 - x^4 + 8))

Sympy [F]

\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=\int \frac {\left (2 x^{6} + 1\right ) \sqrt {x^{6} - x^{2} - 1}}{8 x^{12} - 16 x^{6} - x^{4} + 8}\, dx \]

[In]

integrate((x**6-x**2-1)**(1/2)*(2*x**6+1)/(8*x**12-16*x**6-x**4+8),x)

[Out]

Integral((2*x**6 + 1)*sqrt(x**6 - x**2 - 1)/(8*x**12 - 16*x**6 - x**4 + 8), x)

Maxima [F]

\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - x^{2} - 1}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8} \,d x } \]

[In]

integrate((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x, algorithm="maxima")

[Out]

integrate((2*x^6 + 1)*sqrt(x^6 - x^2 - 1)/(8*x^12 - 16*x^6 - x^4 + 8), x)

Giac [F]

\[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - x^{2} - 1}}{8 \, x^{12} - 16 \, x^{6} - x^{4} + 8} \,d x } \]

[In]

integrate((x^6-x^2-1)^(1/2)*(2*x^6+1)/(8*x^12-16*x^6-x^4+8),x, algorithm="giac")

[Out]

integrate((2*x^6 + 1)*sqrt(x^6 - x^2 - 1)/(8*x^12 - 16*x^6 - x^4 + 8), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx=\int -\frac {\left (2\,x^6+1\right )\,\sqrt {x^6-x^2-1}}{-8\,x^{12}+16\,x^6+x^4-8} \,d x \]

[In]

int(-((2*x^6 + 1)*(x^6 - x^2 - 1)^(1/2))/(x^4 + 16*x^6 - 8*x^12 - 8),x)

[Out]

int(-((2*x^6 + 1)*(x^6 - x^2 - 1)^(1/2))/(x^4 + 16*x^6 - 8*x^12 - 8), x)