Integrand size = 66, antiderivative size = 85 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx=-\frac {1}{3} \arctan \left (\frac {x}{\sqrt {1-x^6}}\right )-\frac {1}{3} \arctan \left (\frac {x \sqrt {1-x^6}}{-1+x^2+x^6}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x \sqrt {1-x^6}}{-1-x^2+x^6}\right )}{\sqrt {3}} \]
-1/3*arctan(x/(-x^6+1)^(1/2))-1/3*arctan(x*(-x^6+1)^(1/2)/(x^6+x^2-1))-1/3 *arctanh(3^(1/2)*x*(-x^6+1)^(1/2)/(x^6-x^2-1))*3^(1/2)
Result contains complex when optimal does not.
Time = 8.75 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx=\frac {1}{3} \left (-\arctan \left (\frac {x}{\sqrt {1-x^6}}\right )+\left (1+i \sqrt {3}\right ) \arctan \left (\frac {\left (1-i \sqrt {3}\right ) x}{2 \sqrt {1-x^6}}\right )+\left (1-i \sqrt {3}\right ) \arctan \left (\frac {\left (1+i \sqrt {3}\right ) x}{2 \sqrt {1-x^6}}\right )\right ) \]
Integrate[(Sqrt[1 - x^6]*(1 + 2*x^6)*(1 + x^2 - x^4 - 2*x^6 - x^8 + x^12)) /((-1 + x^6)*(-1 + 2*x^6 - 3*x^12 + x^18)),x]
(-ArcTan[x/Sqrt[1 - x^6]] + (1 + I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/(2* Sqrt[1 - x^6])] + (1 - I*Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/(2*Sqrt[1 - x ^6])])/3
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.
| \(\displaystyle \int \frac {\sqrt {1-x^6} \left (2 x^6+1\right ) \left (x^{12}-x^8-2 x^6-x^4+x^2+1\right )}{\left (x^6-1\right ) \left (x^{18}-3 x^{12}+2 x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\int -\frac {\left (2 x^6+1\right ) \left (x^{12}-x^8-2 x^6-x^4+x^2+1\right )}{\sqrt {1-x^6} \left (-x^{18}+3 x^{12}-2 x^6+1\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\left (2 x^6+1\right ) \left (x^{12}-x^8-2 x^6-x^4+x^2+1\right )}{\sqrt {1-x^6} \left (-x^{18}+3 x^{12}-2 x^6+1\right )}dx\) |
| \(\Big \downarrow \) 2461 |
| \(\displaystyle \int \left (\frac {\left (x^4-x^2-1\right ) \left (2 x^6+1\right ) \left (x^{12}-x^8-2 x^6-x^4+x^2+1\right )}{3 \sqrt {1-x^6} \left (x^6-x^2-1\right )}+\frac {\left (2 x^6+1\right ) \left (-x^{10}+x^8-x^6+3 x^4-2 x^2+2\right ) \left (x^{12}-x^8-2 x^6-x^4+x^2+1\right )}{3 \sqrt {1-x^6} \left (x^{12}+x^8-2 x^6+x^4-x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\sqrt {1-x^6} \left (x^6-x^2-1\right )}dx+\frac {2}{3} \int \frac {x^2}{\sqrt {1-x^6} \left (x^6-x^2-1\right )}dx+4 \int \frac {1}{\sqrt {1-x^6} \left (x^{12}+x^8-2 x^6+x^4-x^2+1\right )}dx-\frac {10}{3} \int \frac {x^2}{\sqrt {1-x^6} \left (x^{12}+x^8-2 x^6+x^4-x^2+1\right )}dx+\frac {8}{3} \int \frac {x^4}{\sqrt {1-x^6} \left (x^{12}+x^8-2 x^6+x^4-x^2+1\right )}dx-4 \int \frac {x^6}{\sqrt {1-x^6} \left (x^{12}+x^8-2 x^6+x^4-x^2+1\right )}dx+\frac {4}{3} \int \frac {x^8}{\sqrt {1-x^6} \left (x^{12}+x^8-2 x^6+x^4-x^2+1\right )}dx-\frac {x \left (1-x^2\right ) \sqrt {\frac {x^4+x^2+1}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}\) |
Int[(Sqrt[1 - x^6]*(1 + 2*x^6)*(1 + x^2 - x^4 - 2*x^6 - x^8 + x^12))/((-1 + x^6)*(-1 + 2*x^6 - 3*x^12 + x^18)),x]
3.12.53.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u, (Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[ Qx, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Timed out.
\[\int \frac {\sqrt {-x^{6}+1}\, \left (2 x^{6}+1\right ) \left (x^{12}-x^{8}-2 x^{6}-x^{4}+x^{2}+1\right )}{\left (x^{6}-1\right ) \left (x^{18}-3 x^{12}+2 x^{6}-1\right )}d x\]
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 610, normalized size of antiderivative = 7.18 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx=\frac {1}{12} \, \sqrt {2 i \, \sqrt {3} + 2} \log \left (-\frac {4 \, {\left (x^{7} - x^{3} + \sqrt {3} {\left (i \, x^{7} + i \, x^{3} - i \, x\right )} - x\right )} \sqrt {-x^{6} + 1} - {\left (x^{12} - 3 \, x^{8} - 2 \, x^{6} - x^{4} + 3 \, x^{2} + \sqrt {3} {\left (i \, x^{12} + i \, x^{8} - 2 i \, x^{6} - i \, x^{4} - i \, x^{2} + i\right )} + 1\right )} \sqrt {2 i \, \sqrt {3} + 2}}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}\right ) - \frac {1}{12} \, \sqrt {2 i \, \sqrt {3} + 2} \log \left (-\frac {4 \, {\left (x^{7} - x^{3} + \sqrt {3} {\left (i \, x^{7} + i \, x^{3} - i \, x\right )} - x\right )} \sqrt {-x^{6} + 1} + {\left (x^{12} - 3 \, x^{8} - 2 \, x^{6} - x^{4} + 3 \, x^{2} - \sqrt {3} {\left (-i \, x^{12} - i \, x^{8} + 2 i \, x^{6} + i \, x^{4} + i \, x^{2} - i\right )} + 1\right )} \sqrt {2 i \, \sqrt {3} + 2}}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}\right ) - \frac {1}{12} \, \sqrt {-2 i \, \sqrt {3} + 2} \log \left (-\frac {4 \, {\left (x^{7} - x^{3} + \sqrt {3} {\left (-i \, x^{7} - i \, x^{3} + i \, x\right )} - x\right )} \sqrt {-x^{6} + 1} + {\left (x^{12} - 3 \, x^{8} - 2 \, x^{6} - x^{4} + 3 \, x^{2} - \sqrt {3} {\left (i \, x^{12} + i \, x^{8} - 2 i \, x^{6} - i \, x^{4} - i \, x^{2} + i\right )} + 1\right )} \sqrt {-2 i \, \sqrt {3} + 2}}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{12} \, \sqrt {-2 i \, \sqrt {3} + 2} \log \left (-\frac {4 \, {\left (x^{7} - x^{3} + \sqrt {3} {\left (-i \, x^{7} - i \, x^{3} + i \, x\right )} - x\right )} \sqrt {-x^{6} + 1} - {\left (x^{12} - 3 \, x^{8} - 2 \, x^{6} - x^{4} + 3 \, x^{2} + \sqrt {3} {\left (-i \, x^{12} - i \, x^{8} + 2 i \, x^{6} + i \, x^{4} + i \, x^{2} - i\right )} + 1\right )} \sqrt {-2 i \, \sqrt {3} + 2}}{x^{12} + x^{8} - 2 \, x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {-x^{6} + 1} x}{x^{6} + x^{2} - 1}\right ) \]
integrate((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^1 8-3*x^12+2*x^6-1),x, algorithm="fricas")
1/12*sqrt(2*I*sqrt(3) + 2)*log(-(4*(x^7 - x^3 + sqrt(3)*(I*x^7 + I*x^3 - I *x) - x)*sqrt(-x^6 + 1) - (x^12 - 3*x^8 - 2*x^6 - x^4 + 3*x^2 + sqrt(3)*(I *x^12 + I*x^8 - 2*I*x^6 - I*x^4 - I*x^2 + I) + 1)*sqrt(2*I*sqrt(3) + 2))/( x^12 + x^8 - 2*x^6 + x^4 - x^2 + 1)) - 1/12*sqrt(2*I*sqrt(3) + 2)*log(-(4* (x^7 - x^3 + sqrt(3)*(I*x^7 + I*x^3 - I*x) - x)*sqrt(-x^6 + 1) + (x^12 - 3 *x^8 - 2*x^6 - x^4 + 3*x^2 - sqrt(3)*(-I*x^12 - I*x^8 + 2*I*x^6 + I*x^4 + I*x^2 - I) + 1)*sqrt(2*I*sqrt(3) + 2))/(x^12 + x^8 - 2*x^6 + x^4 - x^2 + 1 )) - 1/12*sqrt(-2*I*sqrt(3) + 2)*log(-(4*(x^7 - x^3 + sqrt(3)*(-I*x^7 - I* x^3 + I*x) - x)*sqrt(-x^6 + 1) + (x^12 - 3*x^8 - 2*x^6 - x^4 + 3*x^2 - sqr t(3)*(I*x^12 + I*x^8 - 2*I*x^6 - I*x^4 - I*x^2 + I) + 1)*sqrt(-2*I*sqrt(3) + 2))/(x^12 + x^8 - 2*x^6 + x^4 - x^2 + 1)) + 1/12*sqrt(-2*I*sqrt(3) + 2) *log(-(4*(x^7 - x^3 + sqrt(3)*(-I*x^7 - I*x^3 + I*x) - x)*sqrt(-x^6 + 1) - (x^12 - 3*x^8 - 2*x^6 - x^4 + 3*x^2 + sqrt(3)*(-I*x^12 - I*x^8 + 2*I*x^6 + I*x^4 + I*x^2 - I) + 1)*sqrt(-2*I*sqrt(3) + 2))/(x^12 + x^8 - 2*x^6 + x^ 4 - x^2 + 1)) + 1/6*arctan(2*sqrt(-x^6 + 1)*x/(x^6 + x^2 - 1))
Timed out. \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx=\text {Timed out} \]
integrate((-x**6+1)**(1/2)*(2*x**6+1)*(x**12-x**8-2*x**6-x**4+x**2+1)/(x** 6-1)/(x**18-3*x**12+2*x**6-1),x)
\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx=\int { \frac {{\left (x^{12} - x^{8} - 2 \, x^{6} - x^{4} + x^{2} + 1\right )} {\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{18} - 3 \, x^{12} + 2 \, x^{6} - 1\right )} {\left (x^{6} - 1\right )}} \,d x } \]
integrate((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^1 8-3*x^12+2*x^6-1),x, algorithm="maxima")
integrate((x^12 - x^8 - 2*x^6 - x^4 + x^2 + 1)*(2*x^6 + 1)*sqrt(-x^6 + 1)/ ((x^18 - 3*x^12 + 2*x^6 - 1)*(x^6 - 1)), x)
\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx=\int { \frac {{\left (x^{12} - x^{8} - 2 \, x^{6} - x^{4} + x^{2} + 1\right )} {\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{18} - 3 \, x^{12} + 2 \, x^{6} - 1\right )} {\left (x^{6} - 1\right )}} \,d x } \]
integrate((-x^6+1)^(1/2)*(2*x^6+1)*(x^12-x^8-2*x^6-x^4+x^2+1)/(x^6-1)/(x^1 8-3*x^12+2*x^6-1),x, algorithm="giac")
integrate((x^12 - x^8 - 2*x^6 - x^4 + x^2 + 1)*(2*x^6 + 1)*sqrt(-x^6 + 1)/ ((x^18 - 3*x^12 + 2*x^6 - 1)*(x^6 - 1)), x)
Timed out. \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right ) \left (1+x^2-x^4-2 x^6-x^8+x^{12}\right )}{\left (-1+x^6\right ) \left (-1+2 x^6-3 x^{12}+x^{18}\right )} \, dx=\int -\frac {\left (2\,x^6+1\right )\,\left (x^{12}-x^8-2\,x^6-x^4+x^2+1\right )}{\sqrt {1-x^6}\,\left (x^{18}-3\,x^{12}+2\,x^6-1\right )} \,d x \]
int(-((2*x^6 + 1)*(x^2 - x^4 - 2*x^6 - x^8 + x^12 + 1))/((1 - x^6)^(1/2)*( 2*x^6 - 3*x^12 + x^18 - 1)),x)