Integrand size = 38, antiderivative size = 261 \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right ) \]
-1/4*(4-2*2^(1/2))^(1/2)*arctan((2+2^(1/2))^(1/2)*x*(x^6-1)^(1/4)/(-x^2+(x ^6-1)^(1/2)))+1/4*(4+2*2^(1/2))^(1/2)*arctan((2^(1/2)/(2-2^(1/2))^(1/2)-2/ (2-2^(1/2))^(1/2))*x*(x^6-1)^(1/4)/(-x^2+(x^6-1)^(1/2)))-1/4*(4+2*2^(1/2)) ^(1/2)*arctanh((2-2^(1/2))^(1/2)*x*(x^6-1)^(1/4)/(x^2+(x^6-1)^(1/2)))-1/4* (4-2*2^(1/2))^(1/2)*arctanh((2+2^(1/2))^(1/2)*x*(x^6-1)^(1/4)/(x^2+(x^6-1) ^(1/2)))
Time = 8.57 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.85 \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )}{2 \sqrt {2}} \]
-1/2*(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^6)^(1/4))/(-x^ 2 + Sqrt[-1 + x^6])] + Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^6)^(1/4))/(-x^2 + Sqrt[-1 + x^6])] + Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^6)^(1/4))/(x^2 + Sqrt[-1 + x^6])] + Sqrt[2 - Sqrt[2]] *ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^6)^(1/4))/(x^2 + Sqrt[-1 + x^6])])/S qrt[2]
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.
\(\displaystyle \int \frac {\left (x^6+2\right ) \left (x^6+x^4-1\right )}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{\sqrt [4]{x^6-1}}-\frac {-x^{10}+x^8-3 x^6-2 x^4+3}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \frac {1}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+3 \int \frac {x^6}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx-\int \frac {x^8}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+\int \frac {x^{10}}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+2 \int \frac {x^4}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+\frac {x \sqrt [4]{1-x^6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},x^6\right )}{\sqrt [4]{x^6-1}}\) |
3.28.63.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 16.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.15
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{6}-1\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}}\right )}{4}\) | \(38\) |
trager | \(\text {Expression too large to display}\) | \(681\) |
Result contains complex when optimal does not.
Time = 36.56 (sec) , antiderivative size = 1349, normalized size of antiderivative = 5.17 \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\text {Too large to display} \]
-1/8*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*((-1)^(7/8)*x^6 + (-1)^(3/8)*(x^ 8 - x^2))*sqrt(x^6 - 1) + 2*((I - 1)*x^7 - (I + 1)*x^5 - (I - 1)*x)*(x^6 - 1)^(3/4) - sqrt(2)*(2*(-1)^(5/8)*(x^10 - x^4) + (-1)^(1/8)*(x^12 - x^8 - 2*x^6 + 1)) - 2*(x^6 - 1)^(1/4)*((-1)^(3/4)*(x^9 - x^7 - x^3) - (-1)^(1/4) *(x^9 + x^7 - x^3)))/(x^12 + x^8 - 2*x^6 + 1)) + 1/8*sqrt(2)*(-1)^(1/8)*lo g(8*(2*sqrt(2)*((-1)^(7/8)*x^6 + (-1)^(3/8)*(x^8 - x^2))*sqrt(x^6 - 1) - 2 *((I - 1)*x^7 - (I + 1)*x^5 - (I - 1)*x)*(x^6 - 1)^(3/4) - sqrt(2)*(2*(-1) ^(5/8)*(x^10 - x^4) + (-1)^(1/8)*(x^12 - x^8 - 2*x^6 + 1)) + 2*(x^6 - 1)^( 1/4)*((-1)^(3/4)*(x^9 - x^7 - x^3) - (-1)^(1/4)*(x^9 + x^7 - x^3)))/(x^12 + x^8 - 2*x^6 + 1)) + 1/8*I*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*(I*(-1)^( 7/8)*x^6 + (-1)^(3/8)*(I*x^8 - I*x^2))*sqrt(x^6 - 1) + 2*((I - 1)*x^7 - (I + 1)*x^5 - (I - 1)*x)*(x^6 - 1)^(3/4) + sqrt(2)*(2*(-1)^(5/8)*(I*x^10 - I *x^4) + (-1)^(1/8)*(I*x^12 - I*x^8 - 2*I*x^6 + I)) + 2*(x^6 - 1)^(1/4)*((- 1)^(3/4)*(x^9 - x^7 - x^3) - (-1)^(1/4)*(x^9 + x^7 - x^3)))/(x^12 + x^8 - 2*x^6 + 1)) - 1/8*I*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*(-I*(-1)^(7/8)*x^ 6 + (-1)^(3/8)*(-I*x^8 + I*x^2))*sqrt(x^6 - 1) + 2*((I - 1)*x^7 - (I + 1)* x^5 - (I - 1)*x)*(x^6 - 1)^(3/4) + sqrt(2)*(2*(-1)^(5/8)*(-I*x^10 + I*x^4) + (-1)^(1/8)*(-I*x^12 + I*x^8 + 2*I*x^6 - I)) + 2*(x^6 - 1)^(1/4)*((-1)^( 3/4)*(x^9 - x^7 - x^3) - (-1)^(1/4)*(x^9 + x^7 - x^3)))/(x^12 + x^8 - 2*x^ 6 + 1)) - (1/8*I + 1/8)*(-1)^(1/8)*log(-16*(2*(-(I + 1)*x^7 + (I - 1)*x...
\[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^{6} + 2\right ) \left (x^{6} + x^{4} - 1\right )}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{12} + x^{8} - 2 x^{6} + 1\right )}\, dx \]
Integral((x**6 + 2)*(x**6 + x**4 - 1)/(((x - 1)*(x + 1)*(x**2 - x + 1)*(x* *2 + x + 1))**(1/4)*(x**12 + x**8 - 2*x**6 + 1)), x)
\[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int { \frac {{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int { \frac {{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^6+2\right )\,\left (x^6+x^4-1\right )}{{\left (x^6-1\right )}^{1/4}\,\left (x^{12}+x^8-2\,x^6+1\right )} \,d x \]