Integrand size = 40, antiderivative size = 241 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right ) \]
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+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 = 8.55 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2-\sqrt {1+x^6}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )-\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )}{2 \sqrt {2}} \]
(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(1 + x^6)^(1/4))/(x^2 - Sqr t[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[(Sq rt[2 + Sqrt[2]]*x*(1 + x^6)^(1/4))/(x^2 + Sqrt[1 + x^6])])/(2*Sqrt[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 |
\(\displaystyle -3 \int \frac {1}{\sqrt [4]{x^6+1} \left (x^{12}+x^8+2 x^6+1\right )}dx-3 \int \frac {x^6}{\sqrt [4]{x^6+1} \left (x^{12}+x^8+2 x^6+1\right )}dx-\int \frac {x^8}{\sqrt [4]{x^6+1} \left (x^{12}+x^8+2 x^6+1\right )}dx-\int \frac {x^{10}}{\sqrt [4]{x^6+1} \left (x^{12}+x^8+2 x^6+1\right )}dx+2 \int \frac {x^4}{\sqrt [4]{x^6+1} \left (x^{12}+x^8+2 x^6+1\right )}dx+x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )\) |
3.27.77.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 16.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16
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 | \(\text {Expression too large to display}\) | \(679\) |
Result contains complex when optimal does not.
Time = 37.83 (sec) , antiderivative size = 1319, normalized size of antiderivative = 5.47 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\text {Too large to display} \]
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)...
\[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^{6} - 2\right ) \left (x^{6} - x^{4} + 1\right )}{\sqrt [4]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (x^{12} + x^{8} + 2 x^{6} + 1\right )}\, dx \]
Integral((x**6 - 2)*(x**6 - x**4 + 1)/(((x**2 + 1)*(x**4 - x**2 + 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. \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^6-2\right )\,\left (x^6-x^4+1\right )}{{\left (x^6+1\right )}^{1/4}\,\left (x^{12}+x^8+2\,x^6+1\right )} \,d x \]