Integrand size = 38, antiderivative size = 261 \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right ) \] Output:
-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 = 8.63 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.85 \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )}{2 \sqrt {2}} \] Input:
Integrate[((2 + x^6)*(-1 + x^4 + x^6))/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)),x]
Output:
-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 |
| \(\displaystyle -3 \int \frac {1}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+3 \int \frac {x^6}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx-\int \frac {x^8}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+\int \frac {x^{10}}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+2 \int \frac {x^4}{\sqrt [4]{x^6-1} \left (x^{12}+x^8-2 x^6+1\right )}dx+\frac {x \sqrt [4]{1-x^6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},x^6\right )}{\sqrt [4]{x^6-1}}\) |
Input:
Int[((2 + x^6)*(-1 + x^4 + x^6))/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12 )),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 19.36 (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 | \(\text {Expression too large to display}\) | \(681\) |
Input:
int((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x,method=_RETURNV ERBOSE)
Output:
1/4*sum(1/_R^5*(_R^4+1)*ln((-_R*x+(x^6-1)^(1/4))/x),_R=RootOf(_Z^8+1))
Result contains complex when optimal does not.
Time = 37.46 (sec) , antiderivative size = 1344, normalized size of antiderivative = 5.15 \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\text {Too large to display} \] Input:
integrate((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x, algorith m="fricas")
Output:
(1/8*I - 1/8)*sqrt(2)*(-1/16)^(1/8)*log(-(4*sqrt(2)*((4*I + 4)*(-1/16)^(7/ 8)*x^6 + (-1/16)^(3/8)*(-(I + 1)*x^8 + (I + 1)*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)*(8*(-1/16) ^(5/8)*(-(I - 1)*x^10 + (I - 1)*x^4) - (-1/16)^(1/8)*(-(I - 1)*x^12 + (I - 1)*x^8 + (2*I - 2)*x^6 - I + 1)) + 2*(x^6 - 1)^(1/4)*((-1)^(3/4)*(-I*x^9 + I*x^7 + I*x^3) + (-1)^(1/4)*(-I*x^9 - I*x^7 + I*x^3)))/(x^12 + x^8 - 2*x ^6 + 1)) - (1/8*I + 1/8)*sqrt(2)*(-1/16)^(1/8)*log(-(4*sqrt(2)*(-(4*I - 4) *(-1/16)^(7/8)*x^6 + (-1/16)^(3/8)*((I - 1)*x^8 - (I - 1)*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) *(8*(-1/16)^(5/8)*((I + 1)*x^10 - (I + 1)*x^4) - (-1/16)^(1/8)*((I + 1)*x^ 12 - (I + 1)*x^8 - (2*I + 2)*x^6 + I + 1)) + 2*(x^6 - 1)^(1/4)*((-1)^(3/4) *(I*x^9 - I*x^7 - I*x^3) + (-1)^(1/4)*(I*x^9 + I*x^7 - I*x^3)))/(x^12 + x^ 8 - 2*x^6 + 1)) + (1/8*I + 1/8)*sqrt(2)*(-1/16)^(1/8)*log(-(4*sqrt(2)*((4* I - 4)*(-1/16)^(7/8)*x^6 + (-1/16)^(3/8)*(-(I - 1)*x^8 + (I - 1)*x^2))*sqr t(x^6 - 1) + 2*(-(I + 1)*x^7 + (I - 1)*x^5 + (I + 1)*x)*(x^6 - 1)^(3/4) + sqrt(2)*(8*(-1/16)^(5/8)*(-(I + 1)*x^10 + (I + 1)*x^4) - (-1/16)^(1/8)*(-( I + 1)*x^12 + (I + 1)*x^8 + (2*I + 2)*x^6 - I - 1)) + 2*(x^6 - 1)^(1/4)*(( -1)^(3/4)*(I*x^9 - I*x^7 - I*x^3) + (-1)^(1/4)*(I*x^9 + I*x^7 - I*x^3)))/( x^12 + x^8 - 2*x^6 + 1)) - (1/8*I - 1/8)*sqrt(2)*(-1/16)^(1/8)*log(-(4*sqr t(2)*(-(4*I + 4)*(-1/16)^(7/8)*x^6 + (-1/16)^(3/8)*((I + 1)*x^8 - (I + ...
\[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^{6} + 2\right ) \left (x^{6} + x^{4} - 1\right )}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{12} + x^{8} - 2 x^{6} + 1\right )}\, dx \] Input:
integrate((x**6+2)*(x**6+x**4-1)/(x**6-1)**(1/4)/(x**12+x**8-2*x**6+1),x)
Output:
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)
\[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int { \frac {{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x, algorith m="maxima")
Output:
integrate((x^6 + x^4 - 1)*(x^6 + 2)/((x^12 + x^8 - 2*x^6 + 1)*(x^6 - 1)^(1 /4)), x)
\[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int { \frac {{\left (x^{6} + x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x, algorith m="giac")
Output:
integrate((x^6 + x^4 - 1)*(x^6 + 2)/((x^12 + x^8 - 2*x^6 + 1)*(x^6 - 1)^(1 /4)), x)
Timed out. \[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^6+2\right )\,\left (x^6+x^4-1\right )}{{\left (x^6-1\right )}^{1/4}\,\left (x^{12}+x^8-2\,x^6+1\right )} \,d x \] Input:
int(((x^6 + 2)*(x^4 + x^6 - 1))/((x^6 - 1)^(1/4)*(x^8 - 2*x^6 + x^12 + 1)) ,x)
Output:
int(((x^6 + 2)*(x^4 + x^6 - 1))/((x^6 - 1)^(1/4)*(x^8 - 2*x^6 + x^12 + 1)) , x)
\[ \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {x^{12}}{\left (x^{6}-1\right )^{\frac {1}{4}} x^{12}+\left (x^{6}-1\right )^{\frac {1}{4}} x^{8}-2 \left (x^{6}-1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}-1\right )^{\frac {1}{4}}}d x +\int \frac {x^{10}}{\left (x^{6}-1\right )^{\frac {1}{4}} x^{12}+\left (x^{6}-1\right )^{\frac {1}{4}} x^{8}-2 \left (x^{6}-1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}-1\right )^{\frac {1}{4}}}d x +\int \frac {x^{6}}{\left (x^{6}-1\right )^{\frac {1}{4}} x^{12}+\left (x^{6}-1\right )^{\frac {1}{4}} x^{8}-2 \left (x^{6}-1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}-1\right )^{\frac {1}{4}}}d x +2 \left (\int \frac {x^{4}}{\left (x^{6}-1\right )^{\frac {1}{4}} x^{12}+\left (x^{6}-1\right )^{\frac {1}{4}} x^{8}-2 \left (x^{6}-1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}-1\right )^{\frac {1}{4}}}d x \right )-2 \left (\int \frac {1}{\left (x^{6}-1\right )^{\frac {1}{4}} x^{12}+\left (x^{6}-1\right )^{\frac {1}{4}} x^{8}-2 \left (x^{6}-1\right )^{\frac {1}{4}} x^{6}+\left (x^{6}-1\right )^{\frac {1}{4}}}d x \right ) \] Input:
int((x^6+2)*(x^6+x^4-1)/(x^6-1)^(1/4)/(x^12+x^8-2*x^6+1),x)
Output:
int(x**12/((x**6 - 1)**(1/4)*x**12 + (x**6 - 1)**(1/4)*x**8 - 2*(x**6 - 1) **(1/4)*x**6 + (x**6 - 1)**(1/4)),x) + int(x**10/((x**6 - 1)**(1/4)*x**12 + (x**6 - 1)**(1/4)*x**8 - 2*(x**6 - 1)**(1/4)*x**6 + (x**6 - 1)**(1/4)),x ) + int(x**6/((x**6 - 1)**(1/4)*x**12 + (x**6 - 1)**(1/4)*x**8 - 2*(x**6 - 1)**(1/4)*x**6 + (x**6 - 1)**(1/4)),x) + 2*int(x**4/((x**6 - 1)**(1/4)*x* *12 + (x**6 - 1)**(1/4)*x**8 - 2*(x**6 - 1)**(1/4)*x**6 + (x**6 - 1)**(1/4 )),x) - 2*int(1/((x**6 - 1)**(1/4)*x**12 + (x**6 - 1)**(1/4)*x**8 - 2*(x** 6 - 1)**(1/4)*x**6 + (x**6 - 1)**(1/4)),x)