Integrand size = 45, antiderivative size = 93 \[ \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\frac {1}{5} \left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {-1-x^2+x^3}}\right )+\frac {1}{5} \left (5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {-1-x^2+x^3}}\right ) \]
1/5*(-5-5^(1/2))*arctan((1/2+1/2*5^(1/2))*x/(x^3-x^2-1)^(1/2))+1/5*(5-5^(1 /2))*arctan(2^(1/2)/(3+5^(1/2))^(1/2)*x/(x^3-x^2-1)^(1/2))
Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\frac {1}{5} \left (-\left (\left (-5+\sqrt {5}\right ) \arctan \left (\frac {\left (-1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^3}}\right )\right )-\left (5+\sqrt {5}\right ) \arctan \left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^3}}\right )\right ) \]
(-((-5 + Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^3])]) - ( 5 + Sqrt[5])*ArcTan[((1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^3])])/5
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.
| \(\displaystyle \int \frac {\left (x^3+2\right ) \sqrt {x^3-x^2-1}}{x^6+x^5-x^4-2 x^3-x^2+1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {x^3-x^2-1} x^3}{x^6+x^5-x^4-2 x^3-x^2+1}+\frac {2 \sqrt {x^3-x^2-1}}{x^6+x^5-x^4-2 x^3-x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {\sqrt {x^3-x^2-1}}{x^6+x^5-x^4-2 x^3-x^2+1}dx+\int \frac {x^3 \sqrt {x^3-x^2-1}}{x^6+x^5-x^4-2 x^3-x^2+1}dx\) |
3.13.91.3.1 Defintions of rubi rules used
Time = 11.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(-\frac {\left (\left (-\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}+1\right )}\right )+\left (\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}-1\right )}\right )\right ) \sqrt {5}}{5}\) | \(71\) |
| pseudoelliptic | \(-\frac {\left (\left (-\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}+1\right )}\right )+\left (\sqrt {5}-1\right ) \arctan \left (\frac {2 \sqrt {x^{3}-x^{2}-1}}{x \left (\sqrt {5}-1\right )}\right )\right ) \sqrt {5}}{5}\) | \(71\) |
| trager | \(5 \ln \left (\frac {175 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}-35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}+230 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}+30 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-1}\, x -18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}+35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+72 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}+16 \sqrt {x^{3}-x^{2}-1}\, x +18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 x^{2} \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2}+x^{3}+2 x^{2}-1}\right ) \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+3 \ln \left (\frac {175 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}-35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}+230 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}+30 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-1}\, x -18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}+35 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}+72 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}+16 \sqrt {x^{3}-x^{2}-1}\, x +18 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 x^{2} \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2}+x^{3}+2 x^{2}-1}\right ) \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )-\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {75 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{5} x^{2}+15 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{3}-5 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3} x^{2}+30 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-1}\, x +2 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{3}-15 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{3}-2 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}-x^{2}-1}\, x -2 \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )}{5 x^{2} \operatorname {RootOf}\left (25 \textit {\_Z}^{4}+15 \textit {\_Z}^{2}+1\right )^{2}-x^{3}+x^{2}+1}\right )\) | \(674\) |
| elliptic | \(\text {Expression too large to display}\) | \(2877\) |
-1/5*((-5^(1/2)-1)*arctan(2*(x^3-x^2-1)^(1/2)/x/(5^(1/2)+1))+(5^(1/2)-1)*a rctan(2*(x^3-x^2-1)^(1/2)/x/(5^(1/2)-1)))*5^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 497, normalized size of antiderivative = 5.34 \[ \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} + {\left (2 \, x^{4} + \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {2 \, \sqrt {5} - 6} + 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} - {\left (2 \, x^{4} + \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {2 \, \sqrt {5} - 6} + 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} + {\left (2 \, x^{4} - \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {-2 \, \sqrt {5} - 6} - 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-2 \, \sqrt {5} - 6} \log \left (\frac {2 \, x^{6} - 4 \, x^{5} - 4 \, x^{3} + 4 \, x^{2} - {\left (2 \, x^{4} - \sqrt {5} x^{3} + x^{3} - 2 \, x\right )} \sqrt {x^{3} - x^{2} - 1} \sqrt {-2 \, \sqrt {5} - 6} - 2 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{2}\right )} + 2}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1}\right ) \]
-1/20*sqrt(5)*sqrt(2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 + (2* x^4 + sqrt(5)*x^3 + x^3 - 2*x)*sqrt(x^3 - x^2 - 1)*sqrt(2*sqrt(5) - 6) + 2 *sqrt(5)*(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1)) + 1/2 0*sqrt(5)*sqrt(2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 - (2*x^4 + sqrt(5)*x^3 + x^3 - 2*x)*sqrt(x^3 - x^2 - 1)*sqrt(2*sqrt(5) - 6) + 2*sqr t(5)*(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1)) + 1/20*sq rt(5)*sqrt(-2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 + (2*x^4 - s qrt(5)*x^3 + x^3 - 2*x)*sqrt(x^3 - x^2 - 1)*sqrt(-2*sqrt(5) - 6) - 2*sqrt( 5)*(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1)) - 1/20*sqrt (5)*sqrt(-2*sqrt(5) - 6)*log((2*x^6 - 4*x^5 - 4*x^3 + 4*x^2 - (2*x^4 - sqr t(5)*x^3 + x^3 - 2*x)*sqrt(x^3 - x^2 - 1)*sqrt(-2*sqrt(5) - 6) - 2*sqrt(5) *(x^5 - x^4 - x^2) + 2)/(x^6 + x^5 - x^4 - 2*x^3 - x^2 + 1))
\[ \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\int \frac {\left (x^{3} + 2\right ) \sqrt {x^{3} - x^{2} - 1}}{x^{6} + x^{5} - x^{4} - 2 x^{3} - x^{2} + 1}\, dx \]
\[ \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1} \,d x } \]
\[ \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - 1} {\left (x^{3} + 2\right )}}{x^{6} + x^{5} - x^{4} - 2 \, x^{3} - x^{2} + 1} \,d x } \]
Time = 8.11 (sec) , antiderivative size = 2803, normalized size of antiderivative = 30.14 \[ \int \frac {\left (2+x^3\right ) \sqrt {-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx=\text {Too large to display} \]
symsum(-(2*((x - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^( 1/2))/108 + 29/54)^(1/3)) + ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1 /3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^ (1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^ (1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))/2))^(1/2)*((1/ (9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - x + ((31^(1/2)*108^(1/2))/1 08 + 29/54)^(1/3) + 1/3)/(1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108 ^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/54)^ (1/3))/2))^(1/2)*ellipticPi((1/(6*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3) ) - (3^(1/2)*(1/(9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)* 108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + (3*((31^(1/2)*108^(1/2))/108 + 29/5 4)^(1/3))/2)/(root(z^6 + z^5 - z^4 - 2*z^3 - z^2 + 1, z, k) - (3^(1/2)*(1/ (9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + ( (31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3), asin(((x - (3^(1/2)*(1/( 9*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) - ((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3))*1i)/2 + 1/(18*((31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)) + (( 31^(1/2)*108^(1/2))/108 + 29/54)^(1/3)/2 - 1/3)/(1/(6*((31^(1/2)*108^(1...