Integrand size = 25, antiderivative size = 80 \[ \int \frac {-3+x^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{-2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]
3^(1/2)*arctan(3^(1/2)*(x^2-1)^(1/3)/(-2*x+(x^2-1)^(1/3)))+ln(x+(x^2-1)^(1 /3))-1/2*ln(x^2-x*(x^2-1)^(1/3)+(x^2-1)^(2/3))
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {-3+x^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{-2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]
Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3))/(-2*x + (-1 + x^2)^(1/3))] + Log [x + (-1 + x^2)^(1/3)] - Log[x^2 - x*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)]/ 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.
\(\displaystyle \int \frac {x^2-3}{\sqrt [3]{x^2-1} \left (x^3+x^2-1\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2}{\sqrt [3]{x^2-1} \left (x^3+x^2-1\right )}-\frac {3}{\sqrt [3]{x^2-1} \left (x^3+x^2-1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {x^2}{\sqrt [3]{x^2-1} \left (x^3+x^2-1\right )}dx-3 \int \frac {1}{\sqrt [3]{x^2-1} \left (x^3+x^2-1\right )}dx\) |
3.11.60.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.46 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.55
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 x \left (x^{2}-1\right )^{\frac {2}{3}}-2 \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+x^{3}-x^{2}+1}{x^{3}+x^{2}-1}\right )-\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 x \left (x^{2}-1\right )^{\frac {2}{3}}+3 \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x^{3}+x^{2}-1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 x \left (x^{2}-1\right )^{\frac {2}{3}}+3 \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x^{3}+x^{2}-1}\right )\) | \(284\) |
RootOf(_Z^2+_Z+1)*ln(-(RootOf(_Z^2+_Z+1)*(x^2-1)^(2/3)*x-RootOf(_Z^2+_Z+1) *(x^2-1)^(1/3)*x^2+RootOf(_Z^2+_Z+1)*x^3+2*x*(x^2-1)^(2/3)-2*(x^2-1)^(1/3) *x^2+x^3-x^2+1)/(x^3+x^2-1))-ln((RootOf(_Z^2+_Z+1)^2*x^3+2*RootOf(_Z^2+_Z+ 1)*x^3+RootOf(_Z^2+_Z+1)*x^2-3*x*(x^2-1)^(2/3)+3*(x^2-1)^(1/3)*x^2+2*x^2-R ootOf(_Z^2+_Z+1)-2)/(x^3+x^2-1))*RootOf(_Z^2+_Z+1)-ln((RootOf(_Z^2+_Z+1)^2 *x^3+2*RootOf(_Z^2+_Z+1)*x^3+RootOf(_Z^2+_Z+1)*x^2-3*x*(x^2-1)^(2/3)+3*(x^ 2-1)^(1/3)*x^2+2*x^2-RootOf(_Z^2+_Z+1)-2)/(x^3+x^2-1))
Time = 0.70 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {-3+x^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{3} + 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 8 \, x^{2} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} + 3 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} x^{2} + x^{2} + 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} x - 1}{x^{3} + x^{2} - 1}\right ) \]
-sqrt(3)*arctan((sqrt(3)*x^3 + 2*sqrt(3)*(x^2 - 1)^(1/3)*x^2 + 4*sqrt(3)*( x^2 - 1)^(2/3)*x)/(x^3 - 8*x^2 + 8)) + 1/2*log((x^3 + 3*(x^2 - 1)^(1/3)*x^ 2 + x^2 + 3*(x^2 - 1)^(2/3)*x - 1)/(x^3 + x^2 - 1))
\[ \int \frac {-3+x^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx=\int \frac {x^{2} - 3}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x^{3} + x^{2} - 1\right )}\, dx \]
\[ \int \frac {-3+x^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx=\int { \frac {x^{2} - 3}{{\left (x^{3} + x^{2} - 1\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {-3+x^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx=\int { \frac {x^{2} - 3}{{\left (x^{3} + x^{2} - 1\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {-3+x^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx=\int \frac {x^2-3}{{\left (x^2-1\right )}^{1/3}\,\left (x^3+x^2-1\right )} \,d x \]