Integrand size = 27, antiderivative size = 86 \[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )+\log \left (-x+\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]
-3^(1/2)*arctan((2/3*x*3^(1/2)+1/3*(x^2+1)^(1/3)*3^(1/2))/(x^2+1)^(1/3))+l n(-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 = 72, normalized size of antiderivative = 0.84 \[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )+\log \left (-x+\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]
-(Sqrt[3]*ArcTan[(1 + (2*x)/(1 + x^2)^(1/3))/Sqrt[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 |
\(\displaystyle 3 \int \frac {1}{\sqrt [3]{x^2+1} \left (x^3-x^2-1\right )}dx+\int \frac {x^2}{\sqrt [3]{x^2+1} \left (x^3-x^2-1\right )}dx\) |
3.12.58.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.43 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.21
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 x \left (x^{2}+1\right )^{\frac {2}{3}}-3 \left (x^{2}+1\right )^{\frac {1}{3}} x^{2}-x^{3}-x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{3}-x^{2}-1}\right )-\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x +\left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x \left (x^{2}+1\right )^{\frac {2}{3}}-\left (x^{2}+1\right )^{\frac {1}{3}} x^{2}-x^{2}-1}{x^{3}-x^{2}-1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x +\left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x \left (x^{2}+1\right )^{\frac {2}{3}}-\left (x^{2}+1\right )^{\frac {1}{3}} x^{2}-x^{2}-1}{x^{3}-x^{2}-1}\right )\) | \(276\) |
RootOf(_Z^2+_Z+1)*ln((RootOf(_Z^2+_Z+1)^2*x^3+RootOf(_Z^2+_Z+1)*x^2-3*x*(x ^2+1)^(2/3)-3*(x^2+1)^(1/3)*x^2-x^3-x^2+RootOf(_Z^2+_Z+1)-1)/(x^3-x^2-1))- ln((RootOf(_Z^2+_Z+1)*(x^2+1)^(2/3)*x+(x^2+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+ RootOf(_Z^2+_Z+1)*x^3-x*(x^2+1)^(2/3)-(x^2+1)^(1/3)*x^2-x^2-1)/(x^3-x^2-1) )*RootOf(_Z^2+_Z+1)-ln((RootOf(_Z^2+_Z+1)*(x^2+1)^(2/3)*x+(x^2+1)^(1/3)*Ro otOf(_Z^2+_Z+1)*x^2+RootOf(_Z^2+_Z+1)*x^3-x*(x^2+1)^(2/3)-(x^2+1)^(1/3)*x^ 2-x^2-1)/(x^3-x^2-1))
Time = 0.69 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21 \[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{3} - 2 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 8 \, x^{2} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} - x^{2} + 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} x - 1}{x^{3} - x^{2} - 1}\right ) \]
-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))
\[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx=\int \frac {x^{2} + 3}{\sqrt [3]{x^{2} + 1} \left (x^{3} - x^{2} - 1\right )}\, dx \]
\[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx=\int { \frac {x^{2} + 3}{{\left (x^{3} - x^{2} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx=\int { \frac {x^{2} + 3}{{\left (x^{3} - x^{2} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx=\int -\frac {x^2+3}{{\left (x^2+1\right )}^{1/3}\,\left (-x^3+x^2+1\right )} \,d x \]