Integrand size = 25, antiderivative size = 85 \[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (1+x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )+\log \left (x+\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]
-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))+ 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 = 71, 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 |
\(\displaystyle 3 \int \frac {1}{\sqrt [3]{x^2+1} \left (x^3+x^2+1\right )}dx+\int \frac {x^2}{\sqrt [3]{x^2+1} \left (x^3+x^2+1\right )}dx\) |
3.12.35.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.56 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.38
method | result | size |
trager | \(\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 x \left (x^{2}+1\right )^{\frac {2}{3}}+2 x^{3}-2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x^{3}+x^{2}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 x \left (x^{2}+1\right )^{\frac {2}{3}}+x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{3}+x^{2}+1}\right )\) | \(202\) |
ln(-(RootOf(_Z^2+_Z+1)^2*x^3+3*(x^2+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+3*RootO f(_Z^2+_Z+1)*x^3-RootOf(_Z^2+_Z+1)*x^2-3*x*(x^2+1)^(2/3)+2*x^3-2*x^2-RootO f(_Z^2+_Z+1)-2)/(x^3+x^2+1))+RootOf(_Z^2+_Z+1)*ln((2*RootOf(_Z^2+_Z+1)^2*x ^3-3*(x^2+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+RootOf(_Z^2+_Z+1)*x^3+2*RootOf(_Z ^2+_Z+1)*x^2+3*x*(x^2+1)^(2/3)+x^2+2*RootOf(_Z^2+_Z+1)+1)/(x^3+x^2+1))
Time = 0.72 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.18 \[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (1+x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{3} + 2 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {2}{3}} x}{x^{3} - 8 \, x^{2} - 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} + 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} + x^{2} + 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} x + 1}{x^{3} + x^{2} + 1}\right ) \]
-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. \[ \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (1+x^2+x^3\right )} \, dx=\int \frac {x^2+3}{{\left (x^2+1\right )}^{1/3}\,\left (x^3+x^2+1\right )} \,d x \]