Integrand size = 14, antiderivative size = 399 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=-\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1388, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^3}{2+x^3+x^6} \, dx=-\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+\left (1-i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+\left (1+i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1388
Rubi steps \begin{align*} \text {integral}& = \frac {1}{14} \left (7-i \sqrt {7}\right ) \int \frac {1}{\frac {1}{2}+\frac {i \sqrt {7}}{2}+x^3} \, dx+\frac {1}{14} \left (7+i \sqrt {7}\right ) \int \frac {1}{\frac {1}{2}-\frac {i \sqrt {7}}{2}+x^3} \, dx \\ & = \frac {\left (7-i \sqrt {7}\right ) \int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}+x} \, dx}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \int \frac {2^{2/3} \sqrt [3]{1+i \sqrt {7}}-x}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}+x} \, dx}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {2^{2/3} \sqrt [3]{1-i \sqrt {7}}-x}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}} \\ & = \frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}+2 x}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{14\ 2^{2/3} \sqrt [3]{1+i \sqrt {7}}}-\frac {\left (7+i \sqrt {7}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}+2 x}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{14\ 2^{2/3} \sqrt [3]{1-i \sqrt {7}}} \\ & = \frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{7 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{7 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}} \\ & = -\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\frac {1}{3} \text {RootSum}\left [2+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\) | \(36\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\) | \(36\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.65 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=-\frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (i \, \sqrt {-3} + i\right )} + 196 \, x\right ) + \frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (i \, \sqrt {-3} - i\right )} + 196 \, x\right ) + \frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (-i \, \sqrt {-3} + i\right )} + 196 \, x\right ) - \frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (-i \, \sqrt {-3} - i\right )} + 196 \, x\right ) + \frac {1}{294} \cdot 98^{\frac {2}{3}} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} \log \left (i \cdot 98^{\frac {2}{3}} \sqrt {7} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 98 \, x\right ) + \frac {1}{294} \cdot 98^{\frac {2}{3}} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} \log \left (-i \cdot 98^{\frac {2}{3}} \sqrt {7} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 98 \, x\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.06 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\operatorname {RootSum} {\left (250047 t^{6} + 1323 t^{3} + 2, \left ( t \mapsto t \log {\left (7938 t^{4} + 21 t + x \right )} \right )\right )} \]
[In]
[Out]
\[ \int \frac {x^3}{2+x^3+x^6} \, dx=\int { \frac {x^{3}}{x^{6} + x^{3} + 2} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 9.29 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\frac {\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-196-\sqrt {7}\,28{}\mathrm {i}\right )}^{1/3}}{42}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{42}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84} \]
[In]
[Out]