Integrand size = 27, antiderivative size = 177 \[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{2 \sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1600, 2178, 2177, 245, 2174, 371} \[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((1-x) (x+1)^2\right )}{2 \sqrt [3]{2}} \]
[In]
[Out]
Rule 245
Rule 371
Rule 1600
Rule 2174
Rule 2177
Rule 2178
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx \\ & = \frac {1}{2} \left (1-x^3\right )^{2/3}+\int \frac {x}{\sqrt [3]{1-x^3}} \, dx+\int \frac {1-x}{(1+x) \sqrt [3]{1-x^3}} \, dx \\ & = \frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+2 \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx-\int \frac {1}{\sqrt [3]{1-x^3}} \, dx \\ & = \frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{2 \sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \\ \end{align*}
\[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]
[In]
[Out]
\[\int \frac {\left (x^{2}-x +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}d x\]
[In]
[Out]
\[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}}{x^{3} + 1} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{x + 1}\, dx \]
[In]
[Out]
\[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}}{x^{3} + 1} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}}{x^{3} + 1} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}\,\left (x^2-x+1\right )}{x^3+1} \,d x \]
[In]
[Out]