Integrand size = 31, antiderivative size = 175 \[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=-\frac {3 \left (1-x+x^3\right )^{2/3}}{4 x^2}+\frac {3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1-x+x^3}}\right )}{2\ 2^{2/3}}-\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1-x+x^3}\right )+\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1-x+x^3}+2^{2/3} \sqrt [3]{3} \left (1-x+x^3\right )^{2/3}\right ) \]
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\[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=\int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \left (1-x+x^3\right )^{2/3}}{2 x^3}+\frac {\left (1-x+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1-x+x^3\right )^{2/3}}{2 x}+\frac {\left (-5-x-x^2\right ) \left (1-x+x^3\right )^{2/3}}{2 \left (-2+2 x+x^3\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x^2} \, dx+\frac {1}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x} \, dx+\frac {1}{2} \int \frac {\left (-5-x-x^2\right ) \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx+\frac {3}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x^3} \, dx \\ & = \frac {1}{2} \int \left (-\frac {5 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}-\frac {x \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}-\frac {x^2 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}\right ) \, dx+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (3 \left (1-x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}} \\ & = -\left (\frac {1}{2} \int \frac {x \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx\right )-\frac {1}{2} \int \frac {x^2 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx-\frac {5}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (3 \left (1-x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00 \[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=-\frac {3 \left (1-x+x^3\right )^{2/3}}{4 x^2}+\frac {3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1-x+x^3}}\right )}{2\ 2^{2/3}}-\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1-x+x^3}\right )+\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1-x+x^3}+2^{2/3} \sqrt [3]{3} \left (1-x+x^3\right )^{2/3}\right ) \]
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Time = 16.75 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (2\right )-2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 \left (x^{3}-x +1\right )^{\frac {1}{3}}}{x}\right )+2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2} \ln \left (\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}-x +1\right )^{\frac {1}{3}} x +2 \left (x^{3}-x +1\right )^{\frac {2}{3}}}{x^{2}}\right )-6 \,3^{\frac {1}{6}} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-x +1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{2}-6 \left (x^{3}-x +1\right )^{\frac {2}{3}}}{8 x^{2}}\) | \(167\) |
risch | \(\text {Expression too large to display}\) | \(1033\) |
trager | \(\text {Expression too large to display}\) | \(1234\) |
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Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (133) = 266\).
Time = 9.60 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.43 \[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=-\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{5} - 7 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} - 2 \, x\right )} {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} + 12 \, \left (-9\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{6} + 50 \, x^{5} + 4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{7} + 600 \, x^{6} + 204 \, x^{5} - 408 \, x^{4} + 196 \, x^{3} + 24 \, x^{2} - 24 \, x + 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{7} + 480 \, x^{6} + 12 \, x^{5} - 24 \, x^{4} + 20 \, x^{3} - 24 \, x^{2} + 24 \, x - 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-9\right )^{\frac {2}{3}} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} {\left (x^{3} + 2 \, x - 2\right )} - 36 \, {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} x}{x^{3} + 2 \, x - 2}\right ) + 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {18 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x^{2} + x\right )} {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-9\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{4} + 50 \, x^{3} + 4 \, x^{2} - 8 \, x + 4\right )} - 54 \, {\left (7 \, x^{5} - 4 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, x + 4}\right ) + 36 \, {\left (x^{3} - x + 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \]
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\[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=\int \frac {\left (2 x - 3\right ) \left (x^{3} - x + 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{3} + 2 x - 2\right )}\, dx \]
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\[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - x + 1\right )}^{\frac {2}{3}} {\left (2 \, x - 3\right )}}{{\left (x^{3} + 2 \, x - 2\right )} x^{3}} \,d x } \]
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\[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - x + 1\right )}^{\frac {2}{3}} {\left (2 \, x - 3\right )}}{{\left (x^{3} + 2 \, x - 2\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx=\int \frac {\left (2\,x-3\right )\,{\left (x^3-x+1\right )}^{2/3}}{x^3\,\left (x^3+2\,x-2\right )} \,d x \]
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