Integrand size = 33, antiderivative size = 154 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\frac {3 \left (1-x^3+x^4\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3+x^4}}\right )+2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3+x^4}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3+x^4}-\sqrt [3]{2} \left (1-x^3+x^4\right )^{2/3}\right )}{\sqrt [3]{2}} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (1-x^3+x^4\right )^{2/3}}{x^3}+\frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx \\ & = -\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \left (\frac {3 \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}+\frac {4 x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx+4 \int \frac {x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\frac {3 \left (1-x^3+x^4\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3+x^4}}\right )+2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3+x^4}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3+x^4}-\sqrt [3]{2} \left (1-x^3+x^4\right )^{2/3}\right )}{\sqrt [3]{2}} \]
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Time = 60.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x -2^{\frac {2}{3}} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{4}-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(142\) |
risch | \(\text {Expression too large to display}\) | \(1061\) |
trager | \(\text {Expression too large to display}\) | \(1511\) |
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Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (124) = 248\).
Time = 48.97 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.68 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\frac {2 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{2} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{9} - 4 \, x^{8} - 5 \, x^{7} + 2 \, x^{5} - 4 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{10} - 16 \, x^{9} + 19 \, x^{8} + 2 \, x^{6} - 16 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (x^{12} - 33 \, x^{11} + 111 \, x^{10} - 71 \, x^{9} + 3 \, x^{8} - 66 \, x^{7} + 111 \, x^{6} + 3 \, x^{4} - 33 \, x^{3} + 1\right )}}{3 \, {\left (x^{12} + 3 \, x^{11} - 105 \, x^{10} + 109 \, x^{9} + 3 \, x^{8} + 6 \, x^{7} - 105 \, x^{6} + 3 \, x^{4} + 3 \, x^{3} + 1\right )}}\right ) + 2 \cdot 4^{\frac {1}{3}} x^{2} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{4} + x^{3} + 1\right )}}{x^{4} + x^{3} + 1}\right ) - 4^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} - 5 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {2}{3}} {\left (x^{8} - 16 \, x^{7} + 19 \, x^{6} + 2 \, x^{4} - 16 \, x^{3} + 1\right )} - 24 \, {\left (x^{6} - 2 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}}}{x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 1}\right ) + 9 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{3}} \,d x } \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\int \frac {\left (x^4-3\right )\,{\left (x^4-x^3+1\right )}^{2/3}}{x^3\,\left (x^4+x^3+1\right )} \,d x \]
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