Integrand size = 44, antiderivative size = 167 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3} \left (-1-5 x^3+x^4\right )}{5 x^5}+3 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x-2 \sqrt [3]{2} \sqrt [3]{-1+x^4}}\right )-\sqrt [3]{2} 3^{2/3} \log \left (3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^4}\right )+\left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2-\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^4}+2^{2/3} \sqrt [3]{3} \left (-1+x^4\right )^{2/3}\right ) \]
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\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \left (-1+x^4\right )^{2/3}}{x^6}+\frac {6 \left (-1+x^4\right )^{2/3}}{x^3}+\frac {\left (-1+x^4\right )^{2/3}}{x^2}-\frac {2 (9+8 x) \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}\right ) \, dx \\ & = -\left (2 \int \frac {(9+8 x) \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx\right )+3 \int \frac {\left (-1+x^4\right )^{2/3}}{x^6} \, dx+6 \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1+x^4\right )^{2/3}}{x^2} \, dx \\ & = -\left (2 \int \left (\frac {9 \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}+\frac {8 x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}\right ) \, dx\right )+3 \text {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )+\frac {\left (-1+x^4\right )^{2/3} \int \frac {\left (1-x^4\right )^{2/3}}{x^2} \, dx}{\left (1-x^4\right )^{2/3}}+\frac {\left (3 \left (-1+x^4\right )^{2/3}\right ) \int \frac {\left (1-x^4\right )^{2/3}}{x^6} \, dx}{\left (1-x^4\right )^{2/3}} \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}+4 \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx+\frac {\left (6 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2} \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx+\frac {\left (6 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}+\frac {\left (6 \left (-1+\sqrt {3}\right ) \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2} \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}+\frac {12 x^2}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}-\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}+\frac {4 \sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right ),-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx \\ \end{align*}
Time = 2.60 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3} \left (-1-5 x^3+x^4\right )}{5 x^5}+3 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x-2 \sqrt [3]{2} \sqrt [3]{-1+x^4}}\right )-\sqrt [3]{2} 3^{2/3} \log \left (3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^4}\right )+\left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2-\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^4}+2^{2/3} \sqrt [3]{3} \left (-1+x^4\right )^{2/3}\right ) \]
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Time = 101.72 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {-5 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right )+3 \left (x^{4}-5 x^{3}-1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-15 \,2^{\frac {1}{3}} x^{5} \left (\frac {\left (-\ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{4}-1\right )^{\frac {1}{3}} x +2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2 \left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (2\right )\right ) 3^{\frac {2}{3}}}{6}+\arctan \left (\frac {\left (-\frac {2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{4}-1\right )^{\frac {1}{3}}}{3}+x \right ) \sqrt {3}}{3 x}\right ) 3^{\frac {1}{6}}\right )}{5 x^{5}}\) | \(152\) |
risch | \(\text {Expression too large to display}\) | \(808\) |
trager | \(\text {Expression too large to display}\) | \(1469\) |
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Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (129) = 258\).
Time = 83.49 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.50 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\frac {10 \, \sqrt {3} \left (-18\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {4 \, \sqrt {3} \left (-18\right )^{\frac {2}{3}} {\left (2 \, x^{9} - 3 \, x^{8} - 9 \, x^{7} - 4 \, x^{5} + 3 \, x^{4} + 2 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-18\right )^{\frac {1}{3}} {\left (4 \, x^{10} - 42 \, x^{9} + 9 \, x^{8} - 8 \, x^{6} + 42 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (8 \, x^{12} - 180 \, x^{11} + 216 \, x^{10} + 27 \, x^{9} - 24 \, x^{8} + 360 \, x^{7} - 216 \, x^{6} + 24 \, x^{4} - 180 \, x^{3} - 8\right )}}{3 \, {\left (8 \, x^{12} + 36 \, x^{11} - 432 \, x^{10} + 27 \, x^{9} - 24 \, x^{8} - 72 \, x^{7} + 432 \, x^{6} + 24 \, x^{4} + 36 \, x^{3} - 8\right )}}\right ) + 10 \, \left (-18\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-18\right )^{\frac {2}{3}} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} + 18 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \left (-18\right )^{\frac {1}{3}} {\left (2 \, x^{4} + 3 \, x^{3} - 2\right )}}{2 \, x^{4} + 3 \, x^{3} - 2}\right ) - 5 \, \left (-18\right )^{\frac {1}{3}} x^{5} \log \left (\frac {36 \, \left (-18\right )^{\frac {1}{3}} {\left (x^{5} - 3 \, x^{4} - x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} + \left (-18\right )^{\frac {2}{3}} {\left (4 \, x^{8} - 42 \, x^{7} + 9 \, x^{6} - 8 \, x^{4} + 42 \, x^{3} + 4\right )} + 54 \, {\left (4 \, x^{6} - 3 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{4 \, x^{8} + 12 \, x^{7} + 9 \, x^{6} - 8 \, x^{4} - 12 \, x^{3} + 4}\right ) + 18 \, {\left (x^{4} - 5 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]
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Timed out. \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} - 2\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{4} + 3 \, x^{3} - 2\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} - 2\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{4} + 3 \, x^{3} - 2\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx=\int -\frac {{\left (x^4-1\right )}^{2/3}\,\left (x^4+3\right )\,\left (-2\,x^4+x^3+2\right )}{x^6\,\left (2\,x^4+3\,x^3-2\right )} \,d x \]
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