Integrand size = 24, antiderivative size = 208 \[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{-x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}}\right )}{2\ 2^{2/3}}+\frac {\log \left (-2 x^2+2^{2/3} \sqrt {3} x \sqrt [3]{-x^2+x^4}-\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}\right )}{4\ 2^{2/3}}-\frac {\log \left (2 x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}\right )}{2\ 2^{2/3}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt {3} x \sqrt [3]{-x^2+x^4}+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}\right )}{4\ 2^{2/3}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.40 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.78, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1325, 2038, 636, 633, 242, 225, 2059, 772, 138} \[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=\frac {3 \left (\frac {x^2}{x^2+1}\right )^{2/3} \left (-\frac {1-x^2}{x^2+1}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},\frac {2}{3},\frac {7}{3},\frac {2}{x^2+1},\frac {1}{x^2+1}\right )}{4 \left (x^4-x^2\right )^{2/3}}-\frac {3^{3/4} \sqrt {2-\sqrt {3}} \left (x^2-x^4\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{x^2-x^4}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (x^2-x^4\right )^{2/3}+2^{2/3} \sqrt [3]{x^2-x^4}+1}{\left (-2^{2/3} \sqrt [3]{x^2-x^4}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-2^{2/3} \sqrt [3]{x^2-x^4}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{x^2-x^4}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{2^{2/3} \left (1-2 x^2\right ) \left (x^4-x^2\right )^{2/3} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{x^2-x^4}}{\left (-2^{2/3} \sqrt [3]{x^2-x^4}-\sqrt {3}+1\right )^2}}} \]
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Rule 138
Rule 225
Rule 242
Rule 633
Rule 636
Rule 772
Rule 1325
Rule 2038
Rule 2059
Rubi steps \begin{align*} \text {integral}& = -\left (2 \int \frac {x}{\left (1+x^2\right ) \left (-x^2+x^4\right )^{2/3}} \, dx\right )+\int \frac {x}{\left (-x^2+x^4\right )^{2/3}} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (-x+x^2\right )^{2/3}} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {1}{(1+x) \left (-x+x^2\right )^{2/3}} \, dx,x,x^2\right ) \\ & = \frac {\left (\left (\frac {x^2}{1+x^2}\right )^{2/3} \left (\frac {-1+x^2}{1+x^2}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{x}}{(1-2 x)^{2/3} (1-x)^{2/3}} \, dx,x,\frac {1}{1+x^2}\right )}{\left (\frac {1}{1+x^2}\right )^{4/3} \left (-x^2+x^4\right )^{2/3}}+\frac {\left (x^2-x^4\right )^{2/3} \text {Subst}\left (\int \frac {1}{\left (x-x^2\right )^{2/3}} \, dx,x,x^2\right )}{2 \left (-x^2+x^4\right )^{2/3}} \\ & = \frac {3 \left (\frac {x^2}{1+x^2}\right )^{2/3} \left (-\frac {1-x^2}{1+x^2}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},\frac {2}{3},\frac {7}{3},\frac {2}{1+x^2},\frac {1}{1+x^2}\right )}{4 \left (-x^2+x^4\right )^{2/3}}-\frac {\left (x^2-x^4\right )^{2/3} \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{2/3}} \, dx,x,1-2 x^2\right )}{2^{2/3} \left (-x^2+x^4\right )^{2/3}} \\ & = \frac {3 \left (\frac {x^2}{1+x^2}\right )^{2/3} \left (-\frac {1-x^2}{1+x^2}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},\frac {2}{3},\frac {7}{3},\frac {2}{1+x^2},\frac {1}{1+x^2}\right )}{4 \left (-x^2+x^4\right )^{2/3}}+\frac {\left (3 \sqrt {-\left (1-2 x^2\right )^2} \left (x^2-x^4\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,2^{2/3} \sqrt [3]{-x^2 \left (-1+x^2\right )}\right )}{2\ 2^{2/3} \left (1-2 x^2\right ) \left (-x^2+x^4\right )^{2/3}} \\ & = \frac {3 \left (\frac {x^2}{1+x^2}\right )^{2/3} \left (-\frac {1-x^2}{1+x^2}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},\frac {2}{3},\frac {7}{3},\frac {2}{1+x^2},\frac {1}{1+x^2}\right )}{4 \left (-x^2+x^4\right )^{2/3}}-\frac {3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {-\left (1-2 x^2\right )^2} \left (x^2-x^4\right )^{2/3} \left (1-2^{2/3} \sqrt [3]{x^2 \left (1-x^2\right )}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{x^2 \left (1-x^2\right )}+2 \sqrt [3]{2} \left (x^2 \left (1-x^2\right )\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{x^2 \left (1-x^2\right )}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{x^2 \left (1-x^2\right )}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{x^2 \left (1-x^2\right )}}\right ),-7+4 \sqrt {3}\right )}{2^{2/3} \left (1-2 x^2\right ) \left (-x^2+x^4\right )^{2/3} \sqrt {-1+4 x^2 \left (1-x^2\right )} \sqrt {-\frac {1-2^{2/3} \sqrt [3]{x^2 \left (1-x^2\right )}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{x^2 \left (1-x^2\right )}\right )^2}}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=\frac {x^{4/3} \left (-1+x^2\right )^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )+\log \left (-2 x^{2/3}+2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2}-\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )-2 \log \left (2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )+\log \left (2 x^{2/3}+2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )\right )}{4\ 2^{2/3} \left (x^2 \left (-1+x^2\right )\right )^{2/3}} \]
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Time = 11.69 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {\left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {1}{3}} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+x^{2}\right )}{3 x^{2}}\right )+2 \ln \left (\frac {2^{\frac {2}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {-2^{\frac {2}{3}} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+2 \,2^{\frac {1}{3}} x^{2}+x^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{3}}-\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x^{2}}\right )\right ) 2^{\frac {1}{3}}}{8}\) | \(127\) |
trager | \(\text {Expression too large to display}\) | \(1702\) |
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Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (160) = 320\).
Time = 1.24 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=-\frac {1}{12} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{10} - 33 \, x^{8} + 110 \, x^{6} - 110 \, x^{4} + 33 \, x^{2} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} - 48 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 2 \, x^{6} - 6 \, x^{4} - 2 \, x^{2} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{12} + 42 \, x^{10} - 417 \, x^{8} + 812 \, x^{6} - 417 \, x^{4} + 42 \, x^{2} + 1\right )}\right )}}{6 \, {\left (x^{12} - 102 \, x^{10} + 447 \, x^{8} - 628 \, x^{6} + 447 \, x^{4} - 102 \, x^{2} + 1\right )}}\right ) - \frac {1}{48} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {24 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{2} + 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 32 \, x^{6} + 78 \, x^{4} - 32 \, x^{2} + 1\right )} - 12 \, {\left (x^{6} - 11 \, x^{4} + 11 \, x^{2} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}{x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 2 \, x^{2} + 1\right )} - 12 \, {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}}}{x^{4} + 2 \, x^{2} + 1}\right ) \]
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\[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}{x \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}{{\left (x^{2} + 1\right )} x} \,d x } \]
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\[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}{{\left (x^{2} + 1\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{-x^2+x^4}}{x \left (1+x^2\right )} \, dx=\int \frac {{\left (x^4-x^2\right )}^{1/3}}{x\,\left (x^2+1\right )} \,d x \]
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