Integrand size = 24, antiderivative size = 154 \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {\left (x^2+x^4\right )^{2/3}}{x \left (1+x^2\right )}+\frac {\arctan \left (\frac {3^{2/3} x \sqrt [3]{x^2+x^4}}{\sqrt [3]{3} x^2-\left (x^2+x^4\right )^{2/3}}\right )}{3^{2/3}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [6]{3} x}{\sqrt [3]{x^2+x^4}}\right )}{3 \sqrt [6]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{3} x^2+\frac {\left (x^2+x^4\right )^{2/3}}{\sqrt [6]{3}}}{x \sqrt [3]{x^2+x^4}}\right )}{3 \sqrt [6]{3}} \]
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\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {-1+x^6}{x^{2/3} \sqrt [3]{1+x^2} \left (1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {-1+x^{18}}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}-\frac {2}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{9 \left (1+x^2\right ) \sqrt [3]{1+x^6}}+\frac {2-x^2}{9 \left (1-x^2+x^4\right ) \sqrt [3]{1+x^6}}+\frac {2-x^6}{3 \sqrt [3]{1+x^6} \left (1-x^6+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-x^6}{\sqrt [3]{1+x^6} \left (1-x^6+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {i}{2 (i-x) \sqrt [3]{1+x^6}}+\frac {i}{2 (i+x) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\sqrt [3]{1+x^6} \left (-1-i \sqrt {3}+2 x^6\right )}+\frac {-1+i \sqrt {3}}{\sqrt [3]{1+x^6} \left (-1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (-1-i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (-1+i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = -\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}}+\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}}+\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}} \\ & = -\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1-i \sqrt {3}} \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1-i \sqrt {3}} \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1+i \sqrt {3}} \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1+i \sqrt {3}} \sqrt [3]{x^2+x^4}} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.22 \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {x^{2/3} \left (9 \sqrt [3]{x}-3 \sqrt [3]{3} \sqrt [3]{1+x^2} \arctan \left (\frac {3^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{3} x^{2/3}-\left (1+x^2\right )^{2/3}}\right )+2\ 3^{5/6} \sqrt [3]{1+x^2} \text {arctanh}\left (\frac {\sqrt [6]{3} \sqrt [3]{x}}{\sqrt [3]{1+x^2}}\right )+3^{5/6} \sqrt [3]{1+x^2} \text {arctanh}\left (\frac {3^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}{3 x^{2/3}+3^{2/3} \left (1+x^2\right )^{2/3}}\right )\right )}{9 \sqrt [3]{x^2+x^4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(124)=248\).
Time = 46.41 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.92
method | result | size |
pseudoelliptic | \(\frac {3^{\frac {5}{6}} \ln \left (\frac {-3^{\frac {1}{6}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-2 \,3^{\frac {5}{6}} \ln \left (\frac {3^{\frac {1}{6}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+2 \,3^{\frac {5}{6}} \ln \left (\frac {-3^{\frac {1}{6}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-3^{\frac {5}{6}} \ln \left (\frac {3^{\frac {1}{6}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-6 \,3^{\frac {1}{3}} \arctan \left (\frac {x \sqrt {3}-2 \,3^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{3 x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+6 \,3^{\frac {1}{3}} \arctan \left (\frac {x \sqrt {3}+2 \,3^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{3 x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-18 x}{18 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}\) | \(295\) |
trager | \(\text {Expression too large to display}\) | \(6673\) |
risch | \(\text {Expression too large to display}\) | \(7488\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1438 vs. \(2 (125) = 250\).
Time = 2.03 (sec) , antiderivative size = 1438, normalized size of antiderivative = 9.34 \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
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\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {x^6-1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6+1\right )} \,d x \]
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