Integrand size = 29, antiderivative size = 124 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{1+a^3 x^3+x^6}}\right )}{\sqrt {3} a}+\frac {\log \left (-a x+\sqrt [3]{1+a^3 x^3+x^6}\right )}{3 a}-\frac {\log \left (a^2 x^2+a x \sqrt [3]{1+a^3 x^3+x^6}+\left (1+a^3 x^3+x^6\right )^{2/3}\right )}{6 a} \]
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\[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=\int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {2}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx\right )+\int \frac {1}{\sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ & = -\left (2 \int \left (\frac {1}{3 \left (1+x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {2-x^2}{3 \left (1-x^2+x^4\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx\right )+\frac {\left (\sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}}} \, dx}{\sqrt [3]{1+a^3 x^3+x^6}} \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {2}{3} \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {2}{3} \int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {2}{3} \int \left (\frac {i}{2 (i-x) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {i}{2 (i+x) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx-\frac {2}{3} \int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {1}{3} i \int \frac {1}{(i-x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} i \int \frac {1}{(i+x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {1}{3} i \int \frac {1}{(i-x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} i \int \frac {1}{(i+x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \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+a^3 x^3+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+a^3 x^3+x^6}}\right ) \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \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+a^3 x^3+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+a^3 x^3+x^6}}\right ) \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {1}{3} i \int \frac {1}{(i-x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} i \int \frac {1}{(i+x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{1+a^3 x^3+x^6}}\right )-2 \log \left (a \left (a x-\sqrt [3]{1+a^3 x^3+x^6}\right )\right )+\log \left (a^2 x^2+a x \sqrt [3]{1+a^3 x^3+x^6}+\left (1+a^3 x^3+x^6\right )^{2/3}\right )}{6 a} \]
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Time = 0.72 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (a x +2 \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}\right )}{3 a x}\right )+2 \ln \left (\frac {-a x +\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {a^{2} x^{2}+a x \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}+\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6 a}\) | \(116\) |
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Time = 2.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.19 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} a^{2} x^{2} - 2 \, \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {2}{3}} a x + \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}}{9 \, a^{3} x^{3} + x^{6} + 1}\right ) - \log \left (\frac {x^{6} + 3 \, {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} a^{2} x^{2} - 3 \, {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {2}{3}} a x + 1}{x^{6} + 1}\right )}{6 \, a} \]
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\[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right ) \sqrt [3]{a^{3} x^{3} + x^{6} + 1}}\, dx \]
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\[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=\int { \frac {x^{6} - 1}{{\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 1\right )}} \,d x } \]
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\[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=\int { \frac {x^{6} - 1}{{\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=\int \frac {x^6-1}{\left (x^6+1\right )\,{\left (a^3\,x^3+x^6+1\right )}^{1/3}} \,d x \]
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