Integrand size = 39, antiderivative size = 128 \[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\frac {\left (1-4 x^3+x^6\right ) \sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {4}{3} \arctan \left (\frac {x^3}{\sqrt {2}+\sqrt {2} x^6+\sqrt {2+3 x^6+2 x^{12}}}\right )+\frac {\log (x)}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2}+\sqrt {2} x^6+\sqrt {2+3 x^6+2 x^{12}}\right )}{6 \sqrt {2}} \]
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\[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {2+3 x^6+2 x^{12}}}{x^7}+\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{x^4}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{x}+\frac {4 \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1+x^2\right )}-\frac {4 \left (1+x^2\right ) \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1-x^2+x^4\right )}\right ) \, dx \\ & = \frac {4}{3} \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{1+x^2} \, dx-\frac {4}{3} \int \frac {\left (1+x^2\right ) \sqrt {2+3 x^6+2 x^{12}}}{1-x^2+x^4} \, dx+2 \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{x^4} \, dx-\int \frac {\sqrt {2+3 x^6+2 x^{12}}}{x^7} \, dx+\int \frac {\sqrt {2+3 x^6+2 x^{12}}}{x} \, dx \\ & = -\left (\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {2+3 x+2 x^2}}{x^2} \, dx,x,x^6\right )\right )+\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {2+3 x+2 x^2}}{x} \, dx,x,x^6\right )+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {2+3 x^2+2 x^4}}{x^2} \, dx,x,x^3\right )+\frac {4}{3} \int \left (\frac {i \sqrt {2+3 x^6+2 x^{12}}}{2 (i-x)}+\frac {i \sqrt {2+3 x^6+2 x^{12}}}{2 (i+x)}\right ) \, dx-\frac {4}{3} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {2+3 x^6+2 x^{12}}}{-1-i \sqrt {3}+2 x^2}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {2+3 x^6+2 x^{12}}}{-1+i \sqrt {3}+2 x^2}\right ) \, dx \\ & = \frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx-\frac {1}{12} \text {Subst}\left (\int \frac {-4-3 x}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {1}{12} \text {Subst}\left (\int \frac {3+4 x}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )+\frac {2}{3} \text {Subst}\left (\int \frac {3+4 x^2}{\sqrt {2+3 x^2+2 x^4}} \, dx,x,x^3\right )-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{-1-i \sqrt {3}+2 x^2} \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{-1+i \sqrt {3}+2 x^2} \, dx \\ & = \frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {8}{3} \text {Subst}\left (\int \frac {1-x^2}{\sqrt {2+3 x^2+2 x^4}} \, dx,x,x^3\right )+\frac {14}{3} \text {Subst}\left (\int \frac {1}{\sqrt {2+3 x^2+2 x^4}} \, dx,x,x^3\right )-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right )}+\frac {\sqrt {1+i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right )}+\frac {\sqrt {1-i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx \\ & = \frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {4 x^3 \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1+x^6\right )}-\frac {4 \sqrt {2} \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} E\left (2 \arctan \left (x^3\right )|\frac {1}{8}\right )}{3 \sqrt {2+3 x^6+2 x^{12}}}+\frac {7 \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (x^3\right ),\frac {1}{8}\right )}{3 \sqrt {2} \sqrt {2+3 x^6+2 x^{12}}}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+3 x^6}{\sqrt {2+3 x^6+2 x^{12}}}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+3 x^6}{\sqrt {2+3 x^6+2 x^{12}}}\right )+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,3+4 x^6\right )}{4 \sqrt {14}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,3+4 x^6\right )}{3 \sqrt {14}}+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}} \\ & = \frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {4 x^3 \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1+x^6\right )}-\frac {\text {arcsinh}\left (\frac {3+4 x^6}{\sqrt {7}}\right )}{12 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {4+3 x^6}{2 \sqrt {2} \sqrt {2+3 x^6+2 x^{12}}}\right )}{12 \sqrt {2}}-\frac {4 \sqrt {2} \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} E\left (2 \arctan \left (x^3\right )|\frac {1}{8}\right )}{3 \sqrt {2+3 x^6+2 x^{12}}}+\frac {7 \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (x^3\right ),\frac {1}{8}\right )}{3 \sqrt {2} \sqrt {2+3 x^6+2 x^{12}}}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}} \\ \end{align*}
Time = 2.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\frac {1}{12} \left (\frac {2 \left (1-4 x^3+x^6\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^6}-16 \arctan \left (\frac {x^3}{\sqrt {2}+\sqrt {2} x^6+\sqrt {2+3 x^6+2 x^{12}}}\right )+3 \sqrt {2} \log (x)-\sqrt {2} \log \left (\sqrt {2}+\sqrt {2} x^6+\sqrt {2+3 x^6+2 x^{12}}\right )\right ) \]
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Time = 10.00 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {-\sqrt {2}\, \ln \left (\frac {\sqrt {2}\, x^{6}+\sqrt {\frac {2 x^{12}+3 x^{6}+2}{x^{2}}}\, x +\sqrt {2}}{x^{3}}\right ) x^{5}-8 \arctan \left (\frac {x^{2}}{\sqrt {\frac {2 x^{12}+3 x^{6}+2}{x^{2}}}}\right ) x^{5}+2 \sqrt {\frac {2 x^{12}+3 x^{6}+2}{x^{2}}}\, \left (x^{6}-4 x^{3}+1\right )}{12 x^{5}}\) | \(109\) |
trager | \(\frac {\left (x^{6}-4 x^{3}+1\right ) \sqrt {2 x^{12}+3 x^{6}+2}}{6 x^{6}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-\sqrt {2 x^{12}+3 x^{6}+2}}{x^{3}}\right )}{12}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-\sqrt {2 x^{12}+3 x^{6}+2}}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{3}\) | \(135\) |
risch | \(-\frac {8 x^{15}-2 x^{12}+12 x^{9}-3 x^{6}+8 x^{3}-2}{6 x^{6} \sqrt {2 x^{12}+3 x^{6}+2}}+\frac {\sqrt {2 x^{12}+3 x^{6}+2}}{6}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+\sqrt {2 x^{12}+3 x^{6}+2}}{x^{3}}\right )}{12}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-\sqrt {2 x^{12}+3 x^{6}+2}}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{3}\) | \(165\) |
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Time = 0.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\frac {\sqrt {2} x^{6} \log \left (-\frac {4 \, x^{12} + 7 \, x^{6} - 2 \, \sqrt {2} \sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{6} + 1\right )} + 4}{x^{6}}\right ) - 16 \, x^{6} \arctan \left (\frac {x^{3}}{\sqrt {2 \, x^{12} + 3 \, x^{6} + 2}}\right ) + 4 \, \sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{6} - 4 \, x^{3} + 1\right )}}{24 \, x^{6}} \]
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\[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\int \frac {\left (x - 1\right )^{3} \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )^{3} \sqrt {2 x^{12} + 3 x^{6} + 2}}{x^{7} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
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\[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\int { \frac {\sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{3}}{{\left (x^{6} + 1\right )} x^{7}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\int { \frac {\sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{3}}{{\left (x^{6} + 1\right )} x^{7}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^3\,\left (x^3+1\right )\,\sqrt {2\,x^{12}+3\,x^6+2}}{x^7\,\left (x^6+1\right )} \,d x \]
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