Integrand size = 39, antiderivative size = 112 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=\frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {1}{16} \text {RootSum}\left [5-10 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 615, normalized size of antiderivative = 5.49, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6860, 270, 283, 245, 399, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=-\frac {\left (\sqrt {15}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{16 \sqrt {5}}+\frac {\left (-\sqrt {15}+i\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{16 \sqrt {5}}+\frac {1}{8} \sqrt {3} \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {\sqrt [3]{3} \left (-\sqrt {15}+i\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}-3 i} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{16 \sqrt [6]{5} \left (\sqrt {15}-3 i\right )^{2/3}}+\frac {\sqrt [3]{3} \left (\sqrt {15}+i\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}+3 i} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{16 \sqrt [6]{5} \left (\sqrt {15}+3 i\right )^{2/3}}+\frac {\left (\sqrt {15}+i\right ) \log \left (6 x^3+2 \left (3-i \sqrt {15}\right )\right )}{32 \sqrt [6]{15} \left (\sqrt {15}+3 i\right )^{2/3}}-\frac {\left (-\sqrt {15}+i\right ) \log \left (6 x^3+2 \left (3+i \sqrt {15}\right )\right )}{32 \sqrt [6]{15} \left (\sqrt {15}-3 i\right )^{2/3}}+\frac {3^{5/6} \left (-\sqrt {15}+i\right ) \log \left (-\sqrt [3]{x^3+1}+\frac {\sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}-3 i}}\right )}{32 \sqrt [6]{5} \left (\sqrt {15}-3 i\right )^{2/3}}-\frac {3^{5/6} \left (\sqrt {15}+i\right ) \log \left (-\sqrt [3]{x^3+1}+\frac {\sqrt [6]{15} x}{\sqrt [3]{\sqrt {15}+3 i}}\right )}{32 \sqrt [6]{5} \left (\sqrt {15}+3 i\right )^{2/3}}+\frac {1}{160} \left (15+i \sqrt {15}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{160} \left (15-i \sqrt {15}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {3}{16} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{16 x^2} \]
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Rule 245
Rule 270
Rule 283
Rule 384
Rule 399
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{8 x^3}-\frac {3 \left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8 \left (8+6 x^3+3 x^6\right )}\right ) \, dx \\ & = \frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8+6 x^3+3 x^6} \, dx+\frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {3}{8} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {3}{8} \int \left (\frac {\left (3+i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 x^3}+\frac {\left (3-i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3}\right ) \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{8} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{40} \left (3 \left (15-i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3} \, dx-\frac {1}{40} \left (3 \left (15+i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 x^3} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{8} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{8} \left (3 \left (1-i \sqrt {15}\right )\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (6-2 i \sqrt {15}+6 x^3\right )} \, dx-\frac {1}{80} \left (15-i \sqrt {15}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\frac {1}{8} \left (3 \left (1+i \sqrt {15}\right )\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (6+2 i \sqrt {15}+6 x^3\right )} \, dx-\frac {1}{80} \left (15+i \sqrt {15}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{8} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {\left (i-\sqrt {15}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt {5}}-\frac {\left (i+\sqrt {15}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt {5}}-\frac {\sqrt [3]{3} \left (i-\sqrt {15}\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{15} x}{\sqrt [3]{-3 i+\sqrt {15}} \sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt [6]{5} \left (-3 i+\sqrt {15}\right )^{2/3}}+\frac {\sqrt [3]{3} \left (i+\sqrt {15}\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{15} x}{\sqrt [3]{3 i+\sqrt {15}} \sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt [6]{5} \left (3 i+\sqrt {15}\right )^{2/3}}+\frac {\left (i+\sqrt {15}\right ) \log \left (2 \left (3-i \sqrt {15}\right )+6 x^3\right )}{32 \sqrt [6]{15} \left (3 i+\sqrt {15}\right )^{2/3}}-\frac {\left (i-\sqrt {15}\right ) \log \left (2 \left (3+i \sqrt {15}\right )+6 x^3\right )}{32 \sqrt [6]{15} \left (-3 i+\sqrt {15}\right )^{2/3}}+\frac {3^{5/6} \left (i-\sqrt {15}\right ) \log \left (\frac {\sqrt [6]{15} x}{\sqrt [3]{-3 i+\sqrt {15}}}-\sqrt [3]{1+x^3}\right )}{32 \sqrt [6]{5} \left (-3 i+\sqrt {15}\right )^{2/3}}-\frac {3^{5/6} \left (i+\sqrt {15}\right ) \log \left (\frac {\sqrt [6]{15} x}{\sqrt [3]{3 i+\sqrt {15}}}-\sqrt [3]{1+x^3}\right )}{32 \sqrt [6]{5} \left (3 i+\sqrt {15}\right )^{2/3}}-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{160} \left (15-i \sqrt {15}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{160} \left (15+i \sqrt {15}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=\frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {1}{16} \text {RootSum}\left [5-10 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \]
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Time = 152.86 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{6}-10 \textit {\_Z}^{3}+5\right )}{\sum }\frac {\left (6 \textit {\_R}^{3}-5\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{8 \textit {\_R}^{4}-5 \textit {\_R}}\right ) x^{5}-23 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-8 \left (x^{3}+1\right )^{\frac {2}{3}}}{80 x^{5}}\) | \(85\) |
risch | \(\text {Expression too large to display}\) | \(9297\) |
trager | \(\text {Expression too large to display}\) | \(10408\) |
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Exception generated. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.35 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=\int { \frac {{\left (3 \, x^{6} + 6 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{6} + 6 \, x^{3} + 8\right )} x^{6}} \,d x } \]
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Exception generated. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 5.94 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.35 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (3\,x^6+6\,x^3+4\right )}{x^6\,\left (3\,x^6+6\,x^3+8\right )} \,d x \]
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