Integrand size = 18, antiderivative size = 97 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=\frac {\left (3-x^4\right ) \sqrt [3]{1+x^4}}{8 x^8}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (-1+\sqrt [3]{1+x^4}\right )-\frac {1}{12} \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {457, 79, 43, 59, 632, 210, 31} \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=-\frac {\arctan \left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{x^4+1}}{2 x^4}+\frac {1}{4} \log \left (1-\sqrt [3]{x^4+1}\right )+\frac {3 \left (x^4+1\right )^{4/3}}{8 x^8}-\frac {\log (x)}{3} \]
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Rule 31
Rule 43
Rule 59
Rule 79
Rule 210
Rule 457
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {(-3+x) \sqrt [3]{1+x}}{x^3} \, dx,x,x^4\right ) \\ & = \frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}+\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x^2} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}-\frac {\log (x)}{3}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^4}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^4}\right ) \\ & = -\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}-\frac {\log (x)}{3}+\frac {1}{4} \log \left (1-\sqrt [3]{1+x^4}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^4}\right ) \\ & = -\frac {\sqrt [3]{1+x^4}}{2 x^4}+\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}-\frac {\arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{4} \log \left (1-\sqrt [3]{1+x^4}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=\frac {1}{24} \left (-\frac {3 \left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^8}-4 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^4}}{\sqrt {3}}\right )+4 \log \left (-1+\sqrt [3]{1+x^4}\right )-2 \log \left (1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 4.47 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {x^{8}-2 x^{4}-3}{8 x^{8} \left (x^{4}+1\right )^{\frac {2}{3}}}+\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{4}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{6 \Gamma \left (\frac {2}{3}\right )}\) | \(69\) |
meijerg | \(\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}-\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}\) | \(115\) |
pseudoelliptic | \(\frac {-4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{8}+4 \ln \left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right ) x^{8}-2 \ln \left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right ) x^{8}-3 \left (x^{4}+1\right )^{\frac {1}{3}} x^{4}+9 \left (x^{4}+1\right )^{\frac {1}{3}}}{24 {\left (-1+\left (x^{4}+1\right )^{\frac {1}{3}}\right )}^{2} {\left (1+\left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}\right )}^{2}}\) | \(116\) |
trager | \(-\frac {\left (x^{4}-3\right ) \left (x^{4}+1\right )^{\frac {1}{3}}}{8 x^{8}}+\frac {\ln \left (\frac {-333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-393 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-60 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}-48 \left (x^{4}+1\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+333 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+165 \left (x^{4}+1\right )^{\frac {1}{3}}-384 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-80}{x^{4}}\right )}{6}+\frac {\ln \left (-\frac {-153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+162 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-40 x^{4}+351 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}}+165 \left (x^{4}+1\right )^{\frac {2}{3}}-495 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}}+153 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-48 \left (x^{4}+1\right )^{\frac {1}{3}}+195 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-100}{x^{4}}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2}\) | \(299\) |
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Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=-\frac {4 \, \sqrt {3} x^{8} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 2 \, x^{8} \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, x^{8} \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{24 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 164.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=- \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {8}{3}} \Gamma \left (\frac {5}{3}\right )} + \frac {3 \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {20}{3}} \Gamma \left (\frac {8}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{8 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} - \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} - \frac {1}{12} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} - 4 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{8 \, x^{8}} - \frac {1}{12} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 6.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^9} \, dx=\frac {\ln \left (\frac {{\left (x^4+1\right )}^{1/3}}{16}-\frac {1}{16}\right )}{6}-\frac {\frac {{\left (x^4+1\right )}^{1/3}}{4}+\frac {{\left (x^4+1\right )}^{4/3}}{8}}{2\,x^4-{\left (x^4+1\right )}^2+1}-\frac {{\left (x^4+1\right )}^{1/3}}{4\,x^4}+\ln \left (\frac {3\,{\left (x^4+1\right )}^{1/3}}{4}+\frac {3}{8}-\frac {\sqrt {3}\,3{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (\frac {3\,{\left (x^4+1\right )}^{1/3}}{4}+\frac {3}{8}+\frac {\sqrt {3}\,3{}\mathrm {i}}{8}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]
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