Integrand size = 18, antiderivative size = 16 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=-\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 75} \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=-\frac {3 \left (x^4+1\right )^{4/3}}{8 x^8} \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [3]{1+x} (3+x)}{x^3} \, dx,x,x^4\right ) \\ & = -\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=-\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8} \]
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Time = 0.99 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(-\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{8 x^{8}}\) | \(13\) |
trager | \(-\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{8 x^{8}}\) | \(13\) |
pseudoelliptic | \(-\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{8 x^{8}}\) | \(13\) |
risch | \(-\frac {3 \left (x^{8}+2 x^{4}+1\right )}{8 \left (x^{4}+1\right )^{\frac {2}{3}} x^{8}}\) | \(23\) |
meijerg | \(-\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}-\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}\) | \(115\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=-\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 151.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.38 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=- \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {8}{3}} \Gamma \left (\frac {5}{3}\right )} - \frac {3 \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {20}{3}} \Gamma \left (\frac {8}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.12 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=\frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{8 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} - \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=-\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \]
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Time = 5.16 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx=-\frac {3\,{\left (x^4+1\right )}^{4/3}}{8\,x^8} \]
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