Integrand size = 18, antiderivative size = 16 \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \left (-1+x^3\right )^{3/4}}{3 x^3} \]
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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} \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \left (x^3-1\right )^{3/4}}{3 x^3} \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {-4+x}{\sqrt [4]{-1+x} x^2} \, dx,x,x^3\right ) \\ & = -\frac {4 \left (-1+x^3\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \left (-1+x^3\right )^{3/4}}{3 x^3} \]
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Time = 1.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(-\frac {4 \left (x^{3}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
risch | \(-\frac {4 \left (x^{3}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
pseudoelliptic | \(-\frac {4 \left (x^{3}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
gosper | \(-\frac {4 \left (x^{2}+x +1\right ) \left (x -1\right )}{3 x^{3} \left (x^{3}-1\right )^{\frac {1}{4}}}\) | \(22\) |
meijerg | \(\frac {2 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-\frac {5 \pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {9}{4}\right ], \left [2, 3\right ], x^{3}\right )}{32 \Gamma \left (\frac {3}{4}\right )}-\frac {\left (3-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right ) x^{3}}\right )}{3 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{3}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{6 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}}}\) | \(172\) |
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none
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 3.92 (sec) , antiderivative size = 68, normalized size of antiderivative = 4.25 \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{\frac {3}{4}} \Gamma \left (\frac {5}{4}\right )} + \frac {4 \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{\frac {15}{4}} \Gamma \left (\frac {9}{4}\right )} \]
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \, {\left (x^{3} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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Time = 4.98 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-4+x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx=-\frac {4\,{\left (x^3-1\right )}^{3/4}}{3\,x^3} \]
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