Integrand size = 20, antiderivative size = 18 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (-x+x^4\right )^{7/4}}{21 x^7} \]
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Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2077, 2041, 2039} \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x^4-x\right )^{3/4}}{21 x^3}-\frac {4 \left (x^4-x\right )^{3/4}}{21 x^6} \]
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Rule 2039
Rule 2041
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^6 \sqrt [4]{-x+x^4}}+\frac {1}{x^3 \sqrt [4]{-x+x^4}}\right ) \, dx \\ & = -\int \frac {1}{x^6 \sqrt [4]{-x+x^4}} \, dx+\int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx \\ & = -\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {4 \left (-x+x^4\right )^{3/4}}{9 x^3}-\frac {4}{7} \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx \\ & = -\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^3} \\ \end{align*}
Time = 9.54 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x \left (-1+x^3\right )\right )^{7/4}}{21 x^7} \]
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Time = 0.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
trager | \(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(20\) |
pseudoelliptic | \(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(20\) |
risch | \(\frac {\frac {4}{21} x^{6}-\frac {8}{21} x^{3}+\frac {4}{21}}{x^{5} {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(25\) |
gosper | \(\frac {4 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (x^{3}-1\right )}{21 x^{5} \left (x^{4}-x \right )^{\frac {1}{4}}}\) | \(29\) |
meijerg | \(\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (1+\frac {4 x^{3}}{3}\right ) \left (-x^{3}+1\right )^{\frac {3}{4}}}{21 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {3}{4}}}{9 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\) | \(73\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )}}{21 \, x^{6}} \]
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\[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (14) = 28\).
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}}{9 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {13}{4}}} - \frac {4 \, {\left (4 \, x^{7} - x^{4} - 3 \, x\right )}}{63 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {25}{4}}} \]
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Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} \]
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Time = 5.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=-\frac {4\,{\left (x^4-x\right )}^{3/4}-4\,x^3\,{\left (x^4-x\right )}^{3/4}}{21\,x^6} \]
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