Optimal. Leaf size=27 \[ \frac {2 \left (3+4 x^2\right ) \left (-x^2+x^4\right )^{3/4}}{21 x^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039}
\begin {gather*} \frac {2 \left (x^4-x^2\right )^{3/4}}{7 x^5}+\frac {8 \left (x^4-x^2\right )^{3/4}}{21 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx &=\frac {2 \left (-x^2+x^4\right )^{3/4}}{7 x^5}+\frac {4}{7} \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx\\ &=\frac {2 \left (-x^2+x^4\right )^{3/4}}{7 x^5}+\frac {8 \left (-x^2+x^4\right )^{3/4}}{21 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2 \left (-1+x^2\right )\right )^{3/4} \left (3+4 x^2\right )}{21 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 24, normalized size = 0.89
method | result | size |
trager | \(\frac {2 \left (4 x^{2}+3\right ) \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{21 x^{5}}\) | \(24\) |
risch | \(\frac {-\frac {2}{21} x^{2}-\frac {2}{7}+\frac {8}{21} x^{4}}{x^{3} \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(29\) |
gosper | \(\frac {2 \left (1+x \right ) \left (-1+x \right ) \left (4 x^{2}+3\right )}{21 x^{3} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(30\) |
meijerg | \(-\frac {2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (1+\frac {4 x^{2}}{3}\right ) \left (-x^{2}+1\right )^{\frac {3}{4}}}{7 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {7}{2}}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 29, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (4 \, x^{5} - x^{3} - 3 \, x\right )}}{21 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 23, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (4 \, x^{2} + 3\right )}}{21 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 23, normalized size = 0.85 \begin {gather*} \frac {2}{7} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {7}{4}} - \frac {2}{3} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 35, normalized size = 1.30 \begin {gather*} \frac {8\,x^2\,{\left (x^4-x^2\right )}^{3/4}+6\,{\left (x^4-x^2\right )}^{3/4}}{21\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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