Optimal. Leaf size=16 \[ \frac {\left (x^4-1\right )^{3/4}}{3 x^3} \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} \frac {\left (x^4-1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{-1+x^4}} \, dx &=\frac {\left (-1+x^4\right )^{3/4}}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (x^4-1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^4\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 13, normalized size = 0.81
method | result | size |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {\left (x^{4}-1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
gosper | \(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{3 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(24\) |
meijerg | \(-\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \left (-x^{4}+1\right )^{\frac {3}{4}}}{3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{3}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^4-1\right )}^{3/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 63, normalized size = 3.94 \begin {gather*} \begin {cases} - \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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