Optimal. Leaf size=16 \[ \frac {\left (x^3-1\right )^{4/3}}{4 x^4} \]
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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^3-1\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx &=\frac {\left (-1+x^3\right )^{4/3}}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (x^3-1\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{4/3}}{4 x^4} \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^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \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 {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{5}}\,{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 | \(\frac {\left (x^{3}-1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
gosper | \(\frac {\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{4 x^{4}}\) | \(22\) |
risch | \(\frac {x^{6}-2 x^{3}+1}{4 \left (x^{3}-1\right )^{\frac {2}{3}} x^{4}}\) | \(23\) |
meijerg | \(-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x^{4}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 25, normalized size = 1.56 \begin {gather*} -\frac {{\left (x^3-1\right )}^{1/3}-x^3\,{\left (x^3-1\right )}^{1/3}}{4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.69, size = 129, normalized size = 8.06 \begin {gather*} \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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