Optimal. Leaf size=16 \[ -\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 75}
\begin {gather*} -\frac {3 \left (x^4+1\right )^{4/3}}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 457
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [3]{1+x} (3+x)}{x^3} \, dx,x,x^4\right )\\ &=-\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 13, normalized size = 0.81
method | result | size |
gosper | \(-\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{8 x^{8}}\) | \(13\) |
trager | \(-\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{8 x^{8}}\) | \(13\) |
risch | \(-\frac {3 \left (x^{8}+2 x^{4}+1\right )}{8 \left (x^{4}+1\right )^{\frac {2}{3}} x^{8}}\) | \(23\) |
meijerg | \(-\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}}{12 \Gamma \left (\frac {2}{3}\right )}-\frac {-\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}}{4 \Gamma \left (\frac {2}{3}\right )}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs.
\(2 (12) = 24\).
time = 0.47, size = 50, normalized size = 3.12 \begin {gather*} \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{8 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} - \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3\,{\left (x^4+1\right )}^{4/3}}{8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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