Optimal. Leaf size=16 \[ -\frac {1}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (x^3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {275, 325, 203} \[ -\frac {1}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (x^3\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 275
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1+x^6\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac {1}{3 x^3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^3\right )\\ &=-\frac {1}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {1}{3} \tan ^{-1}\left (\frac {1}{x^3}\right )-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 15, normalized size = 0.94 \[ -\frac {x^{3} \arctan \left (x^{3}\right ) + 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 12, normalized size = 0.75 \[ -\frac {1}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 33, normalized size = 2.06 \[ \frac {\arctan \relax (x )}{3}-\frac {\arctan \left (2 x -\sqrt {3}\right )}{3}-\frac {\arctan \left (2 x +\sqrt {3}\right )}{3}-\frac {1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 12, normalized size = 0.75 \[ -\frac {1}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 12, normalized size = 0.75 \[ -\frac {\mathrm {atan}\left (x^3\right )}{3}-\frac {1}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 14, normalized size = 0.88 \[ - \frac {\operatorname {atan}{\left (x^{3} \right )}}{3} - \frac {1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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