Optimal. Leaf size=16 \[ -\frac {1}{4 x^4}-\frac {1}{4} \tan ^{-1}\left (x^4\right ) \]
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Rubi [A]
time = 0.00, 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 = {281, 331, 209}
\begin {gather*} -\frac {1}{4} \text {ArcTan}\left (x^4\right )-\frac {1}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 281
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (1+x^8\right )} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\frac {1}{4} \tan ^{-1}\left (x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{4 x^4}+\frac {1}{4} \tan ^{-1}\left (\frac {1}{x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 13, normalized size = 0.81
method | result | size |
default | \(-\frac {1}{4 x^{4}}-\frac {\arctan \left (x^{4}\right )}{4}\) | \(13\) |
meijerg | \(-\frac {1}{4 x^{4}}-\frac {\arctan \left (x^{4}\right )}{4}\) | \(13\) |
risch | \(-\frac {1}{4 x^{4}}-\frac {\arctan \left (x^{4}\right )}{4}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{4 \, x^{4}} - \frac {1}{4} \, \arctan \left (x^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 15, normalized size = 0.94 \begin {gather*} -\frac {x^{4} \arctan \left (x^{4}\right ) + 1}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.88 \begin {gather*} - \frac {\operatorname {atan}{\left (x^{4} \right )}}{4} - \frac {1}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.75, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{4 \, x^{4}} - \frac {1}{4} \, \arctan \left (x^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\mathrm {atan}\left (x^4\right )}{4}-\frac {1}{4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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