Optimal. Leaf size=37 \[ -\frac {5}{12 x^6}+\frac {5}{4 x^2}+\frac {1}{4 x^6 \left (1+x^4\right )}+\frac {5}{4} \tan ^{-1}\left (x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {28, 281, 296,
331, 209} \begin {gather*} \frac {5 \text {ArcTan}\left (x^2\right )}{4}-\frac {5}{12 x^6}+\frac {5}{4 x^2}+\frac {1}{4 x^6 \left (x^4+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 209
Rule 281
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (1+2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^7 \left (1+x^4\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 \left (1+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{4 x^6 \left (1+x^4\right )}+\frac {5}{4} \text {Subst}\left (\int \frac {1}{x^4 \left (1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {5}{12 x^6}+\frac {1}{4 x^6 \left (1+x^4\right )}-\frac {5}{4} \text {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {5}{12 x^6}+\frac {5}{4 x^2}+\frac {1}{4 x^6 \left (1+x^4\right )}+\frac {5}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^2\right )\\ &=-\frac {5}{12 x^6}+\frac {5}{4 x^2}+\frac {1}{4 x^6 \left (1+x^4\right )}+\frac {5}{4} \tan ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.89 \begin {gather*} -\frac {1}{6 x^6}+\frac {1}{x^2}+\frac {x^2}{4 \left (1+x^4\right )}-\frac {5}{4} \tan ^{-1}\left (\frac {1}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 28, normalized size = 0.76
method | result | size |
default | \(\frac {x^{2}}{4 x^{4}+4}+\frac {5 \arctan \left (x^{2}\right )}{4}-\frac {1}{6 x^{6}}+\frac {1}{x^{2}}\) | \(28\) |
risch | \(\frac {\frac {5}{4} x^{8}+\frac {5}{6} x^{4}-\frac {1}{6}}{x^{6} \left (x^{4}+1\right )}+\frac {5 \arctan \left (x^{2}\right )}{4}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 30, normalized size = 0.81 \begin {gather*} \frac {15 \, x^{8} + 10 \, x^{4} - 2}{12 \, {\left (x^{10} + x^{6}\right )}} + \frac {5}{4} \, \arctan \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 36, normalized size = 0.97 \begin {gather*} \frac {15 \, x^{8} + 10 \, x^{4} + 15 \, {\left (x^{10} + x^{6}\right )} \arctan \left (x^{2}\right ) - 2}{12 \, {\left (x^{10} + x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 0.78 \begin {gather*} \frac {5 \operatorname {atan}{\left (x^{2} \right )}}{4} + \frac {15 x^{8} + 10 x^{4} - 2}{12 x^{10} + 12 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.48, size = 31, normalized size = 0.84 \begin {gather*} \frac {x^{2}}{4 \, {\left (x^{4} + 1\right )}} + \frac {6 \, x^{4} - 1}{6 \, x^{6}} + \frac {5}{4} \, \arctan \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 30, normalized size = 0.81 \begin {gather*} \frac {5\,\mathrm {atan}\left (x^2\right )}{4}+\frac {\frac {5\,x^8}{4}+\frac {5\,x^4}{6}-\frac {1}{6}}{x^6\,\left (x^4+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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