Optimal. Leaf size=30 \[ -\frac {3}{4 x^2}-\frac {3}{4} \tan ^{-1}\left (x^2\right )+\frac {1}{4 x^2 \left (x^4+1\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {28, 275, 290, 325, 203} \[ \frac {1}{4 x^2 \left (x^4+1\right )}-\frac {3}{4 x^2}-\frac {3}{4} \tan ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 275
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+2 x^4+x^8\right )} \, dx &=\int \frac {1}{x^3 \left (1+x^4\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {1}{4 x^2 \left (1+x^4\right )}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {3}{4 x^2}+\frac {1}{4 x^2 \left (1+x^4\right )}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^2\right )\\ &=-\frac {3}{4 x^2}+\frac {1}{4 x^2 \left (1+x^4\right )}-\frac {3}{4} \tan ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ -\frac {1}{2 x^2}+\frac {3}{4} \tan ^{-1}\left (\frac {1}{x^2}\right )-\frac {x^2}{4 \left (x^4+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 31, normalized size = 1.03 \[ -\frac {3 \, x^{4} + 3 \, {\left (x^{6} + x^{2}\right )} \arctan \left (x^{2}\right ) + 2}{4 \, {\left (x^{6} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 25, normalized size = 0.83 \[ -\frac {3 \, x^{4} + 2}{4 \, {\left (x^{6} + x^{2}\right )}} - \frac {3}{4} \, \arctan \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.83 \[ -\frac {x^{2}}{4 \left (x^{4}+1\right )}-\frac {3 \arctan \left (x^{2}\right )}{4}-\frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 25, normalized size = 0.83 \[ -\frac {3 \, x^{4} + 2}{4 \, {\left (x^{6} + x^{2}\right )}} - \frac {3}{4} \, \arctan \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 25, normalized size = 0.83 \[ -\frac {3\,\mathrm {atan}\left (x^2\right )}{4}-\frac {\frac {3\,x^4}{4}+\frac {1}{2}}{x^6+x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 26, normalized size = 0.87 \[ \frac {- 3 x^{4} - 2}{4 x^{6} + 4 x^{2}} - \frac {3 \operatorname {atan}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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