Optimal. Leaf size=13 \[ -\frac {1}{3 x^3}+\frac {1}{x}+\tan ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {325, 203} \[ -\frac {1}{3 x^3}+\frac {1}{x}+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1+x^2\right )} \, dx &=-\frac {1}{3 x^3}-\int \frac {1}{x^2 \left (1+x^2\right )} \, dx\\ &=-\frac {1}{3 x^3}+\frac {1}{x}+\int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{3 x^3}+\frac {1}{x}+\tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{3 x^3}+\frac {1}{x}+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 19, normalized size = 1.46 \[ \frac {3 \, x^{3} \arctan \relax (x) + 3 \, x^{2} - 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.80, size = 15, normalized size = 1.15 \[ \frac {3 \, x^{2} - 1}{3 \, x^{3}} + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 12, normalized size = 0.92 \[ \arctan \relax (x )+\frac {1}{x}-\frac {1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 15, normalized size = 1.15 \[ \frac {3 \, x^{2} - 1}{3 \, x^{3}} + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 12, normalized size = 0.92 \[ \mathrm {atan}\relax (x)+\frac {x^2-\frac {1}{3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 14, normalized size = 1.08 \[ \operatorname {atan}{\relax (x )} + \frac {3 x^{2} - 1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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