Optimal. Leaf size=31 \[ -\frac {x^2}{\sqrt {x^4+1}}-\frac {1}{2 \sqrt {x^4+1} x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {271, 264} \[ -\frac {x^2}{\sqrt {x^4+1}}-\frac {1}{2 \sqrt {x^4+1} x^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+x^4\right )^{3/2}} \, dx &=-\frac {1}{2 x^2 \sqrt {1+x^4}}-2 \int \frac {x}{\left (1+x^4\right )^{3/2}} \, dx\\ &=-\frac {1}{2 x^2 \sqrt {1+x^4}}-\frac {x^2}{\sqrt {1+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 0.74 \[ -\frac {2 x^4+1}{2 x^2 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 37, normalized size = 1.19 \[ -\frac {2 \, x^{6} + 2 \, x^{2} + {\left (2 \, x^{4} + 1\right )} \sqrt {x^{4} + 1}}{2 \, {\left (x^{6} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 32, normalized size = 1.03 \[ -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} + \frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.65 \[ -\frac {2 x^{4}+1}{2 \sqrt {x^{4}+1}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 25, normalized size = 0.81 \[ -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} - \frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 17, normalized size = 0.55 \[ -\frac {x^4+\frac {1}{2}}{x^2\,\sqrt {x^4+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 42, normalized size = 1.35 \[ - \frac {2 x^{4} \sqrt {x^{4} + 1}}{2 x^{6} + 2 x^{2}} - \frac {\sqrt {x^{4} + 1}}{2 x^{6} + 2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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