Optimal. Leaf size=16 \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]
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Rubi [A] time = 0.0026284, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{1+x^4}} \, dx &=-\frac{\sqrt{1+x^4}}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0021295, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{4}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06402, size = 16, normalized size = 1. \begin{align*} -\frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4882, size = 43, normalized size = 2.69 \begin{align*} -\frac{x^{2} + \sqrt{x^{4} + 1}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.578022, size = 12, normalized size = 0.75 \begin{align*} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22542, size = 12, normalized size = 0.75 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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