Optimal. Leaf size=6 \[ -\tan \left (\frac{1}{x}\right ) \]
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Rubi [A] time = 0.0204549, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4204, 3767, 8} \[ -\tan \left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 4204
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^2\left (\frac{1}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \sec ^2(x) \, dx,x,\frac{1}{x}\right )\\ &=\operatorname{Subst}\left (\int 1 \, dx,x,-\tan \left (\frac{1}{x}\right )\right )\\ &=-\tan \left (\frac{1}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0175814, size = 6, normalized size = 1. \[ -\tan \left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 7, normalized size = 1.2 \begin{align*} -\tan \left ({x}^{-1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.961808, size = 49, normalized size = 8.17 \begin{align*} -\frac{2 \, \sin \left (\frac{2}{x}\right )}{\cos \left (\frac{2}{x}\right )^{2} + \sin \left (\frac{2}{x}\right )^{2} + 2 \, \cos \left (\frac{2}{x}\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11173, size = 27, normalized size = 4.5 \begin{align*} -\frac{\sin \left (\frac{1}{x}\right )}{\cos \left (\frac{1}{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (\frac{1}{x} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07121, size = 8, normalized size = 1.33 \begin{align*} -\tan \left (\frac{1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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