Optimal. Leaf size=6 \[ -\tanh \left (\frac{1}{x}\right ) \]
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Rubi [A] time = 0.0221789, 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 = {5436, 3767, 8} \[ -\tanh \left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5436
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\text{sech}^2\left (\frac{1}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \text{sech}^2(x) \, dx,x,\frac{1}{x}\right )\\ &=-\left (i \operatorname{Subst}\left (\int 1 \, dx,x,-i \tanh \left (\frac{1}{x}\right )\right )\right )\\ &=-\tanh \left (\frac{1}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.01752, size = 6, normalized size = 1. \[ -\tanh \left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 7, normalized size = 1.2 \begin{align*} -\tanh \left ({x}^{-1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1392, size = 16, normalized size = 2.67 \begin{align*} \frac{2}{e^{\frac{2}{x}} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99213, size = 80, normalized size = 13.33 \begin{align*} \frac{2}{\cosh \left (\frac{1}{x}\right )^{2} + 2 \, \cosh \left (\frac{1}{x}\right ) \sinh \left (\frac{1}{x}\right ) + \sinh \left (\frac{1}{x}\right )^{2} + 1} \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{\operatorname{sech}^{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 = 1.18398, size = 16, normalized size = 2.67 \begin{align*} \frac{2}{e^{\frac{2}{x}} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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