Integrand size = 10, antiderivative size = 6 \[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=-\tanh \left (\frac {1}{x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5544, 3852, 8} \[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=-\tanh \left (\frac {1}{x}\right ) \]
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Rule 8
Rule 3852
Rule 5544
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \text {sech}^2(x) \, dx,x,\frac {1}{x}\right ) \\ & = -\left (i \text {Subst}\left (\int 1 \, dx,x,-i \tanh \left (\frac {1}{x}\right )\right )\right ) \\ & = -\tanh \left (\frac {1}{x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=-\tanh \left (\frac {1}{x}\right ) \]
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Time = 0.53 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(-\tanh \left (\frac {1}{x}\right )\) | \(7\) |
default | \(-\tanh \left (\frac {1}{x}\right )\) | \(7\) |
risch | \(\frac {2}{{\mathrm e}^{\frac {2}{x}}+1}\) | \(13\) |
parallelrisch | \(-\frac {2 \tanh \left (\frac {1}{2 x}\right )}{1+\tanh \left (\frac {1}{2 x}\right )^{2}}\) | \(21\) |
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (6) = 12\).
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 4.67 \[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=\frac {2}{\cosh \left (\frac {1}{x}\right )^{2} + 2 \, \cosh \left (\frac {1}{x}\right ) \sinh \left (\frac {1}{x}\right ) + \sinh \left (\frac {1}{x}\right )^{2} + 1} \]
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\[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=\int \frac {\operatorname {sech}^{2}{\left (\frac {1}{x} \right )}}{x^{2}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 2.00 \[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=\frac {2}{e^{\frac {2}{x}} + 1} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 2.00 \[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=\frac {2}{e^{\frac {2}{x}} + 1} \]
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Time = 2.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 2.00 \[ \int \frac {\text {sech}^2\left (\frac {1}{x}\right )}{x^2} \, dx=\frac {2}{{\mathrm {e}}^{2/x}+1} \]
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