Integrand size = 2, antiderivative size = 3 \[ \int \text {sech}(x) \, dx=\arctan (\sinh (x)) \]
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Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3855} \[ \int \text {sech}(x) \, dx=\arctan (\sinh (x)) \]
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Rule 3855
Rubi steps \begin{align*} \text {integral}& = \arctan (\sinh (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \text {sech}(x) \, dx=\arctan (\sinh (x)) \]
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Time = 0.07 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33
method | result | size |
lookup | \(\arctan \left (\sinh \left (x \right )\right )\) | \(4\) |
default | \(\arctan \left (\sinh \left (x \right )\right )\) | \(4\) |
risch | \(i \ln \left ({\mathrm e}^{x}+i\right )-i \ln \left ({\mathrm e}^{x}-i\right )\) | \(20\) |
parallelrisch | \(-i \left (\ln \left (\tanh \left (\frac {x}{2}\right )-i\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+i\right )\right )\) | \(23\) |
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Leaf count of result is larger than twice the leaf count of optimal. 8 vs. \(2 (3) = 6\).
Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 2.67 \[ \int \text {sech}(x) \, dx=2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 7 vs. \(2 (3) = 6\).
Time = 0.17 (sec) , antiderivative size = 7, normalized size of antiderivative = 2.33 \[ \int \text {sech}(x) \, dx=2 \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \text {sech}(x) \, dx=\arctan \left (\sinh \left (x\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.67 \[ \int \text {sech}(x) \, dx=2 \, \arctan \left (e^{x}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.67 \[ \int \text {sech}(x) \, dx=2\,\mathrm {atan}\left ({\mathrm {e}}^x\right ) \]
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