Integrand size = 6, antiderivative size = 11 \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{b} \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3855} \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{b} \]
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Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan (\sinh (a+b x))}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{b} \]
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\arctan \left (\sinh \left (b x +a \right )\right )}{b}\) | \(12\) |
default | \(\frac {\arctan \left (\sinh \left (b x +a \right )\right )}{b}\) | \(12\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{b}\) | \(34\) |
parallelrisch | \(-\frac {i \left (\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-i\right )-\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i\right )\right )}{b}\) | \(36\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \text {sech}(a+b x) \, dx=\frac {2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).
Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \text {sech}(a+b x) \, dx=\begin {cases} \frac {2 \operatorname {atan}{\left (\tanh {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b} & \text {for}\: b \neq 0 \\x \operatorname {sech}{\left (a \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan \left (\sinh \left (b x + a\right )\right )}{b} \]
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Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \text {sech}(a+b x) \, dx=\frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \]
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Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \text {sech}(a+b x) \, dx=\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}} \]
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