Integrand size = 12, antiderivative size = 11 \[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\frac {\arcsin (\tanh (a+b x))}{b} \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4207, 222} \[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\frac {\arcsin (\tanh (a+b x))}{b} \]
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Rule 222
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (a+b x)\right )}{b} \\ & = \frac {\arcsin (\tanh (a+b x))}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(29\) vs. \(2(11)=22\).
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.64 \[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\frac {\arctan (\sinh (a+b x)) \cosh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{b} \]
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Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 130, normalized size of antiderivative = 11.82
method | result | size |
risch | \(\frac {i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{b}-\frac {i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{b}\) | \(130\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\frac {2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \]
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\[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\int \sqrt {\operatorname {sech}^{2}{\left (a + b x \right )}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\frac {\arctan \left (\sinh \left (b x + a\right )\right )}{b} \]
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none
Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \]
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Timed out. \[ \int \sqrt {\text {sech}^2(a+b x)} \, dx=\int \sqrt {\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}} \,d x \]
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