Integrand size = 8, antiderivative size = 34 \[ \int \text {sech}^3(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{2 b}+\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3855} \[ \int \text {sech}^3(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{2 b}+\frac {\tanh (a+b x) \text {sech}(a+b x)}{2 b} \]
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Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b}+\frac {1}{2} \int \text {sech}(a+b x) \, dx \\ & = \frac {\arctan (\sinh (a+b x))}{2 b}+\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \text {sech}^3(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{2 b}+\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
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Time = 0.67 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2}+\arctan \left ({\mathrm e}^{b x +a}\right )}{b}\) | \(27\) |
default | \(\frac {\frac {\operatorname {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2}+\arctan \left ({\mathrm e}^{b x +a}\right )}{b}\) | \(27\) |
risch | \(\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{2 b}\) | \(68\) |
parallelrisch | \(\frac {i \left (-1-\cosh \left (2 b x +2 a \right )\right ) \ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-i\right )+i \left (1+\cosh \left (2 b x +2 a \right )\right ) \ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i\right )+2 \sinh \left (b x +a \right )}{2 b \left (1+\cosh \left (2 b x +2 a \right )\right )}\) | \(84\) |
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Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (30) = 60\).
Time = 0.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 7.85 \[ \int \text {sech}^3(a+b x) \, dx=\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
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\[ \int \text {sech}^3(a+b x) \, dx=\int \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \text {sech}^3(a+b x) \, dx=-\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac {e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int \text {sech}^3(a+b x) \, dx=\frac {\pi + \frac {4 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{4 \, b} \]
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Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.38 \[ \int \text {sech}^3(a+b x) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}+\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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