Integrand size = 12, antiderivative size = 22 \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}} \]
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Time = 0.01 (sec) , antiderivative size = 22, 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, 197} \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}} \]
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Rule 197
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\tanh (a+b x)\right )}{b} \\ & = \frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(20)=40\).
Time = 0.46 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.41
method | result | size |
risch | \(\frac {{\mathrm e}^{2 b x +2 a}}{2 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}-\frac {1}{2 b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}}\) | \(97\) |
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none
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\frac {\sinh \left (b x + a\right )}{b} \]
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\begin {cases} \frac {\tanh {\left (a + b x \right )}}{b \sqrt {\operatorname {sech}^{2}{\left (a + b x \right )}}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {\operatorname {sech}^{2}{\left (a \right )}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\frac {e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \]
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Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx=\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )\,\sqrt {\frac {4\,{\mathrm {e}}^{2\,a+2\,b\,x}}{{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^2}}}{4\,b} \]
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