Integrand size = 12, antiderivative size = 40 \[ \int \text {sech}^2(a+b x)^{3/2} \, dx=\frac {\arcsin (\tanh (a+b x))}{2 b}+\frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 201, 222} \[ \int \text {sech}^2(a+b x)^{3/2} \, dx=\frac {\arcsin (\tanh (a+b x))}{2 b}+\frac {\tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{2 b} \]
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Rule 201
Rule 222
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\tanh (a+b x)\right )}{b} \\ & = \frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (a+b x)\right )}{2 b} \\ & = \frac {\arcsin (\tanh (a+b x))}{2 b}+\frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \text {sech}^2(a+b x)^{3/2} \, dx=\frac {\text {sech}(a+b x) (\arctan (\sinh (a+b x))+\text {sech}(a+b x) \tanh (a+b x))}{2 b \sqrt {\text {sech}^2(a+b x)}} \]
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Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 183, normalized size of antiderivative = 4.58
method | result | size |
risch | \(\frac {\sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left ({\mathrm e}^{2 b x +2 a}-1\right )}{\left (1+{\mathrm e}^{2 b x +2 a}\right ) b}+\frac {i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{2 b}\) | \(183\) |
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Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 267, normalized size of antiderivative = 6.68 \[ \int \text {sech}^2(a+b x)^{3/2} \, dx=\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
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\[ \int \text {sech}^2(a+b x)^{3/2} \, dx=\int \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.62 \[ \int \text {sech}^2(a+b x)^{3/2} \, dx=-\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac {e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.90 \[ \int \text {sech}^2(a+b x)^{3/2} \, dx=\frac {\pi + \frac {4 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{4 \, b} \]
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Timed out. \[ \int \text {sech}^2(a+b x)^{3/2} \, dx=\int {\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{3/2} \,d x \]
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