Integrand size = 12, antiderivative size = 65 \[ \int \text {sech}^2(a+b x)^{5/2} \, dx=\frac {3 \arcsin (\tanh (a+b x))}{8 b}+\frac {3 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{8 b}+\frac {\text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{4 b} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 201, 222} \[ \int \text {sech}^2(a+b x)^{5/2} \, dx=\frac {3 \arcsin (\tanh (a+b x))}{8 b}+\frac {\tanh (a+b x) \text {sech}^2(a+b x)^{3/2}}{4 b}+\frac {3 \tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{8 b} \]
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Rule 201
Rule 222
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,\tanh (a+b x)\right )}{b} \\ & = \frac {\text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{4 b}+\frac {3 \text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\tanh (a+b x)\right )}{4 b} \\ & = \frac {3 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{8 b}+\frac {\text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{4 b}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (a+b x)\right )}{8 b} \\ & = \frac {3 \arcsin (\tanh (a+b x))}{8 b}+\frac {3 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{8 b}+\frac {\text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{4 b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \text {sech}^2(a+b x)^{5/2} \, dx=\frac {\text {sech}(a+b x) \left (3 \arctan (\sinh (a+b x))+3 \text {sech}(a+b x) \tanh (a+b x)+2 \text {sech}^3(a+b x) \tanh (a+b x)\right )}{8 b \sqrt {\text {sech}^2(a+b x)}} \]
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Result contains complex when optimal does not.
Time = 6.34 (sec) , antiderivative size = 208, normalized size of antiderivative = 3.20
method | result | size |
risch | \(\frac {\sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 b x +6 a}+11 \,{\mathrm e}^{4 b x +4 a}-11 \,{\mathrm e}^{2 b x +2 a}-3\right )}{4 \left (1+{\mathrm e}^{2 b x +2 a}\right )^{3} b}+\frac {3 i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{8 b}-\frac {3 i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{8 b}\) | \(208\) |
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Leaf count of result is larger than twice the leaf count of optimal. 812 vs. \(2 (55) = 110\).
Time = 0.26 (sec) , antiderivative size = 812, normalized size of antiderivative = 12.49 \[ \int \text {sech}^2(a+b x)^{5/2} \, dx=\text {Too large to display} \]
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\[ \int \text {sech}^2(a+b x)^{5/2} \, dx=\int \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (55) = 110\).
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.72 \[ \int \text {sech}^2(a+b x)^{5/2} \, dx=-\frac {3 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} + \frac {3 \, e^{\left (-b x - a\right )} + 11 \, e^{\left (-3 \, b x - 3 \, a\right )} - 11 \, e^{\left (-5 \, b x - 5 \, a\right )} - 3 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.57 \[ \int \text {sech}^2(a+b x)^{5/2} \, dx=\frac {3 \, \pi + \frac {4 \, {\left (3 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 20 \, e^{\left (b x + a\right )} - 20 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{2}} + 6 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{16 \, b} \]
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Timed out. \[ \int \text {sech}^2(a+b x)^{5/2} \, dx=\int {\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{5/2} \,d x \]
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