Integrand size = 12, antiderivative size = 102 \[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d} \]
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Time = 0.05 (sec) , antiderivative size = 102, 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 = {3853, 3856, 2719} \[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {6 b^3 \sinh (c+d x) \sqrt {b \text {sech}(c+d x)}}{5 d}+\frac {2 b \sinh (c+d x) (b \text {sech}(c+d x))^{5/2}}{5 d} \]
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Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d}+\frac {1}{5} \left (3 b^2\right ) \int (b \text {sech}(c+d x))^{3/2} \, dx \\ & = \frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d}-\frac {1}{5} \left (3 b^4\right ) \int \frac {1}{\sqrt {b \text {sech}(c+d x)}} \, dx \\ & = \frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d}-\frac {\left (3 b^4\right ) \int \sqrt {\cosh (c+d x)} \, dx}{5 \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}} \\ & = \frac {6 i b^4 E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )}{5 d \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}}+\frac {6 b^3 \sqrt {b \text {sech}(c+d x)} \sinh (c+d x)}{5 d}+\frac {2 b (b \text {sech}(c+d x))^{5/2} \sinh (c+d x)}{5 d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\frac {b^2 (b \text {sech}(c+d x))^{3/2} \left (6 i \cosh ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} i (c+d x)\right |2\right )+3 \sinh (2 (c+d x))+2 \tanh (c+d x)\right )}{5 d} \]
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\[\int \left (b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {7}{2}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.69 \[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\frac {2 \, {\left (3 \, \sqrt {2} {\left (b^{3} \cosh \left (d x + c\right )^{4} + 4 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{3} \sinh \left (d x + c\right )^{4} + 2 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b^{3} \cosh \left (d x + c\right )^{3} + b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (3 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 3 \, b^{3} \sinh \left (d x + c\right )^{5} + 8 \, b^{3} \cosh \left (d x + c\right )^{3} + b^{3} \cosh \left (d x + c\right ) + 2 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{2} + 4 \, b^{3}\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} + 4 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (15 \, b^{3} \cosh \left (d x + c\right )^{4} + 24 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3}\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}\right )}}{5 \, {\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\int { \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]
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\[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\int { \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (b \text {sech}(c+d x))^{7/2} \, dx=\int {\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{7/2} \,d x \]
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