Integrand size = 12, antiderivative size = 104 \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=-\frac {10 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{21 b^4 d}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 104, 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 = {3854, 3856, 2720} \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=-\frac {10 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{21 b^4 d}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}} \]
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Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {5 \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx}{7 b^2} \\ & = \frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}+\frac {5 \int \sqrt {b \text {sech}(c+d x)} \, dx}{21 b^4} \\ & = \frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}}+\frac {\left (5 \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}\right ) \int \frac {1}{\sqrt {\cosh (c+d x)}} \, dx}{21 b^4} \\ & = -\frac {10 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{21 b^4 d}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\frac {\sqrt {b \text {sech}(c+d x)} \left (-40 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right )+26 \sinh (2 (c+d x))+3 \sinh (4 (c+d x))\right )}{84 b^4 d} \]
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\[\int \frac {1}{\left (b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 483, normalized size of antiderivative = 4.64 \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\frac {80 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (3 \, \cosh \left (d x + c\right )^{8} + 24 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, \sinh \left (d x + c\right )^{8} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{2} + 13\right )} \sinh \left (d x + c\right )^{6} + 26 \, \cosh \left (d x + c\right )^{6} + 12 \, {\left (14 \, \cosh \left (d x + c\right )^{3} + 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 30 \, {\left (7 \, \cosh \left (d x + c\right )^{4} + 13 \, \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (21 \, \cosh \left (d x + c\right )^{5} + 65 \, \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{6} + 195 \, \cosh \left (d x + c\right )^{4} - 13\right )} \sinh \left (d x + c\right )^{2} - 26 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (6 \, \cosh \left (d x + c\right )^{7} + 39 \, \cosh \left (d x + c\right )^{5} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 3\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}}}{168 \, {\left (b^{4} d \cosh \left (d x + c\right )^{4} + 4 \, b^{4} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{4} d \sinh \left (d x + c\right )^{4}\right )}} \]
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\[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int \frac {1}{\left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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