Integrand size = 12, antiderivative size = 74 \[ \int (b \text {sech}(c+d x))^{5/2} \, dx=-\frac {2 i b^2 \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{3 d}+\frac {2 b (b \text {sech}(c+d x))^{3/2} \sinh (c+d x)}{3 d} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 74, 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 = {3853, 3856, 2720} \[ \int (b \text {sech}(c+d x))^{5/2} \, dx=\frac {2 b \sinh (c+d x) (b \text {sech}(c+d x))^{3/2}}{3 d}-\frac {2 i b^2 \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{3 d} \]
[In]
[Out]
Rule 2720
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 b (b \text {sech}(c+d x))^{3/2} \sinh (c+d x)}{3 d}+\frac {1}{3} b^2 \int \sqrt {b \text {sech}(c+d x)} \, dx \\ & = \frac {2 b (b \text {sech}(c+d x))^{3/2} \sinh (c+d x)}{3 d}+\frac {1}{3} \left (b^2 \sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}\right ) \int \frac {1}{\sqrt {\cosh (c+d x)}} \, dx \\ & = -\frac {2 i b^2 \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{3 d}+\frac {2 b (b \text {sech}(c+d x))^{3/2} \sinh (c+d x)}{3 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76 \[ \int (b \text {sech}(c+d x))^{5/2} \, dx=\frac {2 b^2 \sqrt {b \text {sech}(c+d x)} \left (-i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right )+\tanh (c+d x)\right )}{3 d} \]
[In]
[Out]
\[\int \left (b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.91 \[ \int (b \text {sech}(c+d x))^{5/2} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )^{2} + b^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )^{2} - b^{2}\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d\right )}} \]
[In]
[Out]
\[ \int (b \text {sech}(c+d x))^{5/2} \, dx=\int \left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
[In]
[Out]
\[ \int (b \text {sech}(c+d x))^{5/2} \, dx=\int { \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
[In]
[Out]
\[ \int (b \text {sech}(c+d x))^{5/2} \, dx=\int { \left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (b \text {sech}(c+d x))^{5/2} \, dx=\int {\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{5/2} \,d x \]
[In]
[Out]