Integrand size = 10, antiderivative size = 66 \[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{3 b}+\frac {2 \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{3 b} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3853, 3856, 2720} \[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=\frac {2 \sinh (a+b x) \text {sech}^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b} \]
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Rule 2720
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{3 b}+\frac {1}{3} \int \sqrt {\text {sech}(a+b x)} \, dx \\ & = \frac {2 \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{3 b}+\frac {1}{3} \left (\sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx \\ & = -\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{3 b}+\frac {2 \text {sech}^{\frac {3}{2}}(a+b x) \sinh (a+b x)}{3 b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.77 \[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=\frac {2 \text {sech}^{\frac {3}{2}}(a+b x) \left (-i \cosh ^{\frac {3}{2}}(a+b x) \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+\sinh (a+b x)\right )}{3 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(216\) vs. \(2(82)=164\).
Time = 1.74 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.29
method | result | size |
default | \(\frac {2 \left (2 \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+\sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}}{3 \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{\frac {3}{2}} \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) b}\) | \(217\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.88 \[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=\frac {2 \, {\left (\sqrt {2} {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}} + {\left (\sqrt {2} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )^{2} + \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b\right )}} \]
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\[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=\int \operatorname {sech}^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]
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\[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=\int { \operatorname {sech}\left (b x + a\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \text {sech}^{\frac {5}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2} \,d x \]
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