Integrand size = 10, antiderivative size = 66 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{3 b}+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}} \]
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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 = {3854, 3856, 2720} \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}-\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b} \]
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Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\frac {1}{3} \int \sqrt {\text {sech}(a+b x)} \, dx \\ & = \frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\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 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {\text {sech}(a+b x)} \left (-2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+\sinh (2 (a+b x))\right )}{3 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(82)=164\).
Time = 1.23 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.64
method | result | size |
default | \(\frac {2 \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (4 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}-6 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+\sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, b}\) | \(174\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.38 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}} + 4 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{6 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
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\[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{\operatorname {sech}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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