Integrand size = 10, antiderivative size = 40 \[ \int \sqrt {\text {sech}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{b} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3856, 2720} \[ \int \sqrt {\text {sech}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{b} \]
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Rule 2720
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \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)}}{b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \sqrt {\text {sech}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(62)=124\).
Time = 0.47 (sec) , antiderivative size = 135, normalized size of antiderivative = 3.38
method | result | size |
default | \(\frac {2 \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, b}\) | \(135\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.60 \[ \int \sqrt {\text {sech}(a+b x)} \, dx=\frac {2 \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \]
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\[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int \sqrt {\operatorname {sech}{\left (a + b x \right )}}\, dx \]
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\[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int { \sqrt {\operatorname {sech}\left (b x + a\right )} \,d x } \]
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\[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int { \sqrt {\operatorname {sech}\left (b x + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int \sqrt {\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]
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