Integrand size = 12, antiderivative size = 42 \[ \int \sqrt {b \text {sech}(c+d x)} \, dx=-\frac {2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{d} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3856, 2720} \[ \int \sqrt {b \text {sech}(c+d x)} \, dx=-\frac {2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{d} \]
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Rule 2720
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \left (\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 \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \sqrt {b \text {sech}(c+d x)} \, dx=-\frac {2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{d} \]
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\[\int \sqrt {b \,\operatorname {sech}\left (d x +c \right )}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64 \[ \int \sqrt {b \text {sech}(c+d x)} \, dx=\frac {2 \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{d} \]
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\[ \int \sqrt {b \text {sech}(c+d x)} \, dx=\int \sqrt {b \operatorname {sech}{\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt {b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {b \text {sech}(c+d x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {b \text {sech}(c+d x)} \, dx=\int \sqrt {\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
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