Optimal. Leaf size=40 \[ -\frac{2 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{b} \]
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Rubi [A] time = 0.0199072, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3771, 2641} \[ -\frac{2 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\text{sech}(a+b x)} \, dx &=\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)} F\left (\left .\frac{1}{2} i (a+b x)\right |2\right ) \sqrt{\text{sech}(a+b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.0292461, size = 40, normalized size = 1. \[ -\frac{2 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.241, size = 135, normalized size = 3.4 \begin{align*} 2\,{\frac{\sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) }{\sqrt{2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sinh \left ( 1/2\,bx+a/2 \right ) \sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\operatorname{sech}\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\operatorname{sech}\left (b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\operatorname{sech}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\operatorname{sech}\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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