Optimal. Leaf size=42 \[ -\frac{2 i E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}} \]
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Rubi [A] time = 0.0220029, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3771, 2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \text{sech}(c+d x)}} \, dx &=\frac{\int \sqrt{\cosh (c+d x)} \, dx}{\sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}}\\ &=-\frac{2 i E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0295799, size = 42, normalized size = 1. \[ -\frac{2 i E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.141, size = 244, normalized size = 5.8 \begin{align*}{\frac{\sqrt{2}}{d}{\frac{1}{\sqrt{{\frac{b{{\rm e}^{dx+c}}}{ \left ({{\rm e}^{dx+c}} \right ) ^{2}+1}}}}}}+{\frac{\sqrt{2}}{d \left ( \left ({{\rm e}^{dx+c}} \right ) ^{2}+1 \right ) } \left ( -2\,{\frac{b \left ({{\rm e}^{dx+c}} \right ) ^{2}+b}{b\sqrt{{{\rm e}^{dx+c}} \left ( b \left ({{\rm e}^{dx+c}} \right ) ^{2}+b \right ) }}}+{i\sqrt{2}\sqrt{-i \left ({{\rm e}^{dx+c}}+i \right ) }\sqrt{i \left ({{\rm e}^{dx+c}}-i \right ) }\sqrt{i{{\rm e}^{dx+c}}} \left ( -2\,i{\it EllipticE} \left ( \sqrt{-i \left ({{\rm e}^{dx+c}}+i \right ) },{\frac{\sqrt{2}}{2}} \right ) +i{\it EllipticF} \left ( \sqrt{-i \left ({{\rm e}^{dx+c}}+i \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{b \left ({{\rm e}^{dx+c}} \right ) ^{3}+b{{\rm e}^{dx+c}}}}}} \right ) \sqrt{b{{\rm e}^{dx+c}} \left ( \left ({{\rm e}^{dx+c}} \right ) ^{2}+1 \right ) }{\frac{1}{\sqrt{{\frac{b{{\rm e}^{dx+c}}}{ \left ({{\rm e}^{dx+c}} \right ) ^{2}+1}}}}}} \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 \frac{1}{\sqrt{b \operatorname{sech}\left (d x + c\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 (\frac{\sqrt{b \operatorname{sech}\left (d x + c\right )}}{b \operatorname{sech}\left (d x + c\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 \frac{1}{\sqrt{b \operatorname{sech}{\left (c + d 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 \frac{1}{\sqrt{b \operatorname{sech}\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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