Optimal. Leaf size=37 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a \text{sech}(c+d x)+a}}\right )}{d} \]
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Rubi [A] time = 0.0185252, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3774, 203} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a \text{sech}(c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \sqrt{a+a \text{sech}(c+d x)} \, dx &=\frac{(2 i a) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{i a \tanh (c+d x)}{\sqrt{a+a \text{sech}(c+d x)}}\right )}{d}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a+a \text{sech}(c+d x)}}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0891638, size = 60, normalized size = 1.62 \[ \frac{\sqrt{2} \sinh ^{-1}\left (\sqrt{2} \sinh \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cosh (c+d x)} \text{sech}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\text{sech}(c+d x)+1)}}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.217, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+a{\rm sech} \left (dx+c\right )}\, dx \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{a \operatorname{sech}\left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55815, size = 1808, normalized size = 48.86 \begin{align*} \text{result too large to display} \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{a \operatorname{sech}{\left (c + d x \right )} + a}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17821, size = 112, normalized size = 3.03 \begin{align*} \frac{\frac{2 \, a \arctan \left (-\frac{\sqrt{a} e^{\left (d x + c\right )} - \sqrt{a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \sqrt{a} \log \left ({\left | -\sqrt{a} e^{\left (d x + c\right )} + \sqrt{a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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