Optimal. Leaf size=70 \[ \frac{2 b \sinh (c+d x) \sqrt{b \text{sech}(c+d x)}}{d}+\frac{2 i b^2 E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}} \]
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Rubi [A] time = 0.0383941, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3771, 2639} \[ \frac{2 b \sinh (c+d x) \sqrt{b \text{sech}(c+d x)}}{d}+\frac{2 i b^2 E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int (b \text{sech}(c+d x))^{3/2} \, dx &=\frac{2 b \sqrt{b \text{sech}(c+d x)} \sinh (c+d x)}{d}-b^2 \int \frac{1}{\sqrt{b \text{sech}(c+d x)}} \, dx\\ &=\frac{2 b \sqrt{b \text{sech}(c+d x)} \sinh (c+d x)}{d}-\frac{b^2 \int \sqrt{\cosh (c+d x)} \, dx}{\sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}}\\ &=\frac{2 i b^2 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)}}+\frac{2 b \sqrt{b \text{sech}(c+d x)} \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0393364, size = 52, normalized size = 0.74 \[ \frac{2 b \sqrt{b \text{sech}(c+d x)} \left (\sinh (c+d x)+i \sqrt{\cosh (c+d x)} E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm sech} \left (dx+c\right ) \right ) ^{{\frac{3}{2}}}\, 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 \left (b \operatorname{sech}\left (d x + c\right )\right )^{\frac{3}{2}}\,{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{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 \left (b \operatorname{sech}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, 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 \left (b \operatorname{sech}\left (d x + c\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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