Optimal. Leaf size=76 \[ \frac{2 \sinh (c+d x)}{3 b d \sqrt{b \text{sech}(c+d x)}}-\frac{2 i \sqrt{\cosh (c+d x)} \text{EllipticF}\left (\frac{1}{2} i (c+d x),2\right ) \sqrt{b \text{sech}(c+d x)}}{3 b^2 d} \]
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Rubi [A] time = 0.0388664, antiderivative size = 76, 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 = {3769, 3771, 2641} \[ \frac{2 \sinh (c+d x)}{3 b d \sqrt{b \text{sech}(c+d x)}}-\frac{2 i \sqrt{\cosh (c+d x)} F\left (\left .\frac{1}{2} i (c+d x)\right |2\right ) \sqrt{b \text{sech}(c+d x)}}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(b \text{sech}(c+d x))^{3/2}} \, dx &=\frac{2 \sinh (c+d x)}{3 b d \sqrt{b \text{sech}(c+d x)}}+\frac{\int \sqrt{b \text{sech}(c+d x)} \, dx}{3 b^2}\\ &=\frac{2 \sinh (c+d x)}{3 b d \sqrt{b \text{sech}(c+d x)}}+\frac{\left (\sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}\right ) \int \frac{1}{\sqrt{\cosh (c+d x)}} \, dx}{3 b^2}\\ &=-\frac{2 i \sqrt{\cosh (c+d x)} F\left (\left .\frac{1}{2} i (c+d x)\right |2\right ) \sqrt{b \text{sech}(c+d x)}}{3 b^2 d}+\frac{2 \sinh (c+d x)}{3 b d \sqrt{b \text{sech}(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.063421, size = 63, normalized size = 0.83 \[ \frac{\text{sech}^2(c+d x) \left (\sinh (2 (c+d x))-2 i \sqrt{\cosh (c+d x)} \text{EllipticF}\left (\frac{1}{2} i (c+d x),2\right )\right )}{3 d (b \text{sech}(c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.083, 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 \frac{1}{\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 (\frac{\sqrt{b \operatorname{sech}\left (d x + c\right )}}{b^{2} \operatorname{sech}\left (d x + c\right )^{2}}, 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}{\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 \frac{1}{\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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