Optimal. Leaf size=15 \[ \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)} \]
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Rubi [A] time = 0.0166341, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 3767, 8} \[ \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sqrt{a \text{sech}^4(x)} \, dx &=\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \text{sech}^2(x) \, dx\\ &=\left (i \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=\cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0051942, size = 15, normalized size = 1. \[ \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.075, size = 29, normalized size = 1.9 \begin{align*} -2\,\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5763, size = 18, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{a}}{e^{\left (-2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10846, size = 230, normalized size = 15.33 \begin{align*} -\frac{2 \, \sqrt{\frac{a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}}{\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )}}{2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (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 \sqrt{a \operatorname{sech}^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17594, size = 18, normalized size = 1.2 \begin{align*} -\frac{2 \, \sqrt{a}}{e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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