Optimal. Leaf size=36 \[ \frac{x \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}+\frac{\tanh (x)}{2 \sqrt{a \text{sech}^4(x)}} \]
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Rubi [A] time = 0.0162538, antiderivative size = 36, 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, 2635, 8} \[ \frac{x \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}+\frac{\tanh (x)}{2 \sqrt{a \text{sech}^4(x)}} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \text{sech}^4(x)}} \, dx &=\frac{\text{sech}^2(x) \int \cosh ^2(x) \, dx}{\sqrt{a \text{sech}^4(x)}}\\ &=\frac{\tanh (x)}{2 \sqrt{a \text{sech}^4(x)}}+\frac{\text{sech}^2(x) \int 1 \, dx}{2 \sqrt{a \text{sech}^4(x)}}\\ &=\frac{x \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}}+\frac{\tanh (x)}{2 \sqrt{a \text{sech}^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0220314, size = 23, normalized size = 0.64 \[ \frac{\tanh (x)+x \text{sech}^2(x)}{2 \sqrt{a \text{sech}^4(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.085, size = 89, normalized size = 2.5 \begin{align*}{\frac{{{\rm e}^{2\,x}}x}{2\, \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{{{\rm e}^{4\,x}}}{8\, \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{1}{8\, \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.72467, size = 41, normalized size = 1.14 \begin{align*} -\frac{{\left (\sqrt{a} e^{\left (-4 \, x\right )} - \sqrt{a}\right )} e^{\left (2 \, x\right )}}{8 \, a} + \frac{x}{2 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33109, size = 771, normalized size = 21.42 \begin{align*} \frac{{\left ({\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{4} + \cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, x \cosh \left (x\right )^{2} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} + 2 \, x\right )} e^{\left (4 \, x\right )} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} + 2 \, x\right )} \sinh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right )^{2} - 1\right )} e^{\left (4 \, x\right )} + 2 \,{\left (\cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 4 \,{\left (\cosh \left (x\right )^{3} + 2 \, x \cosh \left (x\right ) +{\left (\cosh \left (x\right )^{3} + 2 \, x \cosh \left (x\right )\right )} e^{\left (4 \, x\right )} + 2 \,{\left (\cosh \left (x\right )^{3} + 2 \, x \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) - 1\right )} \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}} e^{\left (2 \, x\right )}}{8 \,{\left (a \cosh \left (x\right )^{2} e^{\left (2 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + a e^{\left (2 \, x\right )} \sinh \left (x\right )^{2}\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{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 = 1.13167, size = 38, normalized size = 1.06 \begin{align*} -\frac{{\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - 4 \, x - e^{\left (2 \, x\right )}}{8 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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