Optimal. Leaf size=13 \[ \frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}} \]
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Rubi [A] time = 0.0285025, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 191} \[ \frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \text{sech}^2(x)}} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0064179, size = 13, normalized size = 1. \[ \frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 58, normalized size = 4.5 \begin{align*}{\frac{{{\rm e}^{2\,x}}}{2\,{{\rm e}^{2\,x}}+2}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{1}{2\,{{\rm e}^{2\,x}}+2}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.87257, size = 23, normalized size = 1.77 \begin{align*} -\frac{e^{\left (-x\right )}}{2 \, \sqrt{a}} + \frac{e^{x}}{2 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0295, size = 247, normalized size = 19. \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \,{\left (a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.695431, size = 15, normalized size = 1.15 \begin{align*} \frac{\tanh{\left (x \right )}}{\sqrt{a} \sqrt{\operatorname{sech}^{2}{\left (x \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14099, size = 19, normalized size = 1.46 \begin{align*} -\frac{e^{\left (-x\right )} - e^{x}}{2 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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