Optimal. Leaf size=11 \[ \frac{\tanh (x)}{\sqrt{\text{sech}^2(x)}} \]
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Rubi [A] time = 0.0205086, antiderivative size = 11, 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 = {3657, 4122, 191} \[ \frac{\tanh (x)}{\sqrt{\text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-\tanh ^2(x)}} \, dx &=\int \frac{1}{\sqrt{\text{sech}^2(x)}} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{\sqrt{\text{sech}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0069988, size = 11, normalized size = 1. \[ \frac{\tanh (x)}{\sqrt{\text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 14, normalized size = 1.3 \begin{align*}{\tanh \left ( x \right ){\frac{1}{\sqrt{1- \left ( \tanh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54002, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22183, size = 12, normalized size = 1.09 \begin{align*} \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.592725, size = 12, normalized size = 1.09 \begin{align*} \frac{\tanh{\left (x \right )}}{\sqrt{1 - \tanh ^{2}{\left (x \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13941, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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