Optimal. Leaf size=13 \[ \frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}} \]
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Rubi [A] time = 0.0210973, antiderivative size = 13, 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 = {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 = 0.0067899, size = 13, normalized size = 1. \[ \frac{\tanh (x)}{\sqrt{-\text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 12, normalized size = 0.9 \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 [B] time = 1.5597, size = 34, normalized size = 2.62 \begin{align*} -\frac{e^{\left (-2 \, x\right )}}{2 \, \sqrt{-e^{\left (-2 \, x\right )}}} + \frac{1}{2 \, \sqrt{-e^{\left (-2 \, x\right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28971, size = 4, normalized size = 0.31 \begin{align*} 0 \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{\tanh ^{2}{\left (x \right )} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.18516, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{2} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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