Optimal. Leaf size=31 \[ \frac{\sqrt{-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.0176316, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3658, 3475} \[ \frac{\sqrt{-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \sqrt{-\tanh ^2(c+d x)} \, dx &=\left (\coth (c+d x) \sqrt{-\tanh ^2(c+d x)}\right ) \int \tanh (c+d x) \, dx\\ &=\frac{\coth (c+d x) \log (\cosh (c+d x)) \sqrt{-\tanh ^2(c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.0330789, size = 31, normalized size = 1. \[ \frac{\sqrt{-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 45, normalized size = 1.5 \begin{align*} -{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) +\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{2\,d\tanh \left ( dx+c \right ) }\sqrt{- \left ( \tanh \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.62238, size = 38, normalized size = 1.23 \begin{align*} -\frac{i \,{\left (d x + c\right )}}{d} - \frac{i \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.34377, size = 55, normalized size = 1.77 \begin{align*} \frac{-i \, d x + i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} \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{- \tanh ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.18398, size = 73, normalized size = 2.35 \begin{align*} \frac{i \,{\left (d x + c\right )} \mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) \mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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