Optimal. Leaf size=31 \[ \frac{\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt{-\tanh ^2(c+d x)}} \]
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Rubi [A] time = 0.0174017, 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} \[ \frac{\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt{-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\tanh ^2(c+d x)}} \, dx &=\frac{\tanh (c+d x) \int \coth (c+d x) \, dx}{\sqrt{-\tanh ^2(c+d x)}}\\ &=\frac{\log (\sinh (c+d x)) \tanh (c+d x)}{d \sqrt{-\tanh ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0792873, size = 39, normalized size = 1.26 \[ \frac{\tanh (c+d x) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{d \sqrt{-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 52, normalized size = 1.7 \begin{align*} -{\frac{\tanh \left ( dx+c \right ) \left ( \ln \left ( \tanh \left ( dx+c \right ) +1 \right ) -2\,\ln \left ( \tanh \left ( dx+c \right ) \right ) +\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) \right ) }{2\,d}{\frac{1}{\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 = 1.57261, size = 61, normalized size = 1.97 \begin{align*} \frac{i \,{\left (d x + c\right )}}{d} + \frac{i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.32129, size = 54, normalized size = 1.74 \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 \frac{1}{\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 = 1.22931, size = 85, normalized size = 2.74 \begin{align*} \frac{\frac{-2 i \, d x - 2 i \, c}{\mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} + \frac{2 i \, \log \left (-i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )}{\mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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