Optimal. Leaf size=31 \[ \frac{\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt{b \coth ^2(c+d x)}} \]
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Rubi [A] time = 0.0204262, 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} \[ \frac{\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt{b \coth ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \coth ^2(c+d x)}} \, dx &=\frac{\coth (c+d x) \int \tanh (c+d x) \, dx}{\sqrt{b \coth ^2(c+d x)}}\\ &=\frac{\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt{b \coth ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0594719, size = 31, normalized size = 1. \[ \frac{\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt{b \coth ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 52, normalized size = 1.7 \begin{align*} -{\frac{{\rm coth} \left (dx+c\right ) \left ( \ln \left ({\rm coth} \left (dx+c\right )+1 \right ) -2\,\ln \left ({\rm coth} \left (dx+c\right ) \right ) +\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) \right ) }{2\,d}{\frac{1}{\sqrt{b \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62442, size = 46, normalized size = 1.48 \begin{align*} -\frac{d x + c}{\sqrt{b} d} - \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{\sqrt{b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92024, size = 309, normalized size = 9.97 \begin{align*} -\frac{{\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x -{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \log \left (\frac{2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )} \sqrt{\frac{b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{b d e^{\left (2 \, d x + 2 \, c\right )} + b 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{b \coth ^{2}{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19993, size = 81, normalized size = 2.61 \begin{align*} -\frac{\frac{d x + c}{\sqrt{b} \mathrm{sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac{\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{\sqrt{b} \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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