Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.0394633, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3205, 63, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 3205
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\coth (c+d x)}{\sqrt{a \cosh ^2(c+d x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a x}} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a \cosh ^2(c+d x)}\right )}{a d}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}\\ \end{align*}
Mathematica [A] time = 0.0661061, size = 49, normalized size = 1.58 \[ \frac{\cosh (c+d x) \left (\log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )\right )}{d \sqrt{a \cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 31, normalized size = 1. \begin{align*} -{\frac{\cosh \left ( dx+c \right ){\it Artanh} \left ( \cosh \left ( dx+c \right ) \right ) }{d}{\frac{1}{\sqrt{a \left ( \cosh \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 = 1.71249, size = 54, normalized size = 1.74 \begin{align*} -\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{\sqrt{a} d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{\sqrt{a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85583, size = 455, normalized size = 14.68 \begin{align*} \left [\frac{\sqrt{a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} \log \left (\frac{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1}\right )}{a d e^{\left (2 \, d x + 2 \, c\right )} + a d}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} \sqrt{-a}}{a \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + a \cosh \left (d x + c\right ) +{\left (a e^{\left (2 \, d x + 2 \, c\right )} + a\right )} \sinh \left (d x + c\right )}\right )}{a d}\right ] \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{\coth{\left (c + d x \right )}}{\sqrt{a \cosh ^{2}{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20626, size = 46, normalized size = 1.48 \begin{align*} -\frac{\frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{\sqrt{a}} - \frac{\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{\sqrt{a}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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