Optimal. Leaf size=60 \[ \frac{\coth (c+d x)}{2 d \sqrt{-\tanh ^2(c+d x)}}-\frac{\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt{-\tanh ^2(c+d x)}} \]
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Rubi [A] time = 0.0309542, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 3475} \[ \frac{\coth (c+d x)}{2 d \sqrt{-\tanh ^2(c+d x)}}-\frac{\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt{-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\left (-\tanh ^2(c+d x)\right )^{3/2}} \, dx &=-\frac{\tanh (c+d x) \int \coth ^3(c+d x) \, dx}{\sqrt{-\tanh ^2(c+d x)}}\\ &=\frac{\coth (c+d x)}{2 d \sqrt{-\tanh ^2(c+d x)}}-\frac{\tanh (c+d x) \int \coth (c+d x) \, dx}{\sqrt{-\tanh ^2(c+d x)}}\\ &=\frac{\coth (c+d x)}{2 d \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 = 0.126368, size = 51, normalized size = 0.85 \[ \frac{\coth (c+d x)-2 \tanh (c+d x) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{2 d \sqrt{-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 79, normalized size = 1.3 \begin{align*} -{\frac{\tanh \left ( dx+c \right ) \left ( \ln \left ( \tanh \left ( dx+c \right ) +1 \right ) \left ( \tanh \left ( dx+c \right ) \right ) ^{2}-2\,\ln \left ( \tanh \left ( dx+c \right ) \right ) \left ( \tanh \left ( dx+c \right ) \right ) ^{2}+\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) \left ( \tanh \left ( dx+c \right ) \right ) ^{2}+1 \right ) }{2\,d} \left ( - \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.6302, size = 115, normalized size = 1.92 \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} - \frac{2 i \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.27635, size = 252, normalized size = 4.2 \begin{align*} \frac{-i \, d x e^{\left (4 \, d x + 4 \, c\right )} - i \, d x +{\left (2 i \, d x - 2 i\right )} e^{\left (2 \, d x + 2 \, c\right )} +{\left (i \, e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d e^{\left (2 \, d x + 2 \, c\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}{\left (- \tanh ^{2}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.26639, size = 151, normalized size = 2.52 \begin{align*} \frac{i \, \log \left (i \, e^{\left (2 \, d x + 2 \, c\right )}\right )}{2 \, d \mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} - \frac{i \, \log \left (-i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )}{d \mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} + \frac{2 i \, e^{\left (2 \, d x + 2 \, c\right )}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2} \mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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