Optimal. Leaf size=88 \[ -\frac{\coth ^3(c+d x)}{4 d \sqrt{-\tanh ^2(c+d x)}}-\frac{\coth (c+d x)}{2 d \sqrt{-\tanh ^2(c+d x)}}+\frac{\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt{-\tanh ^2(c+d x)}} \]
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Rubi [A] time = 0.0480801, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 3475} \[ -\frac{\coth ^3(c+d x)}{4 d \sqrt{-\tanh ^2(c+d x)}}-\frac{\coth (c+d x)}{2 d \sqrt{-\tanh ^2(c+d x)}}+\frac{\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt{-\tanh ^2(c+d x)}} \]
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 )^{5/2}} \, dx &=\frac{\tanh (c+d x) \int \coth ^5(c+d x) \, dx}{\sqrt{-\tanh ^2(c+d x)}}\\ &=-\frac{\coth ^3(c+d x)}{4 d \sqrt{-\tanh ^2(c+d x)}}+\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{\coth ^3(c+d x)}{4 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{\coth ^3(c+d x)}{4 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 = 0.235704, size = 63, normalized size = 0.72 \[ \frac{-\coth ^3(c+d x)-2 \coth (c+d x)+4 \tanh (c+d x) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{4 d \sqrt{-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 91, normalized size = 1. \begin{align*} -{\frac{\tanh \left ( dx+c \right ) \left ( 2\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) \left ( \tanh \left ( dx+c \right ) \right ) ^{4}-4\,\ln \left ( \tanh \left ( dx+c \right ) \right ) \left ( \tanh \left ( dx+c \right ) \right ) ^{4}+2\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) \left ( \tanh \left ( dx+c \right ) \right ) ^{4}+2\, \left ( \tanh \left ( dx+c \right ) \right ) ^{2}+1 \right ) }{4\,d} \left ( - \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.60669, size = 177, normalized size = 2.01 \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{4 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 4 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 i \, e^{\left (-6 \, d x - 6 \, c\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.32849, size = 466, normalized size = 5.3 \begin{align*} \frac{i \, d x e^{\left (8 \, d x + 8 \, c\right )} + i \, d x +{\left (-4 i \, d x + 4 i\right )} e^{\left (6 \, d x + 6 \, c\right )} +{\left (6 i \, d x - 4 i\right )} e^{\left (4 \, d x + 4 \, c\right )} +{\left (-4 i \, d x + 4 i\right )} e^{\left (2 \, d x + 2 \, c\right )} +{\left (-i \, e^{\left (8 \, d x + 8 \, c\right )} + 4 i \, e^{\left (6 \, d x + 6 \, c\right )} - 6 i \, e^{\left (4 \, d x + 4 \, c\right )} + 4 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 (8 \, d x + 8 \, c\right )} - 4 \, d e^{\left (6 \, d x + 6 \, c\right )} + 6 \, d e^{\left (4 \, d x + 4 \, c\right )} - 4 \, 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{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.38535, size = 182, normalized size = 2.07 \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 ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{d \mathrm{sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} + \frac{-4 i \, e^{\left (6 \, d x + 6 \, c\right )} + 4 i \, e^{\left (4 \, d x + 4 \, c\right )} - 4 i \, e^{\left (2 \, d x + 2 \, c\right )}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4} \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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