Optimal. Leaf size=55 \[ \frac{\tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}-\frac{\coth (a+b x) \text{csch}(a+b x)}{8 b} \]
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Rubi [A] time = 0.0596215, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2611, 3768, 3770} \[ \frac{\tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}-\frac{\coth (a+b x) \text{csch}(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \coth ^2(a+b x) \text{csch}^3(a+b x) \, dx &=-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}+\frac{1}{4} \int \text{csch}^3(a+b x) \, dx\\ &=-\frac{\coth (a+b x) \text{csch}(a+b x)}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}-\frac{1}{8} \int \text{csch}(a+b x) \, dx\\ &=\frac{\tanh ^{-1}(\cosh (a+b x))}{8 b}-\frac{\coth (a+b x) \text{csch}(a+b x)}{8 b}-\frac{\coth (a+b x) \text{csch}^3(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.044009, size = 95, normalized size = 1.73 \[ -\frac{\text{csch}^4\left (\frac{1}{2} (a+b x)\right )}{64 b}-\frac{\text{csch}^2\left (\frac{1}{2} (a+b x)\right )}{32 b}+\frac{\text{sech}^4\left (\frac{1}{2} (a+b x)\right )}{64 b}-\frac{\text{sech}^2\left (\frac{1}{2} (a+b x)\right )}{32 b}-\frac{\log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 58, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{\cosh \left ( bx+a \right ) }{3\, \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}}-{\frac{{\rm coth} \left (bx+a\right )}{3} \left ( -{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (bx+a\right )}{8}} \right ) }+{\frac{{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09227, size = 174, normalized size = 3.16 \begin{align*} \frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} - \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} + \frac{e^{\left (-b x - a\right )} + 7 \, e^{\left (-3 \, b x - 3 \, a\right )} + 7 \, e^{\left (-5 \, b x - 5 \, a\right )} + e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b{\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02739, size = 3066, normalized size = 55.75 \begin{align*} \text{result too large to display} \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 \coth ^{2}{\left (a + b x \right )} \operatorname{csch}^{3}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.223, size = 108, normalized size = 1.96 \begin{align*} -\frac{\frac{2 \,{\left (e^{\left (7 \, b x + 7 \, a\right )} + 7 \, e^{\left (5 \, b x + 5 \, a\right )} + 7 \, e^{\left (3 \, b x + 3 \, a\right )} + e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} - \log \left (e^{\left (b x + a\right )} + 1\right ) + \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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