Optimal. Leaf size=15 \[ -\frac{\text{csch}^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0200289, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2606, 30} \[ -\frac{\text{csch}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \coth (a+b x) \text{csch}^2(a+b x) \, dx &=\frac{\operatorname{Subst}(\int x \, dx,x,-i \text{csch}(a+b x))}{b}\\ &=-\frac{\text{csch}^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0131857, size = 15, normalized size = 1. \[ -\frac{\text{csch}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 14, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03158, size = 18, normalized size = 1.2 \begin{align*} -\frac{\coth \left (b x + a\right )^{2}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81544, size = 232, normalized size = 15.47 \begin{align*} -\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right ) + 3 \,{\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.92465, size = 22, normalized size = 1.47 \begin{align*} \begin{cases} - \frac{\operatorname{csch}^{2}{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\x \coth{\left (a \right )} \operatorname{csch}^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17034, size = 36, normalized size = 2.4 \begin{align*} -\frac{2 \, e^{\left (2 \, b x + 2 \, a\right )}}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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