Optimal. Leaf size=11 \[ -\frac{\coth (a+b x)}{b} \]
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Rubi [A] time = 0.009889, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767, 8} \[ -\frac{\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \text{csch}^2(a+b x) \, dx &=-\frac{i \operatorname{Subst}(\int 1 \, dx,x,-i \coth (a+b x))}{b}\\ &=-\frac{\coth (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0099317, size = 11, normalized size = 1. \[ -\frac{\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 12, normalized size = 1.1 \begin{align*} -{\frac{{\rm coth} \left (bx+a\right )}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990234, size = 24, normalized size = 2.18 \begin{align*} \frac{2}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.5139, size = 111, normalized size = 10.09 \begin{align*} -\frac{2}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \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 \operatorname{csch}^{2}{\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 = 1.1387, size = 24, normalized size = 2.18 \begin{align*} -\frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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