Optimal. Leaf size=24 \[ -\frac{\text{csch}(a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0222952, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2621, 321, 207} \[ -\frac{\text{csch}(a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \text{csch}^2(a+b x) \text{sech}(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=-\frac{\text{csch}(a+b x)}{b}-\frac{i \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=-\frac{\tan ^{-1}(\sinh (a+b x))}{b}-\frac{\text{csch}(a+b x)}{b}\\ \end{align*}
Mathematica [C] time = 0.0185736, size = 29, normalized size = 1.21 \[ -\frac{\text{csch}(a+b x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\sinh ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 27, normalized size = 1.1 \begin{align*} -{\frac{1}{b\sinh \left ( bx+a \right ) }}-2\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50841, size = 58, normalized size = 2.42 \begin{align*} \frac{2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac{2 \, e^{\left (-b x - a\right )}}{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 = 2.13367, size = 305, normalized size = 12.71 \begin{align*} -\frac{2 \,{\left ({\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{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 )} \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21202, size = 78, normalized size = 3.25 \begin{align*} -\frac{\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{2 \, b} - \frac{2}{b{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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