Optimal. Leaf size=36 \[ \frac{\text{csch}(a-c) \log (\sinh (b x+c))}{b}-\frac{\text{csch}(a-c) \log (\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0237326, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5645, 3475} \[ \frac{\text{csch}(a-c) \log (\sinh (b x+c))}{b}-\frac{\text{csch}(a-c) \log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 5645
Rule 3475
Rubi steps
\begin{align*} \int \text{csch}(a+b x) \text{csch}(c+b x) \, dx &=-(\text{csch}(a-c) \int \coth (a+b x) \, dx)+\text{csch}(a-c) \int \coth (c+b x) \, dx\\ &=-\frac{\text{csch}(a-c) \log (\sinh (a+b x))}{b}+\frac{\text{csch}(a-c) \log (\sinh (c+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.220831, size = 28, normalized size = 0.78 \[ -\frac{\text{csch}(a-c) (\log (\sinh (a+b x))-\log (\sinh (b x+c)))}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 79, normalized size = 2.2 \begin{align*} 2\,{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{a+c}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }}-2\,{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ){{\rm e}^{a+c}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15714, size = 180, normalized size = 5. \begin{align*} -\frac{2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x - a\right )} + 1\right )}{b{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} - \frac{2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x - a\right )} - 1\right )}{b{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} + \frac{2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{b{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} + \frac{2 \, e^{\left (a + c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{b{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14778, size = 491, normalized size = 13.64 \begin{align*} -\frac{2 \,{\left ({\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac{2 \,{\left (\cosh \left (-a + c\right ) \sinh \left (b x + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) -{\left (\cosh \left (-a + c\right ) + \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )}\right ) -{\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac{2 \, \sinh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )\right )}}{b \cosh \left (-a + c\right )^{2} - 2 \, b \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + b \sinh \left (-a + c\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}{\left (a + b x \right )} \operatorname{csch}{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20016, size = 147, normalized size = 4.08 \begin{align*} \frac{2 \, e^{\left (a + c\right )} \log \left (\frac{{\left | -{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |} + 2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} - e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |}}{{\left |{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |} + 2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} - e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |}}\right )}{b{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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