Optimal. Leaf size=36 \[ -\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{\sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b} \]
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Rubi [A] time = 0.0306922, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5625, 2606, 8, 3770} \[ -\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{\sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b} \]
Antiderivative was successfully verified.
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Rule 5625
Rule 2606
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \cosh (a+b x) \text{csch}^2(c+b x) \, dx &=\cosh (a-c) \int \coth (c+b x) \text{csch}(c+b x) \, dx+\sinh (a-c) \int \text{csch}(c+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cosh (c+b x)) \sinh (a-c)}{b}-\frac{(i \cosh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(c+b x))}{b}\\ &=-\frac{\cosh (a-c) \text{csch}(c+b x)}{b}-\frac{\tanh ^{-1}(\cosh (c+b x)) \sinh (a-c)}{b}\\ \end{align*}
Mathematica [C] time = 0.0682005, size = 90, normalized size = 2.5 \[ -\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{2 i \sinh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac{b x}{2}\right )+\cosh (c) \cosh \left (\frac{b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac{b x}{2}\right )-i \sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 170, normalized size = 4.7 \begin{align*}{\frac{{{\rm e}^{bx+a}} \left ({{\rm e}^{2\,a}}+{{\rm e}^{2\,c}} \right ) }{b \left ( -{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) }}-{\frac{\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}+{\frac{\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}}+{\frac{\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}-{\frac{\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0882, size = 142, normalized size = 3.94 \begin{align*} -\frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac{{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-b x - a\right )}}{b{\left (e^{\left (-2 \, b x\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 = 1.8243, size = 1709, normalized size = 47.47 \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 \cosh{\left (a + b x \right )} \operatorname{csch}^{2}{\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 = 1.19119, size = 143, normalized size = 3.97 \begin{align*} -\frac{{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right ) -{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right ) + \frac{2 \,{\left (e^{\left (b x + 2 \, a\right )} + e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{e^{\left (2 \, b x + 2 \, c\right )} - 1}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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