Optimal. Leaf size=46 \[ -\frac{\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}-\frac{\sinh (a-c) \text{csch}(b x+c)}{b}+\frac{\cosh (a+b x)}{b} \]
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Rubi [A] time = 0.0460025, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5622, 5621, 2638, 3770, 2606, 8} \[ -\frac{\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}-\frac{\sinh (a-c) \text{csch}(b x+c)}{b}+\frac{\cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5622
Rule 5621
Rule 2638
Rule 3770
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \coth ^2(c+b x) \sinh (a+b x) \, dx &=\sinh (a-c) \int \coth (c+b x) \text{csch}(c+b x) \, dx+\int \cosh (a+b x) \coth (c+b x) \, dx\\ &=\cosh (a-c) \int \text{csch}(c+b x) \, dx-\frac{(i \sinh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(c+b x))}{b}+\int \sinh (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cosh (c+b x)) \cosh (a-c)}{b}+\frac{\cosh (a+b x)}{b}-\frac{\text{csch}(c+b x) \sinh (a-c)}{b}\\ \end{align*}
Mathematica [C] time = 0.0942187, size = 110, normalized size = 2.39 \[ -\frac{\sinh (a-c) \text{csch}(b x+c)}{b}-\frac{2 i \cosh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac{b x}{2}\right )+\cosh (c) \cosh \left (\frac{b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac{b x}{2}\right )-i \sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b}+\frac{\sinh (a) \sinh (b x)}{b}+\frac{\cosh (a) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 197, normalized size = 4.3 \begin{align*}{\frac{{{\rm e}^{bx+a}}}{2\,b}}+{\frac{{{\rm e}^{-bx-a}}}{2\,b}}+{\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 = 1.17949, size = 189, normalized size = 4.11 \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{e^{\left (-b x - a\right )}}{2 \, b} - \frac{{\left (3 \, e^{\left (2 \, a\right )} - 2 \, e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (2 \, c\right )}}{2 \, b{\left (e^{\left (-b x - a + 2 \, c\right )} - e^{\left (-3 \, b x - a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40071, size = 3363, normalized size = 73.11 \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 \sinh{\left (a + b x \right )} \coth ^{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 = 1.18693, size = 163, normalized size = 3.54 \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{{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, c\right )} + 1\right )} e^{\left (-a\right )}}{e^{\left (3 \, b x + 2 \, c\right )} - e^{\left (b x\right )}} - e^{\left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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