Optimal. Leaf size=39 \[ -\frac{\cosh (a-c) \text{csch}^2(b x+c)}{2 b}-\frac{\sinh (a-c) \coth (b x+c)}{b} \]
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Rubi [A] time = 0.0430111, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5625, 2606, 30, 3767, 8} \[ -\frac{\cosh (a-c) \text{csch}^2(b x+c)}{2 b}-\frac{\sinh (a-c) \coth (b x+c)}{b} \]
Antiderivative was successfully verified.
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Rule 5625
Rule 2606
Rule 30
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \cosh (a+b x) \text{csch}^3(c+b x) \, dx &=\cosh (a-c) \int \coth (c+b x) \text{csch}^2(c+b x) \, dx+\sinh (a-c) \int \text{csch}^2(c+b x) \, dx\\ &=\frac{\cosh (a-c) \operatorname{Subst}(\int x \, dx,x,-i \text{csch}(c+b x))}{b}-\frac{(i \sinh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+b x))}{b}\\ &=-\frac{\cosh (a-c) \text{csch}^2(c+b x)}{2 b}-\frac{\coth (c+b x) \sinh (a-c)}{b}\\ \end{align*}
Mathematica [A] time = 0.169359, size = 35, normalized size = 0.9 \[ -\frac{\text{csch}(c) \text{csch}^2(b x+c) (\sinh (a)-\sinh (a-c) \cosh (2 b x+c))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 59, normalized size = 1.5 \begin{align*}{\frac{ \left ( -2\,{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ){{\rm e}^{3\,a-c}}}{b \left ( -{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06859, size = 178, normalized size = 4.56 \begin{align*} \frac{2 \, e^{\left (-2 \, b x + 3 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} - e^{\left (-4 \, b x + a\right )} - e^{\left (a + 4 \, c\right )}\right )}} + \frac{e^{\left (2 \, a + 3 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} - e^{\left (-4 \, b x + a\right )} - e^{\left (a + 4 \, c\right )}\right )}} - \frac{e^{\left (5 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} - e^{\left (-4 \, b x + a\right )} - e^{\left (a + 4 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83185, size = 620, normalized size = 15.9 \begin{align*} -\frac{2 \,{\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) +{\left (\cosh \left (-a + c\right ) - 2 \, \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )\right )}}{b \cosh \left (b x + c\right )^{3} \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} +{\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{3} + 3 \,{\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} -{\left (b \cosh \left (b x + c\right )^{3} - b \cosh \left (b x + c\right )\right )} \sinh \left (-a + c\right )^{2} + 3 \,{\left (b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} - b \cosh \left (-a + c\right )^{2} -{\left (b \cosh \left (b x + c\right )^{2} - b\right )} \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )} \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}^{3}{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1989, size = 69, normalized size = 1.77 \begin{align*} -\frac{{\left (2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} - e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{b{\left (e^{\left (2 \, b x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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