Optimal. Leaf size=26 \[ \frac {\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5626, 3475, 8} \[ \frac {\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3475
Rule 5626
Rubi steps
\begin {align*} \int \text {csch}(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int 1 \, dx+\sinh (a-c) \int \coth (c+b x) \, dx\\ &=x \cosh (a-c)+\frac {\log (\sinh (c+b x)) \sinh (a-c)}{b}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 26, normalized size = 1.00 \[ \frac {\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 86, normalized size = 3.31 \[ \frac {2 \, b x + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )}{2 \, {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 51, normalized size = 1.96 \[ \frac {2 \, b x e^{\left (-a + c\right )} + {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, c\right )} - 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 150, normalized size = 5.77 \[ x \,{\mathrm e}^{a -c}+{\mathrm e}^{-a -c} {\mathrm e}^{2 c} x -{\mathrm e}^{-a -c} {\mathrm e}^{2 a} x +\frac {{\mathrm e}^{-a -c} {\mathrm e}^{2 c} a}{b}-\frac {{\mathrm e}^{-a -c} {\mathrm e}^{2 a} a}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 84, normalized size = 3.23 \[ \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac {{\left (b x + a\right )} e^{\left (a - c\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 65, normalized size = 2.50 \[ x\,{\mathrm {e}}^{c-a}+\frac {{\mathrm {e}}^{2\,c-2\,a}\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )\,\left (2\,b\,{\mathrm {e}}^{3\,a-3\,c}-2\,b\,{\mathrm {e}}^{a-c}\right )}{4\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\left (a + b x \right )} \operatorname {csch}{\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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