Optimal. Leaf size=36 \[ -\frac {\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}-\frac {\sinh (a-c) \text {csch}(b x+c)}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {5626, 2606, 8, 3770} \[ -\frac {\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}-\frac {\sinh (a-c) \text {csch}(b x+c)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 3770
Rule 5626
Rubi steps
\begin {align*} \int \text {csch}^2(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int \text {csch}(c+b x) \, dx+\sinh (a-c) \int \coth (c+b x) \text {csch}(c+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\cosh (c+b x)) \cosh (a-c)}{b}-\frac {(i \sinh (a-c)) \operatorname {Subst}(\int 1 \, dx,x,-i \text {csch}(c+b x))}{b}\\ &=-\frac {\tanh ^{-1}(\cosh (c+b x)) \cosh (a-c)}{b}-\frac {\text {csch}(c+b x) \sinh (a-c)}{b}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 90, normalized size = 2.50 \[ -\frac {\sinh (a-c) \text {csch}(b x+c)}{b}-\frac {2 i \cosh (a-c) \tan ^{-1}\left (\frac {(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac {b x}{2}\right )+\cosh (c) \cosh \left (\frac {b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac {b x}{2}\right )-i \sinh (c) \cosh \left (\frac {b x}{2}\right )}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 617, normalized size = 17.14 \[ \frac {4 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - 2 \, \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - 2 \, {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right ) - {\left ({\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )^{2} + {\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 )} \sinh \left (b x + c\right )^{2} + {\left (\cosh \left (b x + c\right )^{2} - 1\right )} \sinh \left (-a + c\right )^{2} - \cosh \left (-a + c\right )^{2} - 2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 2 \, {\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) - \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) - 1\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) + 1\right ) + {\left ({\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )^{2} + {\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 )} \sinh \left (b x + c\right )^{2} + {\left (\cosh \left (b x + c\right )^{2} - 1\right )} \sinh \left (-a + c\right )^{2} - \cosh \left (-a + c\right )^{2} - 2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 2 \, {\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) - \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) - 1\right )} \log \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right ) - 1\right ) - 2 \, {\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 )} \sinh \left (b x + c\right )}{2 \, {\left (b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) + {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )^{2} - b \cosh \left (-a + c\right ) + 2 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) - {\left (b \cosh \left (b x + c\right )^{2} - b\right )} \sinh \left (-a + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 104, normalized size = 2.89 \[ -\frac {{\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right ) - {\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (e^{\left (b x + 2 \, a\right )} - e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{e^{\left (2 \, b x + 2 \, c\right )} - 1}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 172, normalized size = 4.78 \[ \frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right )}{b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 103, normalized size = 2.86 \[ -\frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-b x - a\right )}}{b {\left (e^{\left (-2 \, b x\right )} - e^{\left (2 \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 156, normalized size = 4.33 \[ \frac {{\mathrm {e}}^{a+b\,x}\,\left ({\mathrm {e}}^{2\,a-2\,c}-1\right )}{b\,\left ({\mathrm {e}}^{2\,a-2\,c}-{\mathrm {e}}^{2\,a+2\,b\,x}\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {-b^2}+{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {-b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left (2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left (2\,{\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{4\,a-4\,c}+1\right )}}{\sqrt {-b^2}} \]
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 \[ \int \sinh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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