Optimal. Leaf size=29 \[ \frac {\cosh (a+b x)}{b}-\frac {\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 29, 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 = {5623, 2638, 3770} \[ \frac {\cosh (a+b x)}{b}-\frac {\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3770
Rule 5623
Rubi steps
\begin {align*} \int \cosh (a+b x) \tanh (c+b x) \, dx &=-(\sinh (a-c) \int \text {sech}(c+b x) \, dx)+\int \sinh (a+b x) \, dx\\ &=\frac {\cosh (a+b x)}{b}-\frac {\tan ^{-1}(\sinh (c+b x)) \sinh (a-c)}{b}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 86, normalized size = 2.97 \[ -\frac {2 \sinh (a-c) \tan ^{-1}\left (\frac {(\cosh (c)-\sinh (c)) \left (\sinh (c) \cosh \left (\frac {b x}{2}\right )+\cosh (c) \sinh \left (\frac {b x}{2}\right )\right )}{\cosh (c) \cosh \left (\frac {b x}{2}\right )-\sinh (c) \cosh \left (\frac {b x}{2}\right )}\right )}{b}+\frac {\sinh (a) \sinh (b x)}{b}+\frac {\cosh (a) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 327, normalized size = 11.28 \[ \frac {\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right )^{2} + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} + 2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right ) - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )\right )} \arctan \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right )\right ) + 2 \, {\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right ) + 1}{2 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right ) + {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 53, normalized size = 1.83 \[ -\frac {2 \, {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 167, normalized size = 5.76 \[ \frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 59, normalized size = 2.03 \[ \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{b} + \frac {e^{\left (b x + a\right )}}{2 \, b} + \frac {e^{\left (-b x - a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 133, normalized size = 4.59 \[ \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {b^2}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}-2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left ({\mathrm {e}}^{4\,a-4\,c}-2\,{\mathrm {e}}^{2\,a-2\,c}+1\right )}}{\sqrt {b^2}} \]
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 \[ \int \cosh {\left (a + b x \right )} \tanh {\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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