Optimal. Leaf size=23 \[ \frac{\sinh (a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0154926, antiderivative size = 23, normalized size of antiderivative = 1., 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 = {2592, 321, 203} \[ \frac{\sinh (a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2592
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \sinh (a+b x) \tanh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac{\sinh (a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac{\tan ^{-1}(\sinh (a+b x))}{b}+\frac{\sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0116213, size = 23, normalized size = 1. \[ \frac{\sinh (a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 24, normalized size = 1. \begin{align*}{\frac{\sinh \left ( bx+a \right ) }{b}}-2\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6059, size = 55, normalized size = 2.39 \begin{align*} \frac{2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac{e^{\left (b x + a\right )}}{2 \, b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08583, size = 254, normalized size = 11.04 \begin{align*} -\frac{4 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - \cosh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2} + 1}{2 \,{\left (b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )\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 \sinh ^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16673, size = 51, normalized size = 2.22 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} + \frac{e^{\left (b x + a\right )}}{2 \, b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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