Optimal. Leaf size=49 \[ \frac{3 \sinh (a+b x)}{2 b}-\frac{3 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{\sinh (a+b x) \tanh ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0297894, antiderivative size = 49, normalized size of antiderivative = 1., 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 = {2592, 288, 321, 203} \[ \frac{3 \sinh (a+b x)}{2 b}-\frac{3 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{\sinh (a+b x) \tanh ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2592
Rule 288
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \sinh (a+b x) \tanh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac{\sinh (a+b x) \tanh ^2(a+b x)}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{2 b}\\ &=\frac{3 \sinh (a+b x)}{2 b}-\frac{\sinh (a+b x) \tanh ^2(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{2 b}\\ &=-\frac{3 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac{3 \sinh (a+b x)}{2 b}-\frac{\sinh (a+b x) \tanh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0537199, size = 48, normalized size = 0.98 \[ \frac{\sinh (a+b x) \tanh ^2(a+b x)}{b}-\frac{3 \left (\tan ^{-1}(\sinh (a+b x))-\tanh (a+b x) \text{sech}(a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 70, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{b \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}+3\,{\frac{\sinh \left ( bx+a \right ) }{b \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}-{\frac{3\,{\rm sech} \left (bx+a\right )\tanh \left ( bx+a \right ) }{2\,b}}-3\,{\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 [B] time = 1.59881, size = 123, normalized size = 2.51 \begin{align*} \frac{3 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} + \frac{4 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + 1}{2 \, b{\left (e^{\left (-b x - a\right )} + 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96743, size = 1289, normalized size = 26.31 \begin{align*} \frac{\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 3 \,{\left (5 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{4} + 3 \, \cosh \left (b x + a\right )^{4} + 4 \,{\left (5 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \,{\left (5 \, \cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 6 \,{\left (\cosh \left (b x + a\right )^{5} + 5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{5} + 2 \,{\left (5 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )^{3} + 2 \,{\left (5 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} +{\left (5 \, \cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 3 \, \cosh \left (b x + a\right )^{2} + 6 \,{\left (\cosh \left (b x + a\right )^{5} + 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1}{2 \,{\left (b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} + 2 \, b \cosh \left (b x + a\right )^{3} + 2 \,{\left (5 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{3} + 2 \,{\left (5 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + b \cosh \left (b x + a\right ) +{\left (5 \, b \cosh \left (b x + a\right )^{4} + 6 \, b \cosh \left (b x + a\right )^{2} + b\right )} \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{\left (a + b x \right )} \tanh ^{3}{\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 = 1.20499, size = 88, normalized size = 1.8 \begin{align*} \frac{\frac{2 \,{\left (e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} - 6 \, \arctan \left (e^{\left (b x + a\right )}\right ) + e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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