Optimal. Leaf size=28 \[ \frac{\cosh ^2(a+b x)}{2 b}-\frac{\log (\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.024513, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2590, 14} \[ \frac{\cosh ^2(a+b x)}{2 b}-\frac{\log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 14
Rubi steps
\begin{align*} \int \sinh ^2(a+b x) \tanh (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{x} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x}-x\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac{\cosh ^2(a+b x)}{2 b}-\frac{\log (\cosh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0174879, size = 25, normalized size = 0.89 \[ -\frac{\log (\cosh (a+b x))-\frac{1}{2} \cosh ^2(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{2\,b}}-{\frac{\ln \left ( \cosh \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.56552, size = 76, normalized size = 2.71 \begin{align*} -\frac{b x + a}{b} + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac{\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.077, size = 551, normalized size = 19.68 \begin{align*} \frac{8 \, b x \cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (4 \, b x + 3 \, \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - 8 \,{\left (\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}\right )} \log \left (\frac{2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \,{\left (4 \, b x \cosh \left (b x + a\right ) + \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right ) + 1}{8 \,{\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\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 ^{3}{\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 [B] time = 1.17456, size = 92, normalized size = 3.29 \begin{align*} -\frac{{\left (4 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} + \frac{b x + a}{b} + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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