Optimal. Leaf size=27 \[ \frac{\text{sech}^3(a+b x)}{3 b}-\frac{\text{sech}(a+b x)}{b} \]
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Rubi [A] time = 0.0244432, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2606} \[ \frac{\text{sech}^3(a+b x)}{3 b}-\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2606
Rubi steps
\begin{align*} \int \text{sech}(a+b x) \tanh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=-\frac{\text{sech}(a+b x)}{b}+\frac{\text{sech}^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0311728, size = 27, normalized size = 1. \[ \frac{\text{sech}^3(a+b x)}{3 b}-\frac{\text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 50, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}}+{\frac{2\, \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3\,\cosh \left ( bx+a \right ) }}-{\frac{2\,\cosh \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02041, size = 200, normalized size = 7.41 \begin{align*} -\frac{2 \, e^{\left (-b x - a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} - \frac{4 \, e^{\left (-3 \, b x - 3 \, a\right )}}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} - \frac{2 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74834, size = 467, normalized size = 17.3 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, \sinh \left (b x + a\right )^{3} +{\left (9 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) + 5 \, \cosh \left (b x + a\right )\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.28128, size = 41, normalized size = 1.52 \begin{align*} \begin{cases} - \frac{\tanh ^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}}{3 b} - \frac{2 \operatorname{sech}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x \tanh ^{3}{\left (a \right )} \operatorname{sech}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24873, size = 66, normalized size = 2.44 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} + 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (b x + a\right )}\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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