Optimal. Leaf size=55 \[ \frac{\tan ^{-1}(\sinh (a+b x))}{8 b}-\frac{\tanh (a+b x) \text{sech}^3(a+b x)}{4 b}+\frac{\tanh (a+b x) \text{sech}(a+b x)}{8 b} \]
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Rubi [A] time = 0.0451404, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2611, 3768, 3770} \[ \frac{\tan ^{-1}(\sinh (a+b x))}{8 b}-\frac{\tanh (a+b x) \text{sech}^3(a+b x)}{4 b}+\frac{\tanh (a+b x) \text{sech}(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \text{sech}^3(a+b x) \tanh ^2(a+b x) \, dx &=-\frac{\text{sech}^3(a+b x) \tanh (a+b x)}{4 b}+\frac{1}{4} \int \text{sech}^3(a+b x) \, dx\\ &=\frac{\text{sech}(a+b x) \tanh (a+b x)}{8 b}-\frac{\text{sech}^3(a+b x) \tanh (a+b x)}{4 b}+\frac{1}{8} \int \text{sech}(a+b x) \, dx\\ &=\frac{\tan ^{-1}(\sinh (a+b x))}{8 b}+\frac{\text{sech}(a+b x) \tanh (a+b x)}{8 b}-\frac{\text{sech}^3(a+b x) \tanh (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0201913, size = 55, normalized size = 1. \[ \frac{\tan ^{-1}(\sinh (a+b x))}{8 b}-\frac{\tanh (a+b x) \text{sech}^3(a+b x)}{4 b}+\frac{\tanh (a+b x) \text{sech}(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 69, normalized size = 1.3 \begin{align*} -{\frac{\sinh \left ( bx+a \right ) }{3\,b \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}}+{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{3}\tanh \left ( bx+a \right ) }{12\,b}}+{\frac{{\rm sech} \left (bx+a\right )\tanh \left ( bx+a \right ) }{8\,b}}+{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.8487, size = 149, normalized size = 2.71 \begin{align*} -\frac{\arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} + \frac{e^{\left (-b x - a\right )} - 7 \, e^{\left (-3 \, b x - 3 \, a\right )} + 7 \, e^{\left (-5 \, b x - 5 \, a\right )} - e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b{\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87445, size = 2236, normalized size = 40.65 \begin{align*} \text{result too large to display} \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 \tanh ^{2}{\left (a + b x \right )} \operatorname{sech}^{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 = 1.21273, size = 90, normalized size = 1.64 \begin{align*} \frac{\frac{e^{\left (7 \, b x + 7 \, a\right )} - 7 \, e^{\left (5 \, b x + 5 \, a\right )} + 7 \, e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{4}} + \arctan \left (e^{\left (b x + a\right )}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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