Optimal. Leaf size=24 \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac{x \text{sech}(a+b x)}{b} \]
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Rubi [A] time = 0.0190941, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5418, 3770} \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac{x \text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5418
Rule 3770
Rubi steps
\begin{align*} \int x \text{sech}(a+b x) \tanh (a+b x) \, dx &=-\frac{x \text{sech}(a+b x)}{b}+\frac{\int \text{sech}(a+b x) \, dx}{b}\\ &=\frac{\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac{x \text{sech}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0457742, size = 32, normalized size = 1.33 \[ \frac{2 \tan ^{-1}\left (\tanh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b^2}-\frac{x \text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.039, size = 59, normalized size = 2.5 \begin{align*} -2\,{\frac{x{{\rm e}^{bx+a}}}{b \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }}+{\frac{i\ln \left ({{\rm e}^{bx+a}}+i \right ) }{{b}^{2}}}-{\frac{i\ln \left ({{\rm e}^{bx+a}}-i \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78737, size = 50, normalized size = 2.08 \begin{align*} -\frac{2 \, x e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b} + \frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77433, size = 327, normalized size = 13.62 \begin{align*} -\frac{2 \,{\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right ) -{\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} + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} + b^{2}} \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 x \sinh{\left (a + b x \right )} \operatorname{sech}^{2}{\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 = 1.31225, size = 95, normalized size = 3.96 \begin{align*} -\frac{2 \,{\left (\pi + b x e^{\left (b x + a\right )} + \pi e^{\left (2 \, b x + 2 \, a\right )} - \arctan \left (e^{\left (b x + a\right )}\right ) e^{\left (2 \, b x + 2 \, a\right )} - \arctan \left (e^{\left (b x + a\right )}\right )\right )}}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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