Optimal. Leaf size=30 \[ \frac{\tanh (a+b x)}{2 b^2}-\frac{x \text{sech}^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0304105, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5418, 3767, 8} \[ \frac{\tanh (a+b x)}{2 b^2}-\frac{x \text{sech}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5418
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int x \text{sech}^2(a+b x) \tanh (a+b x) \, dx &=-\frac{x \text{sech}^2(a+b x)}{2 b}+\frac{\int \text{sech}^2(a+b x) \, dx}{2 b}\\ &=-\frac{x \text{sech}^2(a+b x)}{2 b}+\frac{i \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (a+b x))}{2 b^2}\\ &=-\frac{x \text{sech}^2(a+b x)}{2 b}+\frac{\tanh (a+b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.072022, size = 30, normalized size = 1. \[ \frac{\tanh (a+b x)}{2 b^2}-\frac{x \text{sech}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 43, normalized size = 1.4 \begin{align*} -{\frac{2\,bx{{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,bx+2\,a}}+1}{{b}^{2} \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06165, size = 177, normalized size = 5.9 \begin{align*} -\frac{2 \, b x e^{\left (4 \, b x + 4 \, a\right )} +{\left (4 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 1}{2 \,{\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} + \frac{2 \, b x e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (2 \, b x + 2 \, a\right )} - 1}{2 \,{\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75086, size = 270, normalized size = 9. \begin{align*} -\frac{2 \,{\left (b x \sinh \left (b x + a\right ) +{\left (b x + 1\right )} \cosh \left (b x + a\right )\right )}}{b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) +{\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} + b^{2}\right )} \sinh \left (b x + a\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 x \sinh{\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 [B] time = 1.25154, size = 248, normalized size = 8.27 \begin{align*} -\frac{4 \, b x e^{\left (2 \, b x + 2 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + e^{\left (4 \, b x + 4 \, a\right )} \log \left (-e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (-e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, e^{\left (2 \, b x + 2 \, a\right )} - \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + \log \left (-e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2}{2 \,{\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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