Optimal. Leaf size=29 \[ \frac{(c+d x) \tanh (a+b x)}{b}-\frac{d \log (\cosh (a+b x))}{b^2} \]
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Rubi [A] time = 0.0303165, antiderivative size = 29, 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 = {4184, 3475} \[ \frac{(c+d x) \tanh (a+b x)}{b}-\frac{d \log (\cosh (a+b x))}{b^2} \]
Antiderivative was successfully verified.
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Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int (c+d x) \text{sech}^2(a+b x) \, dx &=\frac{(c+d x) \tanh (a+b x)}{b}-\frac{d \int \tanh (a+b x) \, dx}{b}\\ &=-\frac{d \log (\cosh (a+b x))}{b^2}+\frac{(c+d x) \tanh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0829271, size = 51, normalized size = 1.76 \[ -\frac{d \log (\cosh (a+b x))}{b^2}+\frac{c \tanh (a+b x)}{b}+\frac{d x \tanh (a)}{b}+\frac{d x \text{sech}(a) \sinh (b x) \text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 57, normalized size = 2. \begin{align*} 2\,{\frac{dx}{b}}+2\,{\frac{da}{{b}^{2}}}-2\,{\frac{dx+c}{b \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }}-{\frac{d\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08853, size = 97, normalized size = 3.34 \begin{align*} d{\left (\frac{2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b} - \frac{\log \left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}\right )}{b^{2}}\right )} + \frac{2 \, c}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35126, size = 429, normalized size = 14.79 \begin{align*} \frac{2 \, b d x \cosh \left (b x + a\right )^{2} + 4 \, b d x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 2 \, b d x \sinh \left (b x + a\right )^{2} - 2 \, b c -{\left (d \cosh \left (b x + a\right )^{2} + 2 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d \sinh \left (b x + a\right )^{2} + d\right )} \log \left (\frac{2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\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 \left (c + d 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 = 1.15591, size = 105, normalized size = 3.62 \begin{align*} \frac{2 \, b d x e^{\left (2 \, b x + 2 \, a\right )} - d e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - 2 \, b c - d \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\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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