Optimal. Leaf size=61 \[ -\frac{i d \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac{i d \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac{2 (c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.037233, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4180, 2279, 2391} \[ -\frac{i d \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac{i d \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac{2 (c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4180
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int (c+d x) \text{sech}(a+b x) \, dx &=\frac{2 (c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{(i d) \int \log \left (1-i e^{a+b x}\right ) \, dx}{b}+\frac{(i d) \int \log \left (1+i e^{a+b x}\right ) \, dx}{b}\\ &=\frac{2 (c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{(i d) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac{(i d) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}\\ &=\frac{2 (c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{i d \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}+\frac{i d \text{Li}_2\left (i e^{a+b x}\right )}{b^2}\\ \end{align*}
Mathematica [B] time = 0.0792382, size = 129, normalized size = 2.11 \[ \frac{-i d \left (\text{PolyLog}\left (2,-i e^{a+b x}\right )-\text{PolyLog}\left (2,i e^{a+b x}\right )\right )+b c \tan ^{-1}(\sinh (a+b x))-\frac{1}{2} d (-2 i a-2 i b x+\pi ) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )+\frac{1}{2} (\pi -2 i a) d \log \left (\cot \left (\frac{1}{4} (2 i a+2 i b x+\pi )\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 449, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, d \int \frac{x e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} + 1}\,{d x} - \frac{2 \, c \arctan \left (e^{\left (-b x - a\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1461, size = 463, normalized size = 7.59 \begin{align*} \frac{i \, d{\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - i \, d{\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) +{\left (i \, b c - i \, a d\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) +{\left (-i \, b c + i \, a d\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) +{\left (-i \, b d x - i \, a d\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) +{\left (i \, b d x + i \, a d\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right )}{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}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \operatorname{sech}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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