Optimal. Leaf size=61 \[ -\frac {i d \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}+\frac {i d \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {2 (c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2279
Rule 2391
Rule 4180
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*}
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Mathematica [B] time = 0.15, size = 127, normalized size = 2.08 \[ \frac {b c \tan ^{-1}(\sinh (a+b x))+\frac {1}{2} d \left (-2 i \left (\text {Li}_2\left (-i e^{a+b x}\right )-\text {Li}_2\left (i e^{a+b x}\right )\right )-\left ((-2 i a-2 i b x+\pi ) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )\right )+(\pi -2 i a) \log \left (\cot \left (\frac {1}{4} (2 i a+2 i b x+\pi )\right )\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 157, normalized size = 2.57 \[ \frac {i \, d {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - i \, d {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\left (i \, b c - i \, a d\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + {\left (-i \, b c + i \, a d\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + {\left (-i \, b d x - i \, a d\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + {\left (i \, b d x + i \, a d\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \operatorname {sech}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 147, normalized size = 2.41 \[ \frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{b}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {i d \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {i d \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 c \arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, d \int \frac {x e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} + 1}\,{d x} - \frac {2 \, c \arctan \left (e^{\left (-b x - a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {c+d\,x}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right ) \operatorname {sech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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