Optimal. Leaf size=61 \[ \frac {2 (c+d x) \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {i d \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {i d \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^2} \]
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Rubi [A]
time = 0.03, 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 = {4265, 2317,
2438} \begin {gather*} \frac {2 (c+d x) \text {ArcTan}\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 {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4265
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) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac {(i d) \text {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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(127\) vs. \(2(61)=122\).
time = 0.12, size = 127, normalized size = 2.08 \begin {gather*} \frac {b c \text {ArcTan}(\sinh (a+b x))+\frac {1}{2} d \left (-\left ((-2 i a+\pi -2 i b x) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )\right )+(-2 i a+\pi ) \log \left (\cot \left (\frac {1}{4} (2 i a+\pi +2 i b x)\right )\right )-2 i \left (\text {PolyLog}\left (2,-i e^{a+b x}\right )-\text {PolyLog}\left (2,i e^{a+b x}\right )\right )\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 101, normalized size = 1.66
method | result | size |
derivativedivides | \(\frac {\frac {d \left (i \left (b x +a \right ) \left (\ln \left (1-i {\mathrm e}^{b x +a}\right )-\ln \left (1+i {\mathrm e}^{b x +a}\right )\right )-i \dilog \left (1+i {\mathrm e}^{b x +a}\right )+i \dilog \left (1-i {\mathrm e}^{b x +a}\right )\right )}{b}-\frac {2 d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b}+2 c \arctan \left ({\mathrm e}^{b x +a}\right )}{b}\) | \(101\) |
default | \(\frac {\frac {d \left (i \left (b x +a \right ) \left (\ln \left (1-i {\mathrm e}^{b x +a}\right )-\ln \left (1+i {\mathrm e}^{b x +a}\right )\right )-i \dilog \left (1+i {\mathrm e}^{b x +a}\right )+i \dilog \left (1-i {\mathrm e}^{b x +a}\right )\right )}{b}-\frac {2 d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b}+2 c \arctan \left ({\mathrm e}^{b x +a}\right )}{b}\) | \(101\) |
risch | \(\frac {2 c \arctan \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{b}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {i d \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {i d \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(147\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 157 vs. \(2 (48) = 96\).
time = 0.39, size = 157, normalized size = 2.57 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \left (c + d x\right ) \operatorname {sech}{\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {c+d\,x}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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