Optimal. Leaf size=51 \[ -\frac {i \text {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i \text {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2320, 4941,
2438} \begin {gather*} \frac {i \text {Li}_2\left (i e^{-a-b x}\right )}{2 b}-\frac {i \text {Li}_2\left (-i e^{-a-b x}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2438
Rule 4941
Rubi steps
\begin {align*} \int \cot ^{-1}\left (e^{a+b x}\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {\cot ^{-1}(x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{x}\right )}{x} \, dx,x,e^{a+b x}\right )}{2 b}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{x}\right )}{x} \, dx,x,e^{a+b x}\right )}{2 b}\\ &=-\frac {i \text {Li}_2\left (-i e^{-a-b x}\right )}{2 b}+\frac {i \text {Li}_2\left (i e^{-a-b x}\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 83, normalized size = 1.63 \begin {gather*} x \cot ^{-1}\left (e^{a+b x}\right )+\frac {i \left (b x \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )-\text {PolyLog}\left (2,-i e^{a+b x}\right )+\text {PolyLog}\left (2,i e^{a+b x}\right )\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 94 vs. \(2 (41 ) = 82\).
time = 0.10, size = 95, normalized size = 1.86
method | result | size |
derivativedivides | \(\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \mathrm {arccot}\left ({\mathrm e}^{b x +a}\right )-\frac {i \ln \left ({\mathrm e}^{b x +a}\right ) \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2}+\frac {i \ln \left ({\mathrm e}^{b x +a}\right ) \ln \left (1-i {\mathrm e}^{b x +a}\right )}{2}-\frac {i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2}}{b}\) | \(95\) |
default | \(\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \mathrm {arccot}\left ({\mathrm e}^{b x +a}\right )-\frac {i \ln \left ({\mathrm e}^{b x +a}\right ) \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2}+\frac {i \ln \left ({\mathrm e}^{b x +a}\right ) \ln \left (1-i {\mathrm e}^{b x +a}\right )}{2}-\frac {i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2}}{b}\) | \(95\) |
risch | \(\frac {i x \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2}+\frac {\pi x}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2 b}+\frac {i \ln \left (-i {\mathrm e}^{b x +a}\right ) \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right )}{2 b}-\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) x}{2}-\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a}{2 b}+\frac {i \dilog \left (-i {\mathrm e}^{b x +a}\right )}{2 b}+\frac {i a \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2 b}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 63, normalized size = 1.24 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {arccot}\left (e^{\left (b x + a\right )}\right )}{b} + \frac {\pi \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 i \, {\rm Li}_2\left (i \, e^{\left (b x + a\right )} + 1\right ) - 2 i \, {\rm Li}_2\left (-i \, e^{\left (b x + a\right )} + 1\right )}{4 \, b} \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. 103 vs. \(2 (35) = 70\).
time = 5.38, size = 103, normalized size = 2.02 \begin {gather*} \frac {2 \, b x \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) - i \, a \log \left (e^{\left (b x + a\right )} + i\right ) + i \, a \log \left (e^{\left (b x + a\right )} - i\right ) + {\left (-i \, b x - i \, a\right )} \log \left (i \, e^{\left (b x + a\right )} + 1\right ) + {\left (i \, b x + i \, a\right )} \log \left (-i \, e^{\left (b x + a\right )} + 1\right ) + i \, {\rm Li}_2\left (i \, e^{\left (b x + a\right )}\right ) - i \, {\rm Li}_2\left (-i \, e^{\left (b x + a\right )}\right )}{2 \, b} \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 \operatorname {acot}{\left (e^{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 \mathrm {acot}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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