Optimal. Leaf size=79 \[ -\frac{i \text{PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}-\frac{1}{2} i x \log \left (1-i c e^{2 a+2 b x}\right )+x \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{2} i b x^2 \]
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Rubi [A] time = 0.123531, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5190, 2184, 2190, 2279, 2391} \[ -\frac{i \text{PolyLog}\left (2,i c e^{2 a+2 b x}\right )}{4 b}-\frac{1}{2} i x \log \left (1-i c e^{2 a+2 b x}\right )+x \cot ^{-1}(c+(c+i) \coth (a+b x))+\frac{1}{2} i b x^2 \]
Antiderivative was successfully verified.
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Rule 5190
Rule 2184
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \cot ^{-1}(c+(i+c) \coth (a+b x)) \, dx &=x \cot ^{-1}(c+(i+c) \coth (a+b x))+b \int \frac{x}{-i-c e^{2 a+2 b x}} \, dx\\ &=\frac{1}{2} i b x^2+x \cot ^{-1}(c+(i+c) \coth (a+b x))+(i b c) \int \frac{e^{2 a+2 b x} x}{-i-c e^{2 a+2 b x}} \, dx\\ &=\frac{1}{2} i b x^2+x \cot ^{-1}(c+(i+c) \coth (a+b x))-\frac{1}{2} i x \log \left (1-i c e^{2 a+2 b x}\right )+\frac{1}{2} i \int \log \left (1-i c e^{2 a+2 b x}\right ) \, dx\\ &=\frac{1}{2} i b x^2+x \cot ^{-1}(c+(i+c) \coth (a+b x))-\frac{1}{2} i x \log \left (1-i c e^{2 a+2 b x}\right )+\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i c x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=\frac{1}{2} i b x^2+x \cot ^{-1}(c+(i+c) \coth (a+b x))-\frac{1}{2} i x \log \left (1-i c e^{2 a+2 b x}\right )-\frac{i \text{Li}_2\left (i c e^{2 a+2 b x}\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.623475, size = 71, normalized size = 0.9 \[ x \cot ^{-1}(c+(c+i) \coth (a+b x))-\frac{i \left (2 b x \log \left (1+\frac{i e^{-2 (a+b x)}}{c}\right )-\text{PolyLog}\left (2,-\frac{i e^{-2 (a+b x)}}{c}\right )\right )}{4 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.121, size = 1381, normalized size = 17.5 \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 [A] time = 5.90107, size = 108, normalized size = 1.37 \begin{align*} -2 \, b{\left (c + i\right )}{\left (\frac{2 \, x^{2}}{2 i \, c - 2} - \frac{2 \, b x \log \left (-i \, c e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) +{\rm Li}_2\left (i \, c e^{\left (2 \, b x + 2 \, a\right )}\right )}{-2 \, b^{2}{\left (-i \, c + 1\right )}}\right )} + x \operatorname{arccot}\left ({\left (c + i\right )} \coth \left (b x + a\right ) + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.21941, size = 512, normalized size = 6.48 \begin{align*} \frac{i \, b^{2} x^{2} + i \, b x \log \left (\frac{{\left (c e^{\left (2 \, b x + 2 \, a\right )} + i\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{c + i}\right ) - i \, a^{2} +{\left (-i \, b x - i \, a\right )} \log \left (\frac{1}{2} \, \sqrt{4 i \, c} e^{\left (b x + a\right )} + 1\right ) +{\left (-i \, b x - i \, a\right )} \log \left (-\frac{1}{2} \, \sqrt{4 i \, c} e^{\left (b x + a\right )} + 1\right ) + i \, a \log \left (\frac{2 \, c e^{\left (b x + a\right )} + i \, \sqrt{4 i \, c}}{2 \, c}\right ) + i \, a \log \left (\frac{2 \, c e^{\left (b x + a\right )} - i \, \sqrt{4 i \, c}}{2 \, c}\right ) - i \,{\rm Li}_2\left (\frac{1}{2} \, \sqrt{4 i \, c} e^{\left (b x + a\right )}\right ) - i \,{\rm Li}_2\left (-\frac{1}{2} \, \sqrt{4 i \, c} e^{\left (b x + a\right )}\right )}{2 \, b} \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*} - b \left (c^{3} + 3 i c^{2} - 3 c - i\right ) \int \frac{x}{c^{4} e^{2 a} e^{2 b x} + 3 i c^{3} e^{2 a} e^{2 b x} + i c^{3} - 3 c^{2} e^{2 a} e^{2 b x} - 3 c^{2} - i c e^{2 a} e^{2 b x} - 3 i c + 1}\, dx + \frac{i x \log{\left (1 - \frac{i}{c + \frac{c}{e^{2 a} e^{2 b x} - 1} + \frac{c e^{a} e^{b x}}{e^{a} e^{b x} - e^{- a} e^{- b x}} + \frac{i}{e^{2 a} e^{2 b x} - 1} + \frac{i e^{a} e^{b x}}{e^{a} e^{b x} - e^{- a} e^{- b x}}} \right )}}{2} - \frac{i x \log{\left (1 + \frac{i}{c + \frac{c}{e^{2 a} e^{2 b x} - 1} + \frac{c e^{a} e^{b x}}{e^{a} e^{b x} - e^{- a} e^{- b x}} + \frac{i}{e^{2 a} e^{2 b x} - 1} + \frac{i e^{a} e^{b x}}{e^{a} e^{b x} - e^{- a} e^{- b x}}} \right )}}{2} \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 \operatorname{arccot}\left ({\left (c + i\right )} \coth \left (b x + a\right ) + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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