Optimal. Leaf size=73 \[ -\frac{i \text{PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}+\frac{i \text{PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}+x \tan ^{-1}\left (e^{2 a+2 b x}\right )+x \tan ^{-1}(\coth (a+b x)) \]
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Rubi [A] time = 0.043107, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5181, 4180, 2279, 2391} \[ -\frac{i \text{PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}+\frac{i \text{PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}+x \tan ^{-1}\left (e^{2 a+2 b x}\right )+x \tan ^{-1}(\coth (a+b x)) \]
Antiderivative was successfully verified.
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Rule 5181
Rule 4180
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \tan ^{-1}(\coth (a+b x)) \, dx &=x \tan ^{-1}(\coth (a+b x))+b \int x \text{sech}(2 a+2 b x) \, dx\\ &=x \tan ^{-1}\left (e^{2 a+2 b x}\right )+x \tan ^{-1}(\coth (a+b x))-\frac{1}{2} i \int \log \left (1-i e^{2 a+2 b x}\right ) \, dx+\frac{1}{2} i \int \log \left (1+i e^{2 a+2 b x}\right ) \, dx\\ &=x \tan ^{-1}\left (e^{2 a+2 b x}\right )+x \tan ^{-1}(\coth (a+b x))-\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}+\frac{i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=x \tan ^{-1}\left (e^{2 a+2 b x}\right )+x \tan ^{-1}(\coth (a+b x))-\frac{i \text{Li}_2\left (-i e^{2 a+2 b x}\right )}{4 b}+\frac{i \text{Li}_2\left (i e^{2 a+2 b x}\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0546063, size = 132, normalized size = 1.81 \[ x \tan ^{-1}(\coth (a+b x))+\frac{-2 i \left (\text{PolyLog}\left (2,-i e^{2 (a+b x)}\right )-\text{PolyLog}\left (2,i e^{2 (a+b x)}\right )\right )-(-4 i a-4 i b x+\pi ) \left (\log \left (1-i e^{2 (a+b x)}\right )-\log \left (1+i e^{2 (a+b x)}\right )\right )+(\pi -4 i a) \log \left (\cot \left (\frac{1}{4} (4 i a+4 i b x+\pi )\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.118, size = 440, normalized size = 6. \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*} x \arctan \left (e^{\left (2 \, b x + 2 \, a\right )} + 1, e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, b \int \frac{x e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (4 \, b x + 4 \, a\right )} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.44648, size = 1098, normalized size = 15.04 \begin{align*} \frac{2 \, b x \arctan \left (\frac{\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right ) +{\left (i \, b x + i \, a\right )} \log \left (\frac{1}{2} \, \sqrt{4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) +{\left (i \, b x + i \, a\right )} \log \left (-\frac{1}{2} \, \sqrt{4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) +{\left (-i \, b x - i \, a\right )} \log \left (\frac{1}{2} \, \sqrt{-4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) +{\left (-i \, b x - i \, a\right )} \log \left (-\frac{1}{2} \, \sqrt{-4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - i \, a \log \left (i \, \sqrt{4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) - i \, a \log \left (-i \, \sqrt{4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + i \, a \log \left (i \, \sqrt{-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + i \, a \log \left (-i \, \sqrt{-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + i \,{\rm Li}_2\left (\frac{1}{2} \, \sqrt{4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + i \,{\rm Li}_2\left (-\frac{1}{2} \, \sqrt{4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - i \,{\rm Li}_2\left (\frac{1}{2} \, \sqrt{-4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - i \,{\rm Li}_2\left (-\frac{1}{2} \, \sqrt{-4 i}{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\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*} \int \operatorname{atan}{\left (\coth{\left (a + b x \right )} \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 \arctan \left (\coth \left (b x + a\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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