Optimal. Leaf size=73 \[ -\frac {i \text {Li}_2\left (-i e^{2 a+2 b x}\right )}{4 b}+\frac {i \text {Li}_2\left (i e^{2 a+2 b x}\right )}{4 b}+x \tan ^{-1}\left (e^{2 a+2 b x}\right )+x \cot ^{-1}(\tanh (a+b x)) \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 = {5180, 4180, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}+\frac {i \text {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}+x \tan ^{-1}\left (e^{2 a+2 b x}\right )+x \cot ^{-1}(\tanh (a+b x)) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4180
Rule 5180
Rubi steps
\begin {align*} \int \cot ^{-1}(\tanh (a+b x)) \, dx &=x \cot ^{-1}(\tanh (a+b x))+b \int x \text {sech}(2 a+2 b x) \, dx\\ &=x \cot ^{-1}(\tanh (a+b x))+x \tan ^{-1}\left (e^{2 a+2 b x}\right )-\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 \cot ^{-1}(\tanh (a+b x))+x \tan ^{-1}\left (e^{2 a+2 b x}\right )-\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 \cot ^{-1}(\tanh (a+b x))+x \tan ^{-1}\left (e^{2 a+2 b x}\right )-\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*}
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Mathematica [A] time = 0.06, size = 132, normalized size = 1.81 \[ x \cot ^{-1}(\tanh (a+b x))+\frac {-2 i \left (\text {Li}_2\left (-i e^{2 (a+b x)}\right )-\text {Li}_2\left (i e^{2 (a+b x)}\right )\right )-\left ((-4 i a-4 i b x+\pi ) \left (\log \left (1-i e^{2 (a+b x)}\right )-\log \left (1+i e^{2 (a+b x)}\right )\right )\right )+(\pi -4 i a) \log \left (\cot \left (\frac {1}{4} (4 i a+4 i b x+\pi )\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 334, normalized size = 4.58 \[ \frac {2 \, b x \arctan \left (\frac {\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right ) + {\left (i \, b x + i \, a\right )} \log \left (\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (i \, b x + i \, a\right )} \log \left (-\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (-i \, b x - i \, a\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (-i \, b x - i \, a\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - i \, a \log \left (i \, \sqrt {4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) - i \, a \log \left (-i \, \sqrt {4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + i \, a \log \left (i \, \sqrt {-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + i \, a \log \left (-i \, \sqrt {-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + i \, {\rm Li}_2\left (\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + i \, {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - i \, {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - i \, {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \]
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 \[ \int \operatorname {arccot}\left (\tanh \left (b x + a\right )\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 198, normalized size = 2.71 \[ \frac {\arctanh \left (\tanh \left (b x +a \right )\right ) \mathrm {arccot}\left (\tanh \left (b x +a \right )\right )}{b}+\frac {\arctan \left (\tanh \left (b x +a \right )\right ) \arctanh \left (\tanh \left (b x +a \right )\right )}{b}+\frac {\arctan \left (\tanh \left (b x +a \right )\right ) \ln \left (1+\frac {i \left (1+i \tanh \left (b x +a \right )\right )^{2}}{\tanh ^{2}\left (b x +a \right )+1}\right )}{2 b}-\frac {\arctan \left (\tanh \left (b x +a \right )\right ) \ln \left (1-\frac {i \left (1+i \tanh \left (b x +a \right )\right )^{2}}{\tanh ^{2}\left (b x +a \right )+1}\right )}{2 b}-\frac {i \dilog \left (1+\frac {i \left (1+i \tanh \left (b x +a \right )\right )^{2}}{\tanh ^{2}\left (b x +a \right )+1}\right )}{4 b}+\frac {i \dilog \left (1-\frac {i \left (1+i \tanh \left (b x +a \right )\right )^{2}}{\tanh ^{2}\left (b x +a \right )+1}\right )}{4 b} \]
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 \[ x \arctan \left (e^{\left (2 \, b x + 2 \, a\right )} + 1, e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, b \int \frac {x e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (4 \, b x + 4 \, a\right )} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {acot}\left (\mathrm {tanh}\left (a+b\,x\right )\right ) \,d x \]
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 \[ \int \operatorname {acot}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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