Optimal. Leaf size=79 \[ -\frac {i \text {Li}_2\left (-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i \text {Li}_2\left (i e^{2 i (a+b x)}\right )}{4 b}+i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )+x \coth ^{-1}(\cot (a+b x)) \]
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Rubi [A] time = 0.05, antiderivative size = 79, 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 = {6250, 4181, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i \text {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b}+i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )+x \coth ^{-1}(\cot (a+b x)) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4181
Rule 6250
Rubi steps
\begin {align*} \int \coth ^{-1}(\cot (a+b x)) \, dx &=x \coth ^{-1}(\cot (a+b x))-b \int x \sec (2 a+2 b x) \, dx\\ &=x \coth ^{-1}(\cot (a+b x))+i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )+\frac {1}{2} \int \log \left (1-i e^{i (2 a+2 b x)}\right ) \, dx-\frac {1}{2} \int \log \left (1+i e^{i (2 a+2 b x)}\right ) \, dx\\ &=x \coth ^{-1}(\cot (a+b x))+i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )-\frac {i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{4 b}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{4 b}\\ &=x \coth ^{-1}(\cot (a+b x))+i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )-\frac {i \text {Li}_2\left (-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i \text {Li}_2\left (i e^{2 i (a+b x)}\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 78, normalized size = 0.99 \[ \frac {-i \text {Li}_2\left (-i e^{2 i (a+b x)}\right )+i \text {Li}_2\left (i e^{2 i (a+b x)}\right )+4 b x \left (\coth ^{-1}(\cot (a+b x))+i \tan ^{-1}\left (e^{2 i (a+b x)}\right )\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 388, normalized size = 4.91 \[ \frac {4 \, b x \log \left (\frac {\cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ) + 1}{\cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right ) + 1}\right ) + 2 \, a \log \left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + i\right ) - 2 \, a \log \left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + i\right ) - 2 \, {\left (b x + a\right )} \log \left (i \, \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b x + a\right )} \log \left (i \, \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) - 2 \, {\left (b x + a\right )} \log \left (-i \, \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b x + a\right )} \log \left (-i \, \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, a \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + i\right ) - 2 \, a \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + i\right ) + i \, {\rm Li}_2\left (i \, \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right ) + i \, {\rm Li}_2\left (i \, \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right ) - i \, {\rm Li}_2\left (-i \, \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right ) - i \, {\rm Li}_2\left (-i \, \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )}{8 \, b} \]
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 \[ \int \operatorname {arcoth}\left (\cot \left (b x + a\right )\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.55, size = 267, normalized size = 3.38 \[ -\frac {\mathrm {arccoth}\left (\cot \left (b x +a \right )\right ) \pi }{2 b}+\frac {\mathrm {arccoth}\left (\cot \left (b x +a \right )\right ) \mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{b}-\frac {\ln \left (1+\frac {i \left (1+i \cot \left (b x +a \right )\right )^{2}}{\cot ^{2}\left (b x +a \right )+1}\right ) \pi }{4 b}+\frac {\ln \left (1+\frac {i \left (1+i \cot \left (b x +a \right )\right )^{2}}{\cot ^{2}\left (b x +a \right )+1}\right ) \mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{2 b}+\frac {\ln \left (1-\frac {i \left (1+i \cot \left (b x +a \right )\right )^{2}}{\cot ^{2}\left (b x +a \right )+1}\right ) \pi }{4 b}-\frac {\ln \left (1-\frac {i \left (1+i \cot \left (b x +a \right )\right )^{2}}{\cot ^{2}\left (b x +a \right )+1}\right ) \mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{2 b}+\frac {i \dilog \left (1+\frac {i \left (1+i \cot \left (b x +a \right )\right )^{2}}{\cot ^{2}\left (b x +a \right )+1}\right )}{4 b}-\frac {i \dilog \left (1-\frac {i \left (1+i \cot \left (b x +a \right )\right )^{2}}{\cot ^{2}\left (b x +a \right )+1}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 184, normalized size = 2.33 \[ \frac {4 \, {\left (b x + a\right )} \operatorname {arcoth}\left (\frac {1}{\tan \left (b x + a\right )}\right ) + {\left (\arctan \left (\frac {1}{2} \, \tan \left (b x + a\right ) + \frac {1}{2}, \frac {1}{2} \, \tan \left (b x + a\right ) + \frac {1}{2}\right ) - \arctan \left (\frac {1}{2} \, \tan \left (b x + a\right ) - \frac {1}{2}, -\frac {1}{2} \, \tan \left (b x + a\right ) + \frac {1}{2}\right )\right )} \log \left (\tan \left (b x + a\right )^{2} + 1\right ) - {\left (b x + a\right )} \log \left (\frac {1}{2} \, \tan \left (b x + a\right )^{2} + \tan \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (b x + a\right )} \log \left (\frac {1}{2} \, \tan \left (b x + a\right )^{2} - \tan \left (b x + a\right ) + \frac {1}{2}\right ) - i \, {\rm Li}_2\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \tan \left (b x + a\right ) - \frac {1}{2} i + \frac {1}{2}\right ) + i \, {\rm Li}_2\left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \tan \left (b x + a\right ) + \frac {1}{2} i + \frac {1}{2}\right ) + i \, {\rm Li}_2\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \tan \left (b x + a\right ) + \frac {1}{2} i + \frac {1}{2}\right ) - i \, {\rm Li}_2\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \tan \left (b x + a\right ) - \frac {1}{2} i + \frac {1}{2}\right )}{4 \, b} \]
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 \[ \int \mathrm {acoth}\left (\mathrm {cot}\left (a+b\,x\right )\right ) \,d x \]
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 \[ \int \operatorname {acoth}{\left (\cot {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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