Optimal. Leaf size=79 \[ i x \text {ArcTan}\left (e^{2 i (a+b x)}\right )+x \tanh ^{-1}(\cot (a+b x))-\frac {i \text {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i \text {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b} \]
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Rubi [A]
time = 0.03, 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 = {6384, 4266,
2317, 2438} \begin {gather*} i x \text {ArcTan}\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}+x \tanh ^{-1}(\cot (a+b x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4266
Rule 6384
Rubi steps
\begin {align*} \int \tanh ^{-1}(\cot (a+b x)) \, dx &=x \tanh ^{-1}(\cot (a+b x))-b \int x \sec (2 a+2 b x) \, dx\\ &=i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )+x \tanh ^{-1}(\cot (a+b x))+\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\\ &=i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )+x \tanh ^{-1}(\cot (a+b x))-\frac {i \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{4 b}+\frac {i \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{4 b}\\ &=i x \tan ^{-1}\left (e^{2 i (a+b x)}\right )+x \tanh ^{-1}(\cot (a+b x))-\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 = 0.05, size = 127, normalized size = 1.61 \begin {gather*} x \tanh ^{-1}(\cot (a+b x))-\frac {(-4 a+\pi -4 b x) \left (\log \left (1-i e^{-2 i (a+b x)}\right )-\log \left (1+i e^{-2 i (a+b x)}\right )\right )-(-4 a+\pi ) \log \left (\cot \left (a+\frac {\pi }{4}+b x\right )\right )+2 i \left (\text {PolyLog}\left (2,-i e^{-2 i (a+b x)}\right )-\text {PolyLog}\left (2,i e^{-2 i (a+b x)}\right )\right )}{8 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. 144 vs. \(2 (64 ) = 128\).
time = 0.59, size = 145, normalized size = 1.84
method | result | size |
derivativedivides | \(\frac {-\frac {i \arctanh \left (\cot \left (b x +a \right )\right ) \left (\ln \left (1-\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )-\ln \left (1+\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )\right )}{2}+\frac {i \dilog \left (1+\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )}{4}-\frac {i \dilog \left (1-\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )}{4}}{b}\) | \(145\) |
default | \(\frac {-\frac {i \arctanh \left (\cot \left (b x +a \right )\right ) \left (\ln \left (1-\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )-\ln \left (1+\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )\right )}{2}+\frac {i \dilog \left (1+\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )}{4}-\frac {i \dilog \left (1-\frac {i \left (\cot \left (b x +a \right )+1\right )^{2}}{-\left (\cot ^{2}\left (b x +a \right )\right )+1}\right )}{4}}{b}\) | \(145\) |
risch | \(\text {Expression too large to display}\) | \(1161\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 184 vs. \(2 (57) = 114\).
time = 0.51, size = 184, normalized size = 2.33 \begin {gather*} \frac {4 \, {\left (b x + a\right )} \operatorname {artanh}\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} \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. 389 vs. \(2 (57) = 114\).
time = 0.44, size = 389, normalized size = 4.92 \begin {gather*} \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} \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 {atanh}{\left (\cot {\left (a + b x \right )} \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 = -0.01 \begin {gather*} \int \mathrm {atanh}\left (\mathrm {cot}\left (a+b\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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