Optimal. Leaf size=80 \[ -\frac {1}{4} \operatorname {PolyLog}\left (2,\frac {1}{3} e^{2 i x}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,3 e^{2 i x}\right )+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )+x \tan ^{-1}(2 \tan (x)) \]
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Rubi [A] time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5167, 2190, 2279, 2391} \[ -\frac {1}{4} \text {PolyLog}\left (2,\frac {1}{3} e^{2 i x}\right )+\frac {1}{4} \text {PolyLog}\left (2,3 e^{2 i x}\right )+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )+x \tan ^{-1}(2 \tan (x)) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 5167
Rubi steps
\begin {align*} \int \tan ^{-1}(2 \tan (x)) \, dx &=x \tan ^{-1}(2 \tan (x))-3 \int \frac {e^{2 i x} x}{-1+3 e^{2 i x}} \, dx-\int \frac {e^{2 i x} x}{3-e^{2 i x}} \, dx\\ &=x \tan ^{-1}(2 \tan (x))+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )-\frac {1}{2} i \int \log \left (1-3 e^{2 i x}\right ) \, dx+\frac {1}{2} i \int \log \left (1-\frac {1}{3} e^{2 i x}\right ) \, dx\\ &=x \tan ^{-1}(2 \tan (x))+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\log (1-3 x)}{x} \, dx,x,e^{2 i x}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{3}\right )}{x} \, dx,x,e^{2 i x}\right )\\ &=x \tan ^{-1}(2 \tan (x))+\frac {1}{2} i x \log \left (1-3 e^{2 i x}\right )-\frac {1}{2} i x \log \left (1-\frac {1}{3} e^{2 i x}\right )-\frac {1}{4} \text {Li}_2\left (\frac {1}{3} e^{2 i x}\right )+\frac {1}{4} \text {Li}_2\left (3 e^{2 i x}\right )\\ \end {align*}
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Mathematica [B] time = 0.26, size = 262, normalized size = 3.28 \[ x \tan ^{-1}(2 \tan (x))-\frac {1}{4} i \left (i \left (\operatorname {PolyLog}\left (2,\frac {2 \tan (x)-i}{6 \tan (x)+3 i}\right )-\operatorname {PolyLog}\left (2,\frac {6 \tan (x)-3 i}{2 \tan (x)+i}\right )\right )+2 i \cos ^{-1}\left (\frac {5}{3}\right ) \tan ^{-1}(2 \tan (x))+4 i x \tan ^{-1}\left (\frac {\cot (x)}{2}\right )-\log \left (\frac {-4 \tan (x)+4 i}{2 \tan (x)+i}\right ) \left (\cos ^{-1}\left (\frac {5}{3}\right )-2 \tan ^{-1}(2 \tan (x))\right )-\log \left (\frac {4 (\tan (x)+i)}{6 \tan (x)+3 i}\right ) \left (2 \tan ^{-1}(2 \tan (x))+\cos ^{-1}\left (\frac {5}{3}\right )\right )+\log \left (\frac {2 i \sqrt {\frac {2}{3}} e^{-i x}}{\sqrt {3 \cos (2 x)-5}}\right ) \left (2 \tan ^{-1}(2 \tan (x))+2 \tan ^{-1}\left (\frac {\cot (x)}{2}\right )+\cos ^{-1}\left (\frac {5}{3}\right )\right )+\log \left (\frac {2 i \sqrt {\frac {2}{3}} e^{i x}}{\sqrt {3 \cos (2 x)-5}}\right ) \left (-2 \tan ^{-1}(2 \tan (x))-2 \tan ^{-1}\left (\frac {\cot (x)}{2}\right )+\cos ^{-1}\left (\frac {5}{3}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 220, normalized size = 2.75 \[ x \arctan \left (2 \, \tan \relax (x)\right ) - \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \relax (x)^{2} + 3 i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1}\right ) + \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \relax (x)^{2} + i \, \tan \relax (x) + 1\right )}}{3 \, {\left (\tan \relax (x)^{2} + 1\right )}}\right ) - \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \relax (x)^{2} - i \, \tan \relax (x) + 1\right )}}{3 \, {\left (\tan \relax (x)^{2} + 1\right )}}\right ) + \frac {1}{4} i \, x \log \left (\frac {2 \, {\left (2 \, \tan \relax (x)^{2} - 3 i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1}\right ) + \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \relax (x)^{2} + 3 i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1} + 1\right ) - \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \relax (x)^{2} + i \, \tan \relax (x) + 1\right )}}{3 \, {\left (\tan \relax (x)^{2} + 1\right )}} + 1\right ) - \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \relax (x)^{2} - i \, \tan \relax (x) + 1\right )}}{3 \, {\left (\tan \relax (x)^{2} + 1\right )}} + 1\right ) + \frac {1}{8} \, {\rm Li}_2\left (-\frac {2 \, {\left (2 \, \tan \relax (x)^{2} - 3 i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1} + 1\right ) \]
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 \[ \int \arctan \left (2 \, \tan \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 113, normalized size = 1.41 \[ \arctan \left (2 \tan \relax (x )\right ) \arctan \left (\tan \relax (x )\right )+\frac {i \arctan \left (\tan \relax (x )\right ) \ln \left (1-\frac {3 \left (i \tan \relax (x )+1\right )^{2}}{\tan ^{2}\relax (x )+1}\right )}{2}-\frac {i \arctan \left (\tan \relax (x )\right ) \ln \left (1-\frac {\left (i \tan \relax (x )+1\right )^{2}}{3 \left (\tan ^{2}\relax (x )+1\right )}\right )}{2}+\frac {\polylog \left (2, \frac {3 \left (i \tan \relax (x )+1\right )^{2}}{\tan ^{2}\relax (x )+1}\right )}{4}-\frac {\polylog \left (2, \frac {\left (i \tan \relax (x )+1\right )^{2}}{3 \left (\tan ^{2}\relax (x )\right )+3}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 84, normalized size = 1.05 \[ x \arctan \left (2 \, \tan \relax (x)\right ) - \frac {1}{8} \, \log \left (4 \, \tan \relax (x)^{2} + 4\right ) \log \left (4 \, \tan \relax (x)^{2} + 1\right ) + \frac {1}{8} \, \log \left (4 \, \tan \relax (x)^{2} + 1\right ) \log \left (\frac {4}{9} \, \tan \relax (x)^{2} + \frac {4}{9}\right ) - \frac {1}{4} \, {\rm Li}_2\left (2 i \, \tan \relax (x) - 1\right ) + \frac {1}{4} \, {\rm Li}_2\left (\frac {2}{3} i \, \tan \relax (x) + \frac {1}{3}\right ) + \frac {1}{4} \, {\rm Li}_2\left (-\frac {2}{3} i \, \tan \relax (x) + \frac {1}{3}\right ) - \frac {1}{4} \, {\rm Li}_2\left (-2 i \, \tan \relax (x) - 1\right ) \]
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 \[ \int \mathrm {atan}\left (2\,\mathrm {tan}\relax (x)\right ) \,d x \]
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 \[ \int \operatorname {atan}{\left (2 \tan {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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