Optimal. Leaf size=42 \[ \frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+\frac {i x^2}{2}-x \log \left (1+e^{2 i x}\right )+\tan (x) \]
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Rubi [A] time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3719, 2190, 2279, 2391, 3473, 8} \[ \frac {1}{2} i \text {PolyLog}\left (2,-e^{2 i x}\right )+\frac {i x^2}{2}-x \log \left (1+e^{2 i x}\right )+\tan (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3473
Rule 3719
Rubi steps
\begin {align*} \int \left (1+x \tan (x)+\tan ^2(x)\right ) \, dx &=x+\int x \tan (x) \, dx+\int \tan ^2(x) \, dx\\ &=x+\frac {i x^2}{2}+\tan (x)-2 i \int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx-\int 1 \, dx\\ &=\frac {i x^2}{2}-x \log \left (1+e^{2 i x}\right )+\tan (x)+\int \log \left (1+e^{2 i x}\right ) \, dx\\ &=\frac {i x^2}{2}-x \log \left (1+e^{2 i x}\right )+\tan (x)-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac {i x^2}{2}-x \log \left (1+e^{2 i x}\right )+\frac {1}{2} i \text {Li}_2\left (-e^{2 i x}\right )+\tan (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \[ \frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+\frac {i x^2}{2}-x \log \left (1+e^{2 i x}\right )+\tan (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 85, normalized size = 2.02 \[ -\frac {1}{2} \, x \log \left (-\frac {2 \, {\left (i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1}\right ) - \frac {1}{2} \, x \log \left (-\frac {2 \, {\left (-i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1}\right ) - \frac {1}{4} i \, {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1} + 1\right ) + \frac {1}{4} i \, {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \relax (x) - 1\right )}}{\tan \relax (x)^{2} + 1} + 1\right ) + \tan \relax (x) \]
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 \[ \int x \tan \relax (x) + \tan \relax (x)^{2} + 1\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 43, normalized size = 1.02 \[ \frac {i x^{2}}{2}-x \ln \left ({\mathrm e}^{2 i x}+1\right )+\frac {i \polylog \left (2, -{\mathrm e}^{2 i x}\right )}{2}+\frac {2 i}{{\mathrm e}^{2 i x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.17, size = 145, normalized size = 3.45 \[ x + \frac {x^{2} - {\left (2 \, x \cos \left (2 \, x\right ) + 2 i \, x \sin \left (2 \, x\right ) + 2 \, x\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + {\left (x^{2} + 2 i \, x\right )} \cos \left (2 \, x\right ) + {\left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right )} {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) - {\left (-i \, x \cos \left (2 \, x\right ) + x \sin \left (2 \, x\right ) - i \, x\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - {\left (-i \, x^{2} + 2 \, x\right )} \sin \left (2 \, x\right ) + 2 i \, x + 4}{-2 i \, \cos \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right ) - 2 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 32, normalized size = 0.76 \[ \mathrm {tan}\relax (x)+\frac {\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {x\,\left (x+\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
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 \[ \int \left (x \tan {\relax (x )} + \tan ^{2}{\relax (x )} + 1\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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