Optimal. Leaf size=204 \[ -\frac {2 \text {Li}_2\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {2 \text {Li}_2\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}+\frac {x^2}{\sqrt {3} \left (\sqrt {3}+3 i\right )}-\frac {x^2}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}-\frac {2 x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {2 x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2263, 2184, 2190, 2279, 2391} \[ -\frac {2 \text {PolyLog}\left (2,-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {2 \text {PolyLog}\left (2,-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}+\frac {x^2}{\sqrt {3} \left (\sqrt {3}+3 i\right )}-\frac {x^2}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}-\frac {2 x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {2 x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2263
Rule 2279
Rule 2391
Rubi steps
\begin {align*} \int \frac {x}{3+3 e^x+e^{2 x}} \, dx &=-\frac {(2 i) \int \frac {x}{3-i \sqrt {3}+2 e^x} \, dx}{\sqrt {3}}+\frac {(2 i) \int \frac {x}{3+i \sqrt {3}+2 e^x} \, dx}{\sqrt {3}}\\ &=-\frac {x^2}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {x^2}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {(4 i) \int \frac {e^x x}{3-i \sqrt {3}+2 e^x} \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}-\frac {(4 i) \int \frac {e^x x}{3+i \sqrt {3}+2 e^x} \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )}\\ &=-\frac {x^2}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {x^2}{\sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {(2 i) \int \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right ) \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}+\frac {(2 i) \int \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right ) \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )}\\ &=-\frac {x^2}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {x^2}{\sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{3-i \sqrt {3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {3} \left (3-i \sqrt {3}\right )}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{3+i \sqrt {3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {3} \left (3+i \sqrt {3}\right )}\\ &=-\frac {x^2}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {x^2}{\sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {2 \text {Li}_2\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 \text {Li}_2\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 144, normalized size = 0.71 \[ \frac {\left (\sqrt {3}+3 i\right ) \text {Li}_2\left (-\frac {1}{2} i \left (-3 i+\sqrt {3}\right ) e^{-x}\right )+\left (\sqrt {3}-3 i\right ) \text {Li}_2\left (\frac {1}{2} i \left (3 i+\sqrt {3}\right ) e^{-x}\right )-x \left (\left (\sqrt {3}-3 i\right ) \log \left (1+\frac {1}{2} \left (3-i \sqrt {3}\right ) e^{-x}\right )+\left (\sqrt {3}+3 i\right ) \log \left (1+\frac {1}{2} \left (3+i \sqrt {3}\right ) e^{-x}\right )\right )}{6 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 100, normalized size = 0.49 \[ \frac {1}{6} \, x^{2} + \frac {1}{6} \, {\left (i \, \sqrt {3} - 1\right )} {\rm Li}_2\left (-\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x}\right ) + \frac {1}{6} \, {\left (-i \, \sqrt {3} - 1\right )} {\rm Li}_2\left (-\frac {1}{6} \, {\left (-i \, \sqrt {3} + 3\right )} e^{x}\right ) + \frac {1}{6} \, {\left (i \, \sqrt {3} x - x\right )} \log \left (\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x} + 1\right ) + \frac {1}{6} \, {\left (-i \, \sqrt {3} x - x\right )} \log \left (\frac {1}{6} \, {\left (-i \, \sqrt {3} + 3\right )} e^{x} + 1\right ) \]
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 \[ \int \frac {x}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 235, normalized size = 1.15 \[ \frac {x^{2}}{6}+\frac {i \sqrt {3}\, x \ln \left (\frac {-2 \,{\mathrm e}^{x}+i \sqrt {3}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {x \ln \left (\frac {-2 \,{\mathrm e}^{x}+i \sqrt {3}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {i \sqrt {3}\, x \ln \left (\frac {2 \,{\mathrm e}^{x}+i \sqrt {3}+3}{3+i \sqrt {3}}\right )}{6}-\frac {x \ln \left (\frac {2 \,{\mathrm e}^{x}+i \sqrt {3}+3}{3+i \sqrt {3}}\right )}{6}+\frac {i \sqrt {3}\, \dilog \left (\frac {-2 \,{\mathrm e}^{x}+i \sqrt {3}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {\dilog \left (\frac {-2 \,{\mathrm e}^{x}+i \sqrt {3}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {i \sqrt {3}\, \dilog \left (\frac {2 \,{\mathrm e}^{x}+i \sqrt {3}+3}{3+i \sqrt {3}}\right )}{6}-\frac {\dilog \left (\frac {2 \,{\mathrm e}^{x}+i \sqrt {3}+3}{3+i \sqrt {3}}\right )}{6} \]
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 \[ \int \frac {x}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+3} \,d x \]
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 \[ \int \frac {x}{e^{2 x} + 3 e^{x} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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