Optimal. Leaf size=72 \[ -2 x \text {Li}_2\left (-e^x\right )+2 \text {Li}_2\left (-e^x\right )+2 \text {Li}_3\left (-e^x\right )+\frac {x^3}{3}+\frac {x^2}{e^x+1}-x^2-x^2 \log \left (e^x+1\right )+2 x \log \left (e^x+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6688, 2185, 2184, 2190, 2531, 2282, 6589, 2191, 2279, 2391} \[ -2 x \text {PolyLog}\left (2,-e^x\right )+2 \text {PolyLog}\left (2,-e^x\right )+2 \text {PolyLog}\left (3,-e^x\right )+\frac {x^3}{3}+\frac {x^2}{e^x+1}-x^2-x^2 \log \left (e^x+1\right )+2 x \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 6589
Rule 6688
Rubi steps
\begin {align*} \int \frac {x^2}{1+2 e^x+e^{2 x}} \, dx &=\int \frac {x^2}{\left (1+e^x\right )^2} \, dx\\ &=-\int \frac {e^x x^2}{\left (1+e^x\right )^2} \, dx+\int \frac {x^2}{1+e^x} \, dx\\ &=\frac {x^2}{1+e^x}+\frac {x^3}{3}-2 \int \frac {x}{1+e^x} \, dx-\int \frac {e^x x^2}{1+e^x} \, dx\\ &=-x^2+\frac {x^2}{1+e^x}+\frac {x^3}{3}-x^2 \log \left (1+e^x\right )+2 \int \frac {e^x x}{1+e^x} \, dx+2 \int x \log \left (1+e^x\right ) \, dx\\ &=-x^2+\frac {x^2}{1+e^x}+\frac {x^3}{3}+2 x \log \left (1+e^x\right )-x^2 \log \left (1+e^x\right )-2 x \text {Li}_2\left (-e^x\right )-2 \int \log \left (1+e^x\right ) \, dx+2 \int \text {Li}_2\left (-e^x\right ) \, dx\\ &=-x^2+\frac {x^2}{1+e^x}+\frac {x^3}{3}+2 x \log \left (1+e^x\right )-x^2 \log \left (1+e^x\right )-2 x \text {Li}_2\left (-e^x\right )-2 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right )+2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^x\right )\\ &=-x^2+\frac {x^2}{1+e^x}+\frac {x^3}{3}+2 x \log \left (1+e^x\right )-x^2 \log \left (1+e^x\right )+2 \text {Li}_2\left (-e^x\right )-2 x \text {Li}_2\left (-e^x\right )+2 \text {Li}_3\left (-e^x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 57, normalized size = 0.79 \[ -2 (x-1) \text {Li}_2\left (-e^x\right )+2 \text {Li}_3\left (-e^x\right )+\frac {\left (e^x (x-3)+x\right ) x^2}{3 \left (e^x+1\right )}-(x-2) x \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.41, size = 76, normalized size = 1.06 \[ \frac {x^{3} - 6 \, {\left ({\left (x - 1\right )} e^{x} + x - 1\right )} {\rm Li}_2\left (-e^{x}\right ) + {\left (x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \, {\left (x^{2} + {\left (x^{2} - 2 \, x\right )} e^{x} - 2 \, x\right )} \log \left (e^{x} + 1\right ) + 6 \, {\left (e^{x} + 1\right )} {\rm polylog}\left (3, -e^{x}\right )}{3 \, {\left (e^{x} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{e^{\left (2 \, x\right )} + 2 \, e^{x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 65, normalized size = 0.90 \[ \frac {x^{3}}{3}-x^{2} \ln \left ({\mathrm e}^{x}+1\right )-x^{2}+\frac {x^{2}}{{\mathrm e}^{x}+1}-2 x \polylog \left (2, -{\mathrm e}^{x}\right )+2 x \ln \left ({\mathrm e}^{x}+1\right )+2 \polylog \left (2, -{\mathrm e}^{x}\right )+2 \polylog \left (3, -{\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.91, size = 62, normalized size = 0.86 \[ \frac {1}{3} \, x^{3} - x^{2} \log \left (e^{x} + 1\right ) - x^{2} - 2 \, x {\rm Li}_2\left (-e^{x}\right ) + 2 \, x \log \left (e^{x} + 1\right ) + \frac {x^{2}}{e^{x} + 1} + 2 \, {\rm Li}_2\left (-e^{x}\right ) + 2 \, {\rm Li}_{3}(-e^{x}) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x^{2}}{e^{x} + 1} + \int \frac {x \left (x - 2\right )}{e^{x} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________