\(\int \frac {x}{3+3 e^x+e^{2 x}} \, dx\) [513]

Optimal result

Integrand size = 16, antiderivative size = 204 \[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=-\frac {x^2}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {x^2}{\sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )} \]

[Out]

-1/3*x^2/(3*I-3^(1/2))*3^(1/2)+2/3*x*ln(1+2*exp(x)/(3+I*3^(1/2)))/(3*I-3^(1/2))*3^(1/2)+2/3*polylog(2,-2*exp(x
)/(3+I*3^(1/2)))/(3*I-3^(1/2))*3^(1/2)+1/3*x^2*3^(1/2)/(3*I+3^(1/2))-2/3*x*ln(1+2*exp(x)/(3-I*3^(1/2)))*3^(1/2
)/(3*I+3^(1/2))-2/3*polylog(2,-2*exp(x)/(3-I*3^(1/2)))*3^(1/2)/(3*I+3^(1/2))

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2295, 2215, 2221, 2317, 2438} \[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=-\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {2 \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}+\frac {x^2}{\sqrt {3} \left (\sqrt {3}+3 i\right )}-\frac {x^2}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}-\frac {2 x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {2 x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )} \]

[In]

Int[x/(3 + 3*E^x + E^(2*x)),x]

[Out]

-(x^2/(Sqrt[3]*(3*I - Sqrt[3]))) + x^2/(Sqrt[3]*(3*I + Sqrt[3])) - (2*x*Log[1 + (2*E^x)/(3 - I*Sqrt[3])])/(Sqr
t[3]*(3*I + Sqrt[3])) + (2*x*Log[1 + (2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) - (2*PolyLog[2, (-2*E
^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*PolyLog[2, (-2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqr
t[3]))

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2295

Int[((f_.) + (g_.)*(x_))^(m_.)/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*
a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m/(b - q + 2*c*F^u), x], x] - Dist[2*(c/q), Int[(f + g*x)^m/(b + q + 2*c
*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[
m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps \begin{align*} \text {integral}& = -\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) \text {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) \text {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*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.71 \[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=\frac {-x \left (\left (-3 i+\sqrt {3}\right ) \log \left (1+\frac {1}{2} \left (3-i \sqrt {3}\right ) e^{-x}\right )+\left (3 i+\sqrt {3}\right ) \log \left (1+\frac {1}{2} \left (3+i \sqrt {3}\right ) e^{-x}\right )\right )+\left (3 i+\sqrt {3}\right ) \operatorname {PolyLog}\left (2,-\frac {1}{2} i \left (-3 i+\sqrt {3}\right ) e^{-x}\right )+\left (-3 i+\sqrt {3}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} i \left (3 i+\sqrt {3}\right ) e^{-x}\right )}{6 \sqrt {3}} \]

[In]

Integrate[x/(3 + 3*E^x + E^(2*x)),x]

[Out]

(-(x*((-3*I + Sqrt[3])*Log[1 + (3 - I*Sqrt[3])/(2*E^x)] + (3*I + Sqrt[3])*Log[1 + (3 + I*Sqrt[3])/(2*E^x)])) +
(3*I + Sqrt[3])*PolyLog[2, ((-1/2*I)*(-3*I + Sqrt[3]))/E^x] + (-3*I + Sqrt[3])*PolyLog[2, ((I/2)*(3*I + Sqrt[
3]))/E^x])/(6*Sqrt[3])

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {x^{2}}{6}+\frac {i \sqrt {3}\, x \ln \left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {x \ln \left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {i \sqrt {3}\, x \ln \left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}-\frac {x \ln \left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}+\frac {i \sqrt {3}\, \operatorname {dilog}\left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {i \sqrt {3}\, \operatorname {dilog}\left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}\)	\(235\)
risch	\(\frac {x^{2}}{6}+\frac {i \sqrt {3}\, x \ln \left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {x \ln \left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {i \sqrt {3}\, x \ln \left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}-\frac {x \ln \left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}+\frac {i \sqrt {3}\, \operatorname {dilog}\left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \sqrt {3}-2 \,{\mathrm e}^{x}-3}{-3+i \sqrt {3}}\right )}{6}-\frac {i \sqrt {3}\, \operatorname {dilog}\left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \sqrt {3}+2 \,{\mathrm e}^{x}+3}{3+i \sqrt {3}}\right )}{6}\)	\(235\)

[In]

int(x/(3+3*exp(x)+exp(2*x)),x,method=_RETURNVERBOSE)

[Out]

1/6*x^2+1/6*I*3^(1/2)*x*ln((I*3^(1/2)-2*exp(x)-3)/(-3+I*3^(1/2)))-1/6*x*ln((I*3^(1/2)-2*exp(x)-3)/(-3+I*3^(1/2
)))-1/6*I*3^(1/2)*x*ln((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))-1/6*x*ln((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))+1/
6*I*3^(1/2)*dilog((I*3^(1/2)-2*exp(x)-3)/(-3+I*3^(1/2)))-1/6*dilog((I*3^(1/2)-2*exp(x)-3)/(-3+I*3^(1/2)))-1/6*
I*3^(1/2)*dilog((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))-1/6*dilog((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.49 \[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=\frac {1}{6} \, x^{2} + \frac {1}{6} \, {\left (i \, \sqrt {3} - 1\right )} {\rm Li}_2\left (-\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x}\right ) + \frac {1}{6} \, {\left (-i \, \sqrt {3} - 1\right )} {\rm Li}_2\left (-\frac {1}{6} \, {\left (-i \, \sqrt {3} + 3\right )} e^{x}\right ) + \frac {1}{6} \, {\left (i \, \sqrt {3} x - x\right )} \log \left (\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x} + 1\right ) + \frac {1}{6} \, {\left (-i \, \sqrt {3} x - x\right )} \log \left (\frac {1}{6} \, {\left (-i \, \sqrt {3} + 3\right )} e^{x} + 1\right ) \]

[In]

integrate(x/(3+3*exp(x)+exp(2*x)),x, algorithm="fricas")

[Out]

1/6*x^2 + 1/6*(I*sqrt(3) - 1)*dilog(-1/6*(I*sqrt(3) + 3)*e^x) + 1/6*(-I*sqrt(3) - 1)*dilog(-1/6*(-I*sqrt(3) +
3)*e^x) + 1/6*(I*sqrt(3)*x - x)*log(1/6*(I*sqrt(3) + 3)*e^x + 1) + 1/6*(-I*sqrt(3)*x - x)*log(1/6*(-I*sqrt(3)
+ 3)*e^x + 1)

Sympy [F]

\[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=\int \frac {x}{e^{2 x} + 3 e^{x} + 3}\, dx \]

[In]

integrate(x/(3+3*exp(x)+exp(2*x)),x)

[Out]

Integral(x/(exp(2*x) + 3*exp(x) + 3), x)

Maxima [F]

\[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=\int { \frac {x}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3} \,d x } \]

[In]

integrate(x/(3+3*exp(x)+exp(2*x)),x, algorithm="maxima")

[Out]

integrate(x/(e^(2*x) + 3*e^x + 3), x)

Giac [F]

\[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=\int { \frac {x}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3} \,d x } \]

[In]

integrate(x/(3+3*exp(x)+exp(2*x)),x, algorithm="giac")

[Out]

integrate(x/(e^(2*x) + 3*e^x + 3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{3+3 e^x+e^{2 x}} \, dx=\int \frac {x}{{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+3} \,d x \]

[In]

int(x/(exp(2*x) + 3*exp(x) + 3),x)

[Out]

int(x/(exp(2*x) + 3*exp(x) + 3), x)