\(\int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7)}{36-72 e^x+36 e^{2 x}} \, dx\) [6710]

Optimal result

Integrand size = 78, antiderivative size = 31 \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=\frac {x^2 (-1+3 x)^2 \left (x+x^2\right )^2}{36 \left (x-e^x x\right )} \]

[Out]

(-1+3*x)^2*(1/3*x^2+1/3*x)^2*x^2/(-4*exp(x)*x+4*x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(31)=62\).

Time = 1.97 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.61, number of steps used = 138, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.141, Rules used = {6820, 12, 6874, 2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222} \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=\frac {x^7}{4 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}+\frac {x^3}{36 \left (1-e^x\right )} \]

[In]

Int[(3*x^2 - 16*x^3 - 10*x^4 + 72*x^5 + 63*x^6 + E^x*(-3*x^2 + 17*x^3 + 6*x^4 - 74*x^5 - 51*x^6 + 9*x^7))/(36
- 72*E^x + 36*E^(2*x)),x]

[Out]

x^3/(36*(1 - E^x)) - x^4/(9*(1 - E^x)) - x^5/(18*(1 - E^x)) + x^6/(3*(1 - E^x)) + x^7/(4*(1 - E^x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2216

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && 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 2222

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

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 \left (1-2 x-3 x^2\right ) \left (3-10 x-21 x^2-e^x \left (3-11 x-19 x^2+3 x^3\right )\right )}{36 \left (1-e^x\right )^2} \, dx \\ & = \frac {1}{36} \int \frac {x^2 \left (1-2 x-3 x^2\right ) \left (3-10 x-21 x^2-e^x \left (3-11 x-19 x^2+3 x^3\right )\right )}{\left (1-e^x\right )^2} \, dx \\ & = \frac {1}{36} \int \left (\frac {x^3 \left (-1+2 x+3 x^2\right )^2}{\left (-1+e^x\right )^2}+\frac {x^2 \left (-3+17 x+6 x^2-74 x^3-51 x^4+9 x^5\right )}{-1+e^x}\right ) \, dx \\ & = \frac {1}{36} \int \frac {x^3 \left (-1+2 x+3 x^2\right )^2}{\left (-1+e^x\right )^2} \, dx+\frac {1}{36} \int \frac {x^2 \left (-3+17 x+6 x^2-74 x^3-51 x^4+9 x^5\right )}{-1+e^x} \, dx \\ & = \frac {1}{36} \int \left (\frac {x^3}{\left (-1+e^x\right )^2}-\frac {4 x^4}{\left (-1+e^x\right )^2}-\frac {2 x^5}{\left (-1+e^x\right )^2}+\frac {12 x^6}{\left (-1+e^x\right )^2}+\frac {9 x^7}{\left (-1+e^x\right )^2}\right ) \, dx+\frac {1}{36} \int \left (-\frac {3 x^2}{-1+e^x}+\frac {17 x^3}{-1+e^x}+\frac {6 x^4}{-1+e^x}-\frac {74 x^5}{-1+e^x}-\frac {51 x^6}{-1+e^x}+\frac {9 x^7}{-1+e^x}\right ) \, dx \\ & = \frac {1}{36} \int \frac {x^3}{\left (-1+e^x\right )^2} \, dx-\frac {1}{18} \int \frac {x^5}{\left (-1+e^x\right )^2} \, dx-\frac {1}{12} \int \frac {x^2}{-1+e^x} \, dx-\frac {1}{9} \int \frac {x^4}{\left (-1+e^x\right )^2} \, dx+\frac {1}{6} \int \frac {x^4}{-1+e^x} \, dx+\frac {1}{4} \int \frac {x^7}{\left (-1+e^x\right )^2} \, dx+\frac {1}{4} \int \frac {x^7}{-1+e^x} \, dx+\frac {1}{3} \int \frac {x^6}{\left (-1+e^x\right )^2} \, dx+\frac {17}{36} \int \frac {x^3}{-1+e^x} \, dx-\frac {17}{12} \int \frac {x^6}{-1+e^x} \, dx-\frac {37}{18} \int \frac {x^5}{-1+e^x} \, dx \\ & = \frac {x^3}{36}-\frac {17 x^4}{144}-\frac {x^5}{30}+\frac {37 x^6}{108}+\frac {17 x^7}{84}-\frac {x^8}{32}+\frac {1}{36} \int \frac {e^x x^3}{\left (-1+e^x\right )^2} \, dx-\frac {1}{36} \int \frac {x^3}{-1+e^x} \, dx-\frac {1}{18} \int \frac {e^x x^5}{\left (-1+e^x\right )^2} \, dx+\frac {1}{18} \int \frac {x^5}{-1+e^x} \, dx-\frac {1}{12} \int \frac {e^x x^2}{-1+e^x} \, dx-\frac {1}{9} \int \frac {e^x x^4}{\left (-1+e^x\right )^2} \, dx+\frac {1}{9} \int \frac {x^4}{-1+e^x} \, dx+\frac {1}{6} \int \frac {e^x x^4}{-1+e^x} \, dx+\frac {1}{4} \int \frac {e^x x^7}{\left (-1+e^x\right )^2} \, dx-\frac {1}{4} \int \frac {x^7}{-1+e^x} \, dx+\frac {1}{4} \int \frac {e^x x^7}{-1+e^x} \, dx+\frac {1}{3} \int \frac {e^x x^6}{\left (-1+e^x\right )^2} \, dx-\frac {1}{3} \int \frac {x^6}{-1+e^x} \, dx+\frac {17}{36} \int \frac {e^x x^3}{-1+e^x} \, dx-\frac {17}{12} \int \frac {e^x x^6}{-1+e^x} \, dx-\frac {37}{18} \int \frac {e^x x^5}{-1+e^x} \, dx \\ & = \frac {x^3}{36}+\frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4}+\frac {x^7}{4 \left (1-e^x\right )}-\frac {1}{12} x^2 \log \left (1-e^x\right )+\frac {17}{36} x^3 \log \left (1-e^x\right )+\frac {1}{6} x^4 \log \left (1-e^x\right )-\frac {37}{18} x^5 \log \left (1-e^x\right )-\frac {17}{12} x^6 \log \left (1-e^x\right )+\frac {1}{4} x^7 \log \left (1-e^x\right )-\frac {1}{36} \int \frac {e^x x^3}{-1+e^x} \, dx+\frac {1}{18} \int \frac {e^x x^5}{-1+e^x} \, dx+\frac {1}{12} \int \frac {x^2}{-1+e^x} \, dx+\frac {1}{9} \int \frac {e^x x^4}{-1+e^x} \, dx+\frac {1}{6} \int x \log \left (1-e^x\right ) \, dx-\frac {1}{4} \int \frac {e^x x^7}{-1+e^x} \, dx-\frac {5}{18} \int \frac {x^4}{-1+e^x} \, dx-\frac {1}{3} \int \frac {e^x x^6}{-1+e^x} \, dx-\frac {4}{9} \int \frac {x^3}{-1+e^x} \, dx-\frac {2}{3} \int x^3 \log \left (1-e^x\right ) \, dx-\frac {17}{12} \int x^2 \log \left (1-e^x\right ) \, dx+\frac {7}{4} \int \frac {x^6}{-1+e^x} \, dx-\frac {7}{4} \int x^6 \log \left (1-e^x\right ) \, dx+2 \int \frac {x^5}{-1+e^x} \, dx+\frac {17}{2} \int x^5 \log \left (1-e^x\right ) \, dx+\frac {185}{18} \int x^4 \log \left (1-e^x\right ) \, dx \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}-\frac {1}{12} x^2 \log \left (1-e^x\right )+\frac {4}{9} x^3 \log \left (1-e^x\right )+\frac {5}{18} x^4 \log \left (1-e^x\right )-2 x^5 \log \left (1-e^x\right )-\frac {7}{4} x^6 \log \left (1-e^x\right )-\frac {x \text {Li}_2\left (e^x\right )}{6}+\frac {17}{12} x^2 \text {Li}_2\left (e^x\right )+\frac {2}{3} x^3 \text {Li}_2\left (e^x\right )-\frac {185}{18} x^4 \text {Li}_2\left (e^x\right )-\frac {17}{2} x^5 \text {Li}_2\left (e^x\right )+\frac {7}{4} x^6 \text {Li}_2\left (e^x\right )+\frac {1}{12} \int \frac {e^x x^2}{-1+e^x} \, dx+\frac {1}{12} \int x^2 \log \left (1-e^x\right ) \, dx+\frac {1}{6} \int \text {Li}_2\left (e^x\right ) \, dx-\frac {5}{18} \int \frac {e^x x^4}{-1+e^x} \, dx-\frac {5}{18} \int x^4 \log \left (1-e^x\right ) \, dx-\frac {4}{9} \int \frac {e^x x^3}{-1+e^x} \, dx-\frac {4}{9} \int x^3 \log \left (1-e^x\right ) \, dx+\frac {7}{4} \int \frac {e^x x^6}{-1+e^x} \, dx+\frac {7}{4} \int x^6 \log \left (1-e^x\right ) \, dx+2 \int \frac {e^x x^5}{-1+e^x} \, dx+2 \int x^5 \log \left (1-e^x\right ) \, dx-2 \int x^2 \text {Li}_2\left (e^x\right ) \, dx-\frac {17}{6} \int x \text {Li}_2\left (e^x\right ) \, dx-\frac {21}{2} \int x^5 \text {Li}_2\left (e^x\right ) \, dx+\frac {370}{9} \int x^3 \text {Li}_2\left (e^x\right ) \, dx+\frac {85}{2} \int x^4 \text {Li}_2\left (e^x\right ) \, dx \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}-\frac {x \text {Li}_2\left (e^x\right )}{6}+\frac {4}{3} x^2 \text {Li}_2\left (e^x\right )+\frac {10}{9} x^3 \text {Li}_2\left (e^x\right )-10 x^4 \text {Li}_2\left (e^x\right )-\frac {21}{2} x^5 \text {Li}_2\left (e^x\right )-\frac {17 x \text {Li}_3\left (e^x\right )}{6}-2 x^2 \text {Li}_3\left (e^x\right )+\frac {370}{9} x^3 \text {Li}_3\left (e^x\right )+\frac {85}{2} x^4 \text {Li}_3\left (e^x\right )-\frac {21}{2} x^5 \text {Li}_3\left (e^x\right )-\frac {1}{6} \int x \log \left (1-e^x\right ) \, dx+\frac {1}{6} \int x \text {Li}_2\left (e^x\right ) \, dx+\frac {1}{6} \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^x\right )+\frac {10}{9} \int x^3 \log \left (1-e^x\right ) \, dx-\frac {10}{9} \int x^3 \text {Li}_2\left (e^x\right ) \, dx+\frac {4}{3} \int x^2 \log \left (1-e^x\right ) \, dx-\frac {4}{3} \int x^2 \text {Li}_2\left (e^x\right ) \, dx+\frac {17}{6} \int \text {Li}_3\left (e^x\right ) \, dx+4 \int x \text {Li}_3\left (e^x\right ) \, dx-10 \int x^4 \log \left (1-e^x\right ) \, dx+10 \int x^4 \text {Li}_2\left (e^x\right ) \, dx-\frac {21}{2} \int x^5 \log \left (1-e^x\right ) \, dx+\frac {21}{2} \int x^5 \text {Li}_2\left (e^x\right ) \, dx+\frac {105}{2} \int x^4 \text {Li}_3\left (e^x\right ) \, dx-\frac {370}{3} \int x^2 \text {Li}_3\left (e^x\right ) \, dx-170 \int x^3 \text {Li}_3\left (e^x\right ) \, dx \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}+\frac {\text {Li}_3\left (e^x\right )}{6}-\frac {8 x \text {Li}_3\left (e^x\right )}{3}-\frac {10}{3} x^2 \text {Li}_3\left (e^x\right )+40 x^3 \text {Li}_3\left (e^x\right )+\frac {105}{2} x^4 \text {Li}_3\left (e^x\right )+4 x \text {Li}_4\left (e^x\right )-\frac {370}{3} x^2 \text {Li}_4\left (e^x\right )-170 x^3 \text {Li}_4\left (e^x\right )+\frac {105}{2} x^4 \text {Li}_4\left (e^x\right )-\frac {1}{6} \int \text {Li}_2\left (e^x\right ) \, dx-\frac {1}{6} \int \text {Li}_3\left (e^x\right ) \, dx+\frac {8}{3} \int x \text {Li}_2\left (e^x\right ) \, dx+\frac {8}{3} \int x \text {Li}_3\left (e^x\right ) \, dx+\frac {17}{6} \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^x\right )+\frac {10}{3} \int x^2 \text {Li}_2\left (e^x\right ) \, dx+\frac {10}{3} \int x^2 \text {Li}_3\left (e^x\right ) \, dx-4 \int \text {Li}_4\left (e^x\right ) \, dx-40 \int x^3 \text {Li}_2\left (e^x\right ) \, dx-40 \int x^3 \text {Li}_3\left (e^x\right ) \, dx-\frac {105}{2} \int x^4 \text {Li}_2\left (e^x\right ) \, dx-\frac {105}{2} \int x^4 \text {Li}_3\left (e^x\right ) \, dx-210 \int x^3 \text {Li}_4\left (e^x\right ) \, dx+\frac {740}{3} \int x \text {Li}_4\left (e^x\right ) \, dx+510 \int x^2 \text {Li}_4\left (e^x\right ) \, dx \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}+\frac {\text {Li}_3\left (e^x\right )}{6}+\frac {17 \text {Li}_4\left (e^x\right )}{6}+\frac {20 x \text {Li}_4\left (e^x\right )}{3}-120 x^2 \text {Li}_4\left (e^x\right )-210 x^3 \text {Li}_4\left (e^x\right )+\frac {740 x \text {Li}_5\left (e^x\right )}{3}+510 x^2 \text {Li}_5\left (e^x\right )-210 x^3 \text {Li}_5\left (e^x\right )-\frac {1}{6} \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^x\right )-\frac {1}{6} \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^x\right )-\frac {8}{3} \int \text {Li}_3\left (e^x\right ) \, dx-\frac {8}{3} \int \text {Li}_4\left (e^x\right ) \, dx-4 \text {Subst}\left (\int \frac {\text {Li}_4(x)}{x} \, dx,x,e^x\right )-\frac {20}{3} \int x \text {Li}_3\left (e^x\right ) \, dx-\frac {20}{3} \int x \text {Li}_4\left (e^x\right ) \, dx+120 \int x^2 \text {Li}_3\left (e^x\right ) \, dx+120 \int x^2 \text {Li}_4\left (e^x\right ) \, dx+210 \int x^3 \text {Li}_3\left (e^x\right ) \, dx+210 \int x^3 \text {Li}_4\left (e^x\right ) \, dx-\frac {740}{3} \int \text {Li}_5\left (e^x\right ) \, dx+630 \int x^2 \text {Li}_5\left (e^x\right ) \, dx-1020 \int x \text {Li}_5\left (e^x\right ) \, dx \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}+\frac {8 \text {Li}_4\left (e^x\right )}{3}-4 \text {Li}_5\left (e^x\right )+240 x \text {Li}_5\left (e^x\right )+630 x^2 \text {Li}_5\left (e^x\right )-1020 x \text {Li}_6\left (e^x\right )+630 x^2 \text {Li}_6\left (e^x\right )-\frac {8}{3} \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^x\right )-\frac {8}{3} \text {Subst}\left (\int \frac {\text {Li}_4(x)}{x} \, dx,x,e^x\right )+\frac {20}{3} \int \text {Li}_4\left (e^x\right ) \, dx+\frac {20}{3} \int \text {Li}_5\left (e^x\right ) \, dx-240 \int x \text {Li}_4\left (e^x\right ) \, dx-240 \int x \text {Li}_5\left (e^x\right ) \, dx-\frac {740}{3} \text {Subst}\left (\int \frac {\text {Li}_5(x)}{x} \, dx,x,e^x\right )-630 \int x^2 \text {Li}_4\left (e^x\right ) \, dx-630 \int x^2 \text {Li}_5\left (e^x\right ) \, dx+1020 \int \text {Li}_6\left (e^x\right ) \, dx-1260 \int x \text {Li}_6\left (e^x\right ) \, dx \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}-\frac {20 \text {Li}_5\left (e^x\right )}{3}-\frac {740 \text {Li}_6\left (e^x\right )}{3}-1260 x \text {Li}_6\left (e^x\right )-1260 x \text {Li}_7\left (e^x\right )+\frac {20}{3} \text {Subst}\left (\int \frac {\text {Li}_4(x)}{x} \, dx,x,e^x\right )+\frac {20}{3} \text {Subst}\left (\int \frac {\text {Li}_5(x)}{x} \, dx,x,e^x\right )+240 \int \text {Li}_5\left (e^x\right ) \, dx+240 \int \text {Li}_6\left (e^x\right ) \, dx+1020 \text {Subst}\left (\int \frac {\text {Li}_6(x)}{x} \, dx,x,e^x\right )+1260 \int x \text {Li}_5\left (e^x\right ) \, dx+1260 \int x \text {Li}_6\left (e^x\right ) \, dx+1260 \int \text {Li}_7\left (e^x\right ) \, dx \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}-240 \text {Li}_6\left (e^x\right )+1020 \text {Li}_7\left (e^x\right )+240 \text {Subst}\left (\int \frac {\text {Li}_5(x)}{x} \, dx,x,e^x\right )+240 \text {Subst}\left (\int \frac {\text {Li}_6(x)}{x} \, dx,x,e^x\right )-1260 \int \text {Li}_6\left (e^x\right ) \, dx-1260 \int \text {Li}_7\left (e^x\right ) \, dx+1260 \text {Subst}\left (\int \frac {\text {Li}_7(x)}{x} \, dx,x,e^x\right ) \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )}+1260 \text {Li}_7\left (e^x\right )+1260 \text {Li}_8\left (e^x\right )-1260 \text {Subst}\left (\int \frac {\text {Li}_6(x)}{x} \, dx,x,e^x\right )-1260 \text {Subst}\left (\int \frac {\text {Li}_7(x)}{x} \, dx,x,e^x\right ) \\ & = \frac {x^3}{36 \left (1-e^x\right )}-\frac {x^4}{9 \left (1-e^x\right )}-\frac {x^5}{18 \left (1-e^x\right )}+\frac {x^6}{3 \left (1-e^x\right )}+\frac {x^7}{4 \left (1-e^x\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=-\frac {x^3 \left (-1+2 x+3 x^2\right )^2}{36 \left (-1+e^x\right )} \]

[In]

Integrate[(3*x^2 - 16*x^3 - 10*x^4 + 72*x^5 + 63*x^6 + E^x*(-3*x^2 + 17*x^3 + 6*x^4 - 74*x^5 - 51*x^6 + 9*x^7)
)/(36 - 72*E^x + 36*E^(2*x)),x]

[Out]

-1/36*(x^3*(-1 + 2*x + 3*x^2)^2)/(-1 + E^x)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {x^{3} \left (9 x^{4}+12 x^{3}-2 x^{2}-4 x +1\right )}{36 \left ({\mathrm e}^{x}-1\right )}\)	\(32\)
parallelrisch	\(-\frac {9 x^{7}+12 x^{6}-2 x^{5}-4 x^{4}+x^{3}}{36 \left ({\mathrm e}^{x}-1\right )}\)	\(33\)
norman	\(\frac {-\frac {1}{36} x^{3}+\frac {1}{9} x^{4}+\frac {1}{18} x^{5}-\frac {1}{3} x^{6}-\frac {1}{4} x^{7}}{{\mathrm e}^{x}-1}\)	\(34\)

[In]

int(((9*x^7-51*x^6-74*x^5+6*x^4+17*x^3-3*x^2)*exp(x)+63*x^6+72*x^5-10*x^4-16*x^3+3*x^2)/(36*exp(x)^2-72*exp(x)
+36),x,method=_RETURNVERBOSE)

[Out]

-1/36*x^3*(9*x^4+12*x^3-2*x^2-4*x+1)/(exp(x)-1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=-\frac {9 \, x^{7} + 12 \, x^{6} - 2 \, x^{5} - 4 \, x^{4} + x^{3}}{36 \, {\left (e^{x} - 1\right )}} \]

[In]

integrate(((9*x^7-51*x^6-74*x^5+6*x^4+17*x^3-3*x^2)*exp(x)+63*x^6+72*x^5-10*x^4-16*x^3+3*x^2)/(36*exp(x)^2-72*
exp(x)+36),x, algorithm="fricas")

[Out]

-1/36*(9*x^7 + 12*x^6 - 2*x^5 - 4*x^4 + x^3)/(e^x - 1)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=\frac {- 9 x^{7} - 12 x^{6} + 2 x^{5} + 4 x^{4} - x^{3}}{36 e^{x} - 36} \]

[In]

integrate(((9*x**7-51*x**6-74*x**5+6*x**4+17*x**3-3*x**2)*exp(x)+63*x**6+72*x**5-10*x**4-16*x**3+3*x**2)/(36*e
xp(x)**2-72*exp(x)+36),x)

[Out]

(-9*x**7 - 12*x**6 + 2*x**5 + 4*x**4 - x**3)/(36*exp(x) - 36)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=-\frac {9 \, x^{7} + 12 \, x^{6} - 2 \, x^{5} - 4 \, x^{4} + x^{3}}{36 \, {\left (e^{x} - 1\right )}} \]

[In]

integrate(((9*x^7-51*x^6-74*x^5+6*x^4+17*x^3-3*x^2)*exp(x)+63*x^6+72*x^5-10*x^4-16*x^3+3*x^2)/(36*exp(x)^2-72*
exp(x)+36),x, algorithm="maxima")

[Out]

-1/36*(9*x^7 + 12*x^6 - 2*x^5 - 4*x^4 + x^3)/(e^x - 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=-\frac {9 \, x^{7} + 12 \, x^{6} - 2 \, x^{5} - 4 \, x^{4} + x^{3}}{36 \, {\left (e^{x} - 1\right )}} \]

[In]

integrate(((9*x^7-51*x^6-74*x^5+6*x^4+17*x^3-3*x^2)*exp(x)+63*x^6+72*x^5-10*x^4-16*x^3+3*x^2)/(36*exp(x)^2-72*
exp(x)+36),x, algorithm="giac")

[Out]

-1/36*(9*x^7 + 12*x^6 - 2*x^5 - 4*x^4 + x^3)/(e^x - 1)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {3 x^2-16 x^3-10 x^4+72 x^5+63 x^6+e^x \left (-3 x^2+17 x^3+6 x^4-74 x^5-51 x^6+9 x^7\right )}{36-72 e^x+36 e^{2 x}} \, dx=-\frac {x^3\,{\left (3\,x^2+2\,x-1\right )}^2}{36\,\left ({\mathrm {e}}^x-1\right )} \]

[In]

int((3*x^2 - 16*x^3 - 10*x^4 + 72*x^5 + 63*x^6 - exp(x)*(3*x^2 - 17*x^3 - 6*x^4 + 74*x^5 + 51*x^6 - 9*x^7))/(3
6*exp(2*x) - 72*exp(x) + 36),x)

[Out]

-(x^3*(2*x + 3*x^2 - 1)^2)/(36*(exp(x) - 1))