\(\int \frac {-3 x^3+e^3 (10 x^2-x^3-x^4)+e^x (-6 x^3+e^3 (-10 x^2-2 x^3-2 x^4))+e^{2 x} (6 x^3+e^3 (2 x^3+2 x^4))+(-30 x+e^3 (40 x-10 x^2)+e^x (30 x-30 x^2+e^3 (10 x-10 x^3))) \log (3+e^3 (1+x))+(-150+e^3 (-50-50 x)) \log ^2(3+e^3 (1+x))}{3 x^3+e^3 (x^3+x^4)} \, dx\) [4712]

Optimal result

Integrand size = 181, antiderivative size = 26 \[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=-x+\left (-1+e^x-\frac {5 \log \left (3+e^3 (1+x)\right )}{x}\right )^2 \]

[Out]

(exp(x)-1-5*ln((1+x)*exp(3)+3)/x)^2-x

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.93 (sec) , antiderivative size = 365, normalized size of antiderivative = 14.04, number of steps used = 39, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6873, 6874, 2225, 6820, 2209, 2228, 2634, 907, 2465, 2442, 36, 29, 31, 2439, 2438, 2437, 2338, 2445, 2458, 2389, 2379, 2351} \[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=\frac {50 e^6 \operatorname {PolyLog}\left (2,-\frac {e^3 x}{3+e^3}\right )}{\left (3+e^3\right )^2}-\frac {50 e^6 \operatorname {PolyLog}\left (2,\frac {3+e^3}{e^3 x+e^3+3}\right )}{\left (3+e^3\right )^2}+\frac {25 \log ^2\left (e^3 x+e^3+3\right )}{x^2}-2 e^x+e^{2 x}-x+\frac {25 e^6 \log ^2\left (e^3 (x+1)+3\right )}{\left (3+e^3\right )^2}-\frac {10 e^x \log \left (e^3 x+e^3+3\right )}{x}+\frac {10 \left (3-4 e^3\right ) \log \left (e^3 x+e^3+3\right )}{\left (3+e^3\right ) x}-\frac {10 e^3 \log \left (e^3 x+e^3+3\right )}{3+e^3}+\frac {10 e^3 \left (3-4 e^3\right ) \log \left (e^3 x+e^3+3\right )}{\left (3+e^3\right )^2}-\frac {50 e^6 \log \left (3+e^3\right ) \log (x)}{\left (3+e^3\right )^2}+\frac {10 e^3 \log (x)}{3+e^3}-\frac {10 e^3 \left (3-4 e^3\right ) \log (x)}{\left (3+e^3\right )^2}-\frac {50 e^6 \log (x)}{\left (3+e^3\right )^2}+\frac {50 e^3 \left (e^3 x+e^3+3\right ) \log \left (e^3 (x+1)+3\right )}{\left (3+e^3\right )^2 x}+\frac {50 e^6 \log \left (e^3 (x+1)+3\right ) \log \left (1-\frac {3+e^3}{e^3 x+e^3+3}\right )}{\left (3+e^3\right )^2} \]

[In]

Int[(-3*x^3 + E^3*(10*x^2 - x^3 - x^4) + E^x*(-6*x^3 + E^3*(-10*x^2 - 2*x^3 - 2*x^4)) + E^(2*x)*(6*x^3 + E^3*(
2*x^3 + 2*x^4)) + (-30*x + E^3*(40*x - 10*x^2) + E^x*(30*x - 30*x^2 + E^3*(10*x - 10*x^3)))*Log[3 + E^3*(1 + x
)] + (-150 + E^3*(-50 - 50*x))*Log[3 + E^3*(1 + x)]^2)/(3*x^3 + E^3*(x^3 + x^4)),x]

[Out]

-2*E^x + E^(2*x) - x - (50*E^6*Log[x])/(3 + E^3)^2 - (10*E^3*(3 - 4*E^3)*Log[x])/(3 + E^3)^2 + (10*E^3*Log[x])
/(3 + E^3) - (50*E^6*Log[3 + E^3]*Log[x])/(3 + E^3)^2 + (10*E^3*(3 - 4*E^3)*Log[3 + E^3 + E^3*x])/(3 + E^3)^2
- (10*E^3*Log[3 + E^3 + E^3*x])/(3 + E^3) - (10*E^x*Log[3 + E^3 + E^3*x])/x + (10*(3 - 4*E^3)*Log[3 + E^3 + E^
3*x])/((3 + E^3)*x) + (25*Log[3 + E^3 + E^3*x]^2)/x^2 + (50*E^3*(3 + E^3 + E^3*x)*Log[3 + E^3*(1 + x)])/((3 +
E^3)^2*x) + (25*E^6*Log[3 + E^3*(1 + x)]^2)/(3 + E^3)^2 + (50*E^6*Log[3 + E^3*(1 + x)]*Log[1 - (3 + E^3)/(3 +
E^3 + E^3*x)])/(3 + E^3)^2 + (50*E^6*PolyLog[2, -((E^3*x)/(3 + E^3))])/(3 + E^3)^2 - (50*E^6*PolyLog[2, (3 + E
^3)/(3 + E^3 + E^3*x)])/(3 + E^3)^2

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2228

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[g*u^(m + 1)*(F^(c*v)/(b*c*
e*Log[F])), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2437

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

Rule 2439

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

Rule 2442

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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6820

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

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{x^3 \left (3+e^3+e^3 x\right )} \, dx \\ & = \int \left (2 e^{2 x}+\frac {2 e^x \left (-5 e^3 x-3 \left (1+\frac {e^3}{3}\right ) x^2-e^3 x^3+15 \left (1+\frac {e^3}{3}\right ) \log \left (3+e^3 (1+x)\right )-15 x \log \left (3+e^3 (1+x)\right )-5 e^3 x^2 \log \left (3+e^3 (1+x)\right )\right )}{x^2 \left (3+e^3+e^3 x\right )}+\frac {10 e^3 x^2-3 \left (1+\frac {e^3}{3}\right ) x^3-e^3 x^4-30 \left (1-\frac {4 e^3}{3}\right ) x \log \left (3+e^3 (1+x)\right )-10 e^3 x^2 \log \left (3+e^3 (1+x)\right )-150 \left (1+\frac {e^3}{3}\right ) \log ^2\left (3+e^3 (1+x)\right )-50 e^3 x \log ^2\left (3+e^3 (1+x)\right )}{x^3 \left (3+e^3+e^3 x\right )}\right ) \, dx \\ & = 2 \int e^{2 x} \, dx+2 \int \frac {e^x \left (-5 e^3 x-3 \left (1+\frac {e^3}{3}\right ) x^2-e^3 x^3+15 \left (1+\frac {e^3}{3}\right ) \log \left (3+e^3 (1+x)\right )-15 x \log \left (3+e^3 (1+x)\right )-5 e^3 x^2 \log \left (3+e^3 (1+x)\right )\right )}{x^2 \left (3+e^3+e^3 x\right )} \, dx+\int \frac {10 e^3 x^2-3 \left (1+\frac {e^3}{3}\right ) x^3-e^3 x^4-30 \left (1-\frac {4 e^3}{3}\right ) x \log \left (3+e^3 (1+x)\right )-10 e^3 x^2 \log \left (3+e^3 (1+x)\right )-150 \left (1+\frac {e^3}{3}\right ) \log ^2\left (3+e^3 (1+x)\right )-50 e^3 x \log ^2\left (3+e^3 (1+x)\right )}{x^3 \left (3+e^3+e^3 x\right )} \, dx \\ & = e^{2 x}+2 \int \frac {e^x \left (-\frac {x \left (3 x+e^3 \left (5+x+x^2\right )\right )}{3+e^3 (1+x)}-5 (-1+x) \log \left (3+e^3 (1+x)\right )\right )}{x^2} \, dx+\int \frac {-x^2 \left (3 x+e^3 \left (-10+x+x^2\right )\right )-10 \left (3+e^3 (-4+x)\right ) x \log \left (3+e^3 (1+x)\right )-50 \left (3+e^3 (1+x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{x^3 \left (3+e^3+e^3 x\right )} \, dx \\ & = e^{2 x}+2 \int \left (\frac {e^x \left (-5 e^3-\left (3+e^3\right ) x-e^3 x^2\right )}{x \left (3+e^3+e^3 x\right )}+\frac {5 e^x (1-x) \log \left (3+e^3+e^3 x\right )}{x^2}\right ) \, dx+\int \left (\frac {10 e^3-\left (3+e^3\right ) x-e^3 x^2}{x \left (3+e^3+e^3 x\right )}+\frac {10 \left (-3+4 e^3-e^3 x\right ) \log \left (3+e^3+e^3 x\right )}{x^2 \left (3+e^3+e^3 x\right )}-\frac {50 \log ^2\left (3+e^3+e^3 x\right )}{x^3}\right ) \, dx \\ & = e^{2 x}+2 \int \frac {e^x \left (-5 e^3-\left (3+e^3\right ) x-e^3 x^2\right )}{x \left (3+e^3+e^3 x\right )} \, dx+10 \int \frac {e^x (1-x) \log \left (3+e^3+e^3 x\right )}{x^2} \, dx+10 \int \frac {\left (-3+4 e^3-e^3 x\right ) \log \left (3+e^3+e^3 x\right )}{x^2 \left (3+e^3+e^3 x\right )} \, dx-50 \int \frac {\log ^2\left (3+e^3+e^3 x\right )}{x^3} \, dx+\int \frac {10 e^3-\left (3+e^3\right ) x-e^3 x^2}{x \left (3+e^3+e^3 x\right )} \, dx \\ & = e^{2 x}-\frac {10 e^x \log \left (3+e^3+e^3 x\right )}{x}+\frac {25 \log ^2\left (3+e^3+e^3 x\right )}{x^2}+2 \int \left (-e^x-\frac {5 e^{3+x}}{\left (3+e^3\right ) x}+\frac {5 e^{6+x}}{\left (3+e^3\right ) \left (3+e^3+e^3 x\right )}\right ) \, dx+10 \int \frac {e^{3+x}}{x \left (3+e^3+e^3 x\right )} \, dx+10 \int \left (\frac {\left (-3+4 e^3\right ) \log \left (3+e^3+e^3 x\right )}{\left (3+e^3\right ) x^2}-\frac {5 e^6 \log \left (3+e^3+e^3 x\right )}{\left (3+e^3\right )^2 x}+\frac {5 e^9 \log \left (3+e^3+e^3 x\right )}{\left (3+e^3\right )^2 \left (3+e^3+e^3 x\right )}\right ) \, dx-\left (50 e^3\right ) \int \frac {\log \left (3+e^3+e^3 x\right )}{x^2 \left (3+e^3+e^3 x\right )} \, dx+\int \left (-1+\frac {10 e^3}{\left (3+e^3\right ) x}-\frac {10 e^6}{\left (3+e^3\right ) \left (3+e^3+e^3 x\right )}\right ) \, dx \\ & = e^{2 x}-x+\frac {10 e^3 \log (x)}{3+e^3}-\frac {10 e^3 \log \left (3+e^3+e^3 x\right )}{3+e^3}-\frac {10 e^x \log \left (3+e^3+e^3 x\right )}{x}+\frac {25 \log ^2\left (3+e^3+e^3 x\right )}{x^2}-2 \int e^x \, dx+10 \int \left (\frac {e^{3+x}}{\left (3+e^3\right ) x}-\frac {e^{6+x}}{\left (3+e^3\right ) \left (3+e^3+e^3 x\right )}\right ) \, dx-50 \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {-3-e^3}{e^3}+\frac {x}{e^3}\right )^2} \, dx,x,3+e^3+e^3 x\right )-\frac {\left (50 e^6\right ) \int \frac {\log \left (3+e^3+e^3 x\right )}{x} \, dx}{\left (3+e^3\right )^2}+\frac {\left (50 e^9\right ) \int \frac {\log \left (3+e^3+e^3 x\right )}{3+e^3+e^3 x} \, dx}{\left (3+e^3\right )^2}-\frac {10 \int \frac {e^{3+x}}{x} \, dx}{3+e^3}+\frac {10 \int \frac {e^{6+x}}{3+e^3+e^3 x} \, dx}{3+e^3}-\frac {\left (10 \left (3-4 e^3\right )\right ) \int \frac {\log \left (3+e^3+e^3 x\right )}{x^2} \, dx}{3+e^3} \\ & = -2 e^x+e^{2 x}-x-\frac {10 e^3 \text {Ei}(x)}{3+e^3}+\frac {10 e^{2-\frac {3}{e^3}} \text {Ei}\left (\frac {3+e^3+e^3 x}{e^3}\right )}{3+e^3}+\frac {10 e^3 \log (x)}{3+e^3}-\frac {50 e^6 \log \left (3+e^3\right ) \log (x)}{\left (3+e^3\right )^2}-\frac {10 e^3 \log \left (3+e^3+e^3 x\right )}{3+e^3}-\frac {10 e^x \log \left (3+e^3+e^3 x\right )}{x}+\frac {10 \left (3-4 e^3\right ) \log \left (3+e^3+e^3 x\right )}{\left (3+e^3\right ) x}+\frac {25 \log ^2\left (3+e^3+e^3 x\right )}{x^2}-\frac {\left (50 e^6\right ) \int \frac {\log \left (1+\frac {e^3 x}{3+e^3}\right )}{x} \, dx}{\left (3+e^3\right )^2}+\frac {\left (50 e^6\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,3+e^3+e^3 x\right )}{\left (3+e^3\right )^2}+\frac {10 \int \frac {e^{3+x}}{x} \, dx}{3+e^3}-\frac {10 \int \frac {e^{6+x}}{3+e^3+e^3 x} \, dx}{3+e^3}-\frac {50 \text {Subst}\left (\int \frac {\log (x)}{\left (\frac {-3-e^3}{e^3}+\frac {x}{e^3}\right )^2} \, dx,x,3+e^3+e^3 x\right )}{3+e^3}+\frac {\left (50 e^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {-3-e^3}{e^3}+\frac {x}{e^3}\right )} \, dx,x,3+e^3+e^3 x\right )}{3+e^3}-\frac {\left (10 e^3 \left (3-4 e^3\right )\right ) \int \frac {1}{x \left (3+e^3+e^3 x\right )} \, dx}{3+e^3} \\ & = -2 e^x+e^{2 x}-x+\frac {10 e^3 \log (x)}{3+e^3}-\frac {50 e^6 \log \left (3+e^3\right ) \log (x)}{\left (3+e^3\right )^2}-\frac {10 e^3 \log \left (3+e^3+e^3 x\right )}{3+e^3}-\frac {10 e^x \log \left (3+e^3+e^3 x\right )}{x}+\frac {10 \left (3-4 e^3\right ) \log \left (3+e^3+e^3 x\right )}{\left (3+e^3\right ) x}+\frac {25 \log ^2\left (3+e^3+e^3 x\right )}{x^2}+\frac {50 e^3 \left (3+e^3+e^3 x\right ) \log \left (3+e^3 (1+x)\right )}{\left (3+e^3\right )^2 x}+\frac {25 e^6 \log ^2\left (3+e^3 (1+x)\right )}{\left (3+e^3\right )^2}+\frac {50 e^6 \log \left (3+e^3 (1+x)\right ) \log \left (1-\frac {3+e^3}{3+e^3+e^3 x}\right )}{\left (3+e^3\right )^2}+\frac {50 e^6 \text {Li}_2\left (-\frac {e^3 x}{3+e^3}\right )}{\left (3+e^3\right )^2}-\frac {\left (50 e^3\right ) \text {Subst}\left (\int \frac {1}{\frac {-3-e^3}{e^3}+\frac {x}{e^3}} \, dx,x,3+e^3+e^3 x\right )}{\left (3+e^3\right )^2}-\frac {\left (50 e^6\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {-3-e^3}{x}\right )}{x} \, dx,x,3+e^3+e^3 x\right )}{\left (3+e^3\right )^2}-\frac {\left (10 e^3 \left (3-4 e^3\right )\right ) \int \frac {1}{x} \, dx}{\left (3+e^3\right )^2}+\frac {\left (10 e^6 \left (3-4 e^3\right )\right ) \int \frac {1}{3+e^3+e^3 x} \, dx}{\left (3+e^3\right )^2} \\ & = -2 e^x+e^{2 x}-x-\frac {50 e^6 \log (x)}{\left (3+e^3\right )^2}-\frac {10 e^3 \left (3-4 e^3\right ) \log (x)}{\left (3+e^3\right )^2}+\frac {10 e^3 \log (x)}{3+e^3}-\frac {50 e^6 \log \left (3+e^3\right ) \log (x)}{\left (3+e^3\right )^2}+\frac {10 e^3 \left (3-4 e^3\right ) \log \left (3+e^3+e^3 x\right )}{\left (3+e^3\right )^2}-\frac {10 e^3 \log \left (3+e^3+e^3 x\right )}{3+e^3}-\frac {10 e^x \log \left (3+e^3+e^3 x\right )}{x}+\frac {10 \left (3-4 e^3\right ) \log \left (3+e^3+e^3 x\right )}{\left (3+e^3\right ) x}+\frac {25 \log ^2\left (3+e^3+e^3 x\right )}{x^2}+\frac {50 e^3 \left (3+e^3+e^3 x\right ) \log \left (3+e^3 (1+x)\right )}{\left (3+e^3\right )^2 x}+\frac {25 e^6 \log ^2\left (3+e^3 (1+x)\right )}{\left (3+e^3\right )^2}+\frac {50 e^6 \log \left (3+e^3 (1+x)\right ) \log \left (1-\frac {3+e^3}{3+e^3+e^3 x}\right )}{\left (3+e^3\right )^2}+\frac {50 e^6 \text {Li}_2\left (-\frac {e^3 x}{3+e^3}\right )}{\left (3+e^3\right )^2}-\frac {50 e^6 \text {Li}_2\left (\frac {3+e^3}{3+e^3+e^3 x}\right )}{\left (3+e^3\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=-2 e^x+e^{2 x}-x-\frac {10 \left (-1+e^x\right ) \log \left (3+e^3 (1+x)\right )}{x}+\frac {25 \log ^2\left (3+e^3 (1+x)\right )}{x^2} \]

[In]

Integrate[(-3*x^3 + E^3*(10*x^2 - x^3 - x^4) + E^x*(-6*x^3 + E^3*(-10*x^2 - 2*x^3 - 2*x^4)) + E^(2*x)*(6*x^3 +
E^3*(2*x^3 + 2*x^4)) + (-30*x + E^3*(40*x - 10*x^2) + E^x*(30*x - 30*x^2 + E^3*(10*x - 10*x^3)))*Log[3 + E^3*
(1 + x)] + (-150 + E^3*(-50 - 50*x))*Log[3 + E^3*(1 + x)]^2)/(3*x^3 + E^3*(x^3 + x^4)),x]

[Out]

-2*E^x + E^(2*x) - x - (10*(-1 + E^x)*Log[3 + E^3*(1 + x)])/x + (25*Log[3 + E^3*(1 + x)]^2)/x^2

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81

method	result	size
risch	\(\frac {25 \ln \left (\left (1+x \right ) {\mathrm e}^{3}+3\right )^{2}}{x^{2}}-\frac {10 \left ({\mathrm e}^{x}-1\right ) \ln \left (\left (1+x \right ) {\mathrm e}^{3}+3\right )}{x}+{\mathrm e}^{2 x}-x -2 \,{\mathrm e}^{x}\)	\(47\)
parallelrisch	\(\frac {\left (x^{2} {\mathrm e}^{6} {\mathrm e}^{2 x}-x^{3} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{6} {\mathrm e}^{x}-10 \,{\mathrm e}^{6} x \,{\mathrm e}^{x} \ln \left (\left (1+x \right ) {\mathrm e}^{3}+3\right )+2 x^{2} {\mathrm e}^{6}+10 \,{\mathrm e}^{6} x \ln \left (\left (1+x \right ) {\mathrm e}^{3}+3\right )+25 \,{\mathrm e}^{6} \ln \left (\left (1+x \right ) {\mathrm e}^{3}+3\right )^{2}+6 x^{2} {\mathrm e}^{3}\right ) {\mathrm e}^{-6}}{x^{2}}\)	\(109\)

[In]

int((((-50*x-50)*exp(3)-150)*ln((1+x)*exp(3)+3)^2+(((-10*x^3+10*x)*exp(3)-30*x^2+30*x)*exp(x)+(-10*x^2+40*x)*e
xp(3)-30*x)*ln((1+x)*exp(3)+3)+((2*x^4+2*x^3)*exp(3)+6*x^3)*exp(x)^2+((-2*x^4-2*x^3-10*x^2)*exp(3)-6*x^3)*exp(
x)+(-x^4-x^3+10*x^2)*exp(3)-3*x^3)/((x^4+x^3)*exp(3)+3*x^3),x,method=_RETURNVERBOSE)

[Out]

25/x^2*ln((1+x)*exp(3)+3)^2-10*(exp(x)-1)/x*ln((1+x)*exp(3)+3)+exp(2*x)-x-2*exp(x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (26) = 52\).

Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=-\frac {x^{3} - x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{x} + 10 \, {\left (x e^{x} - x\right )} \log \left ({\left (x + 1\right )} e^{3} + 3\right ) - 25 \, \log \left ({\left (x + 1\right )} e^{3} + 3\right )^{2}}{x^{2}} \]

[In]

integrate((((-50*x-50)*exp(3)-150)*log((1+x)*exp(3)+3)^2+(((-10*x^3+10*x)*exp(3)-30*x^2+30*x)*exp(x)+(-10*x^2+
40*x)*exp(3)-30*x)*log((1+x)*exp(3)+3)+((2*x^4+2*x^3)*exp(3)+6*x^3)*exp(x)^2+((-2*x^4-2*x^3-10*x^2)*exp(3)-6*x
^3)*exp(x)+(-x^4-x^3+10*x^2)*exp(3)-3*x^3)/((x^4+x^3)*exp(3)+3*x^3),x, algorithm="fricas")

[Out]

-(x^3 - x^2*e^(2*x) + 2*x^2*e^x + 10*(x*e^x - x)*log((x + 1)*e^3 + 3) - 25*log((x + 1)*e^3 + 3)^2)/x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).

Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=- x + \frac {x e^{2 x} + \left (- 2 x - 10 \log {\left (\left (x + 1\right ) e^{3} + 3 \right )}\right ) e^{x}}{x} + \frac {10 \log {\left (\left (x + 1\right ) e^{3} + 3 \right )}}{x} + \frac {25 \log {\left (\left (x + 1\right ) e^{3} + 3 \right )}^{2}}{x^{2}} \]

[In]

integrate((((-50*x-50)*exp(3)-150)*ln((1+x)*exp(3)+3)**2+(((-10*x**3+10*x)*exp(3)-30*x**2+30*x)*exp(x)+(-10*x*
*2+40*x)*exp(3)-30*x)*ln((1+x)*exp(3)+3)+((2*x**4+2*x**3)*exp(3)+6*x**3)*exp(x)**2+((-2*x**4-2*x**3-10*x**2)*e
xp(3)-6*x**3)*exp(x)+(-x**4-x**3+10*x**2)*exp(3)-3*x**3)/((x**4+x**3)*exp(3)+3*x**3),x)

[Out]

-x + (x*exp(2*x) + (-2*x - 10*log((x + 1)*exp(3) + 3))*exp(x))/x + 10*log((x + 1)*exp(3) + 3)/x + 25*log((x +
1)*exp(3) + 3)**2/x**2

Maxima [F]

\[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=\int { -\frac {3 \, x^{3} + 50 \, {\left ({\left (x + 1\right )} e^{3} + 3\right )} \log \left ({\left (x + 1\right )} e^{3} + 3\right )^{2} + {\left (x^{4} + x^{3} - 10 \, x^{2}\right )} e^{3} - 2 \, {\left (3 \, x^{3} + {\left (x^{4} + x^{3}\right )} e^{3}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (3 \, x^{3} + {\left (x^{4} + x^{3} + 5 \, x^{2}\right )} e^{3}\right )} e^{x} + 10 \, {\left ({\left (x^{2} - 4 \, x\right )} e^{3} + {\left (3 \, x^{2} + {\left (x^{3} - x\right )} e^{3} - 3 \, x\right )} e^{x} + 3 \, x\right )} \log \left ({\left (x + 1\right )} e^{3} + 3\right )}{3 \, x^{3} + {\left (x^{4} + x^{3}\right )} e^{3}} \,d x } \]

[In]

integrate((((-50*x-50)*exp(3)-150)*log((1+x)*exp(3)+3)^2+(((-10*x^3+10*x)*exp(3)-30*x^2+30*x)*exp(x)+(-10*x^2+
40*x)*exp(3)-30*x)*log((1+x)*exp(3)+3)+((2*x^4+2*x^3)*exp(3)+6*x^3)*exp(x)^2+((-2*x^4-2*x^3-10*x^2)*exp(3)-6*x
^3)*exp(x)+(-x^4-x^3+10*x^2)*exp(3)-3*x^3)/((x^4+x^3)*exp(3)+3*x^3),x, algorithm="maxima")

[Out]

((e^3 + 3)*e^(-6)*log(x*e^3 + e^3 + 3) - x*e^(-3))*e^3 - 10*(log(x*e^3 + e^3 + 3)/(e^3 + 3) - log(x)/(e^3 + 3)
)*e^3 + 6*e^(-(e^3 + 3)*e^(-3) - 3)*exp_integral_e(1, -(x*e^3 + e^3 + 3)*e^(-3)) - 6*e^(-2*(e^3 + 3)*e^(-3) -
3)*exp_integral_e(1, -2*(x*e^3 + e^3 + 3)*e^(-3)) - 3*e^(-3)*log(x*e^3 + e^3 + 3) - 10*e^3*log(x)/(e^3 + 3) +
(x^3*(e^6 + 3*e^3)*e^(2*x) - 2*x^3*(e^6 + 3*e^3)*e^x + 25*(x*(e^6 + 3*e^3) + e^6 + 6*e^3 + 9)*log(x*e^3 + e^3
+ 3)^2 + 10*(x^3*e^6 + 2*x^2*(e^6 + 3*e^3) + x*(e^6 + 6*e^3 + 9) - (x^2*(e^6 + 3*e^3) + x*(e^6 + 6*e^3 + 9))*e
^x)*log(x*e^3 + e^3 + 3))/(x^3*(e^6 + 3*e^3) + x^2*(e^6 + 6*e^3 + 9)) + integrate((2*x*e^6 + e^6 + 3*e^3)*e^(2
*x)/(x^2*e^6 + 2*x*(e^6 + 3*e^3) + e^6 + 6*e^3 + 9), x) - 2*integrate(x*e^(x + 6)/(x^2*e^6 + 2*x*(e^6 + 3*e^3)
+ e^6 + 6*e^3 + 9), x) - log(x*e^3 + e^3 + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=-\frac {x^{3} - x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{x} + 10 \, x e^{x} \log \left (x e^{3} + e^{3} + 3\right ) - 10 \, x \log \left (x e^{3} + e^{3} + 3\right ) - 25 \, \log \left (x e^{3} + e^{3} + 3\right )^{2}}{x^{2}} \]

[In]

integrate((((-50*x-50)*exp(3)-150)*log((1+x)*exp(3)+3)^2+(((-10*x^3+10*x)*exp(3)-30*x^2+30*x)*exp(x)+(-10*x^2+
40*x)*exp(3)-30*x)*log((1+x)*exp(3)+3)+((2*x^4+2*x^3)*exp(3)+6*x^3)*exp(x)^2+((-2*x^4-2*x^3-10*x^2)*exp(3)-6*x
^3)*exp(x)+(-x^4-x^3+10*x^2)*exp(3)-3*x^3)/((x^4+x^3)*exp(3)+3*x^3),x, algorithm="giac")

[Out]

-(x^3 - x^2*e^(2*x) + 2*x^2*e^x + 10*x*e^x*log(x*e^3 + e^3 + 3) - 10*x*log(x*e^3 + e^3 + 3) - 25*log(x*e^3 + e
^3 + 3)^2)/x^2

Mupad [F(-1)]

Timed out. \[ \int \frac {-3 x^3+e^3 \left (10 x^2-x^3-x^4\right )+e^x \left (-6 x^3+e^3 \left (-10 x^2-2 x^3-2 x^4\right )\right )+e^{2 x} \left (6 x^3+e^3 \left (2 x^3+2 x^4\right )\right )+\left (-30 x+e^3 \left (40 x-10 x^2\right )+e^x \left (30 x-30 x^2+e^3 \left (10 x-10 x^3\right )\right )\right ) \log \left (3+e^3 (1+x)\right )+\left (-150+e^3 (-50-50 x)\right ) \log ^2\left (3+e^3 (1+x)\right )}{3 x^3+e^3 \left (x^3+x^4\right )} \, dx=\int -\frac {{\mathrm {e}}^3\,\left (x^4+x^3-10\,x^2\right )-\ln \left ({\mathrm {e}}^3\,\left (x+1\right )+3\right )\,\left ({\mathrm {e}}^3\,\left (40\,x-10\,x^2\right )-30\,x+{\mathrm {e}}^x\,\left (30\,x+{\mathrm {e}}^3\,\left (10\,x-10\,x^3\right )-30\,x^2\right )\right )+{\ln \left ({\mathrm {e}}^3\,\left (x+1\right )+3\right )}^2\,\left ({\mathrm {e}}^3\,\left (50\,x+50\right )+150\right )+{\mathrm {e}}^x\,\left ({\mathrm {e}}^3\,\left (2\,x^4+2\,x^3+10\,x^2\right )+6\,x^3\right )-{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^3\,\left (2\,x^4+2\,x^3\right )+6\,x^3\right )+3\,x^3}{{\mathrm {e}}^3\,\left (x^4+x^3\right )+3\,x^3} \,d x \]

[In]

int(-(exp(3)*(x^3 - 10*x^2 + x^4) - log(exp(3)*(x + 1) + 3)*(exp(3)*(40*x - 10*x^2) - 30*x + exp(x)*(30*x + ex
p(3)*(10*x - 10*x^3) - 30*x^2)) + log(exp(3)*(x + 1) + 3)^2*(exp(3)*(50*x + 50) + 150) + exp(x)*(exp(3)*(10*x^
2 + 2*x^3 + 2*x^4) + 6*x^3) - exp(2*x)*(exp(3)*(2*x^3 + 2*x^4) + 6*x^3) + 3*x^3)/(exp(3)*(x^3 + x^4) + 3*x^3),
x)

[Out]

int(-(exp(3)*(x^3 - 10*x^2 + x^4) - log(exp(3)*(x + 1) + 3)*(exp(3)*(40*x - 10*x^2) - 30*x + exp(x)*(30*x + ex
p(3)*(10*x - 10*x^3) - 30*x^2)) + log(exp(3)*(x + 1) + 3)^2*(exp(3)*(50*x + 50) + 150) + exp(x)*(exp(3)*(10*x^
2 + 2*x^3 + 2*x^4) + 6*x^3) - exp(2*x)*(exp(3)*(2*x^3 + 2*x^4) + 6*x^3) + 3*x^3)/(exp(3)*(x^3 + x^4) + 3*x^3),
x)