\(\int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} (2 x+4 x^2+2 x^3)+e^x (6 x-6 x^3-2 x^4)+(6 x-2 x^2-6 x^3+e^x (4 x+6 x^2+2 x^3)) \log (\frac {x}{1+x})+(2 x+2 x^2) \log ^2(\frac {x}{1+x})}{1+x} \, dx\) [7870]

Optimal result

Integrand size = 125, antiderivative size = 24 \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x+x^2 \left (1+e^x-x+\log \left (\frac {x}{1+x}\right )\right )^2 \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(24)=48\).

Time = 1.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 6.21, number of steps used = 58, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6874, 45, 2227, 2207, 2225, 2561, 2384, 2353, 2352, 2549, 2381, 6820, 2230, 2209, 2634} \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x^4-2 e^x x^3-2 x^3-2 x^3 \log \left (\frac {x}{x+1}\right )+2 e^x x^2+e^{2 x} x^2+x^2 \log ^2\left (\frac {x}{x+1}\right )+2 e^x x^2 \log \left (\frac {x}{x+1}\right )-x^2 \log \left (\frac {x}{x+1}\right )+x^2 \left (3 \log \left (\frac {x}{x+1}\right )+1\right )+5 x+4 x \log \left (\frac {x}{x+1}\right )+x \left (2 \log \left (\frac {x}{x+1}\right )+1\right )-x \left (6 \log \left (\frac {x}{x+1}\right )+5\right ) \]

[In]

Int[(1 + 5*x - 6*x^2 - 2*x^3 + 4*x^4 + E^(2*x)*(2*x + 4*x^2 + 2*x^3) + E^x*(6*x - 6*x^3 - 2*x^4) + (6*x - 2*x^
2 - 6*x^3 + E^x*(4*x + 6*x^2 + 2*x^3))*Log[x/(1 + x)] + (2*x + 2*x^2)*Log[x/(1 + x)]^2)/(1 + x),x]

[Out]

5*x + 2*E^x*x^2 + E^(2*x)*x^2 - 2*x^3 - 2*E^x*x^3 + x^4 + 4*x*Log[x/(1 + x)] - x^2*Log[x/(1 + x)] + 2*E^x*x^2*
Log[x/(1 + x)] - 2*x^3*Log[x/(1 + x)] + x^2*Log[x/(1 + x)]^2 + x*(1 + 2*Log[x/(1 + x)]) + x^2*(1 + 3*Log[x/(1
+ x)]) - x*(5 + 6*Log[x/(1 + x)])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2207

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

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 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 2230

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2353

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

Rule 2381

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

Rule 2384

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

Rule 2549

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/b)^m, Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x]
, x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m,
p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rule 2561

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

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{1+x}+\frac {5 x}{1+x}-\frac {6 x^2}{1+x}-\frac {2 x^3}{1+x}+\frac {4 x^4}{1+x}+2 e^{2 x} x (1+x)+\frac {6 x \log \left (\frac {x}{1+x}\right )}{1+x}-\frac {2 x^2 \log \left (\frac {x}{1+x}\right )}{1+x}-\frac {6 x^3 \log \left (\frac {x}{1+x}\right )}{1+x}+2 x \log ^2\left (\frac {x}{1+x}\right )-\frac {2 e^x x \left (-3+3 x^2+x^3-2 \log \left (\frac {x}{1+x}\right )-3 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )\right )}{1+x}\right ) \, dx \\ & = \log (1+x)-2 \int \frac {x^3}{1+x} \, dx+2 \int e^{2 x} x (1+x) \, dx-2 \int \frac {x^2 \log \left (\frac {x}{1+x}\right )}{1+x} \, dx+2 \int x \log ^2\left (\frac {x}{1+x}\right ) \, dx-2 \int \frac {e^x x \left (-3+3 x^2+x^3-2 \log \left (\frac {x}{1+x}\right )-3 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )\right )}{1+x} \, dx+4 \int \frac {x^4}{1+x} \, dx+5 \int \frac {x}{1+x} \, dx-6 \int \frac {x^2}{1+x} \, dx+6 \int \frac {x \log \left (\frac {x}{1+x}\right )}{1+x} \, dx-6 \int \frac {x^3 \log \left (\frac {x}{1+x}\right )}{1+x} \, dx \\ & = \log (1+x)-2 \int \left (1+\frac {1}{-1-x}-x+x^2\right ) \, dx+2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx-2 \int \frac {e^x x \left (-3+3 x^2+x^3-\left (2+3 x+x^2\right ) \log \left (\frac {x}{1+x}\right )\right )}{1+x} \, dx-2 \text {Subst}\left (\int \frac {x^2 \log (x)}{(1-x)^3} \, dx,x,\frac {x}{1+x}\right )+2 \text {Subst}\left (\int \frac {x \log ^2(x)}{(1-x)^3} \, dx,x,\frac {x}{1+x}\right )+4 \int \left (-1+x-x^2+x^3+\frac {1}{1+x}\right ) \, dx+5 \int \left (1+\frac {1}{-1-x}\right ) \, dx-6 \int \left (-1+x+\frac {1}{1+x}\right ) \, dx+6 \text {Subst}\left (\int \frac {x \log (x)}{(1-x)^2} \, dx,x,\frac {x}{1+x}\right )-6 \text {Subst}\left (\int \frac {x^3 \log (x)}{(1-x)^4} \, dx,x,\frac {x}{1+x}\right ) \\ & = 5 x-2 x^3+x^4+6 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )-4 \log (1+x)+2 \int e^{2 x} x \, dx+2 \int e^{2 x} x^2 \, dx-2 \int \left (\frac {e^x x \left (-3+3 x^2+x^3\right )}{1+x}-e^x x (2+x) \log \left (\frac {x}{1+x}\right )\right ) \, dx-2 \text {Subst}\left (\int \frac {x \log (x)}{(1-x)^2} \, dx,x,\frac {x}{1+x}\right )+2 \text {Subst}\left (\int \frac {x^2 (1+3 \log (x))}{(1-x)^3} \, dx,x,\frac {x}{1+x}\right )-6 \text {Subst}\left (\int \frac {1+\log (x)}{1-x} \, dx,x,\frac {x}{1+x}\right )+\text {Subst}\left (\int \frac {x (1+2 \log (x))}{(1-x)^2} \, dx,x,\frac {x}{1+x}\right ) \\ & = 5 x+e^{2 x} x+e^{2 x} x^2-2 x^3+x^4+4 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )+x \left (1+2 \log \left (\frac {x}{1+x}\right )\right )+x^2 \left (1+3 \log \left (\frac {x}{1+x}\right )\right )-10 \log (1+x)-2 \int e^{2 x} x \, dx-2 \int \frac {e^x x \left (-3+3 x^2+x^3\right )}{1+x} \, dx+2 \int e^x x (2+x) \log \left (\frac {x}{1+x}\right ) \, dx+2 \text {Subst}\left (\int \frac {1+\log (x)}{1-x} \, dx,x,\frac {x}{1+x}\right )-6 \text {Subst}\left (\int \frac {\log (x)}{1-x} \, dx,x,\frac {x}{1+x}\right )-\int e^{2 x} \, dx-\text {Subst}\left (\int \frac {3+2 \log (x)}{1-x} \, dx,x,\frac {x}{1+x}\right )-\text {Subst}\left (\int \frac {x (5+6 \log (x))}{(1-x)^2} \, dx,x,\frac {x}{1+x}\right ) \\ & = -\frac {e^{2 x}}{2}+5 x+e^{2 x} x^2-2 x^3+x^4+4 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )+2 e^x x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )+x \left (1+2 \log \left (\frac {x}{1+x}\right )\right )+x^2 \left (1+3 \log \left (\frac {x}{1+x}\right )\right )-x \left (5+6 \log \left (\frac {x}{1+x}\right )\right )-11 \log (1+x)-6 \operatorname {PolyLog}\left (2,\frac {1}{1+x}\right )-2 \int \frac {e^x x}{1+x} \, dx-2 \int \left (-e^x-2 e^x x+2 e^x x^2+e^x x^3+\frac {e^x}{1+x}\right ) \, dx+\int e^{2 x} \, dx+\text {Subst}\left (\int \frac {11+6 \log (x)}{1-x} \, dx,x,\frac {x}{1+x}\right ) \\ & = 5 x+e^{2 x} x^2-2 x^3+x^4+4 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )+2 e^x x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )+x \left (1+2 \log \left (\frac {x}{1+x}\right )\right )+x^2 \left (1+3 \log \left (\frac {x}{1+x}\right )\right )-x \left (5+6 \log \left (\frac {x}{1+x}\right )\right )-6 \operatorname {PolyLog}\left (2,\frac {1}{1+x}\right )+2 \int e^x \, dx-2 \int \left (e^x+\frac {e^x}{-1-x}\right ) \, dx-2 \int e^x x^3 \, dx-2 \int \frac {e^x}{1+x} \, dx+4 \int e^x x \, dx-4 \int e^x x^2 \, dx+6 \text {Subst}\left (\int \frac {\log (x)}{1-x} \, dx,x,\frac {x}{1+x}\right ) \\ & = 2 e^x+5 x+4 e^x x-4 e^x x^2+e^{2 x} x^2-2 x^3-2 e^x x^3+x^4-\frac {2 \operatorname {ExpIntegralEi}(1+x)}{e}+4 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )+2 e^x x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )+x \left (1+2 \log \left (\frac {x}{1+x}\right )\right )+x^2 \left (1+3 \log \left (\frac {x}{1+x}\right )\right )-x \left (5+6 \log \left (\frac {x}{1+x}\right )\right )-2 \int e^x \, dx-2 \int \frac {e^x}{-1-x} \, dx-4 \int e^x \, dx+6 \int e^x x^2 \, dx+8 \int e^x x \, dx \\ & = -4 e^x+5 x+12 e^x x+2 e^x x^2+e^{2 x} x^2-2 x^3-2 e^x x^3+x^4+4 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )+2 e^x x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )+x \left (1+2 \log \left (\frac {x}{1+x}\right )\right )+x^2 \left (1+3 \log \left (\frac {x}{1+x}\right )\right )-x \left (5+6 \log \left (\frac {x}{1+x}\right )\right )-8 \int e^x \, dx-12 \int e^x x \, dx \\ & = -12 e^x+5 x+2 e^x x^2+e^{2 x} x^2-2 x^3-2 e^x x^3+x^4+4 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )+2 e^x x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )+x \left (1+2 \log \left (\frac {x}{1+x}\right )\right )+x^2 \left (1+3 \log \left (\frac {x}{1+x}\right )\right )-x \left (5+6 \log \left (\frac {x}{1+x}\right )\right )+12 \int e^x \, dx \\ & = 5 x+2 e^x x^2+e^{2 x} x^2-2 x^3-2 e^x x^3+x^4+4 x \log \left (\frac {x}{1+x}\right )-x^2 \log \left (\frac {x}{1+x}\right )+2 e^x x^2 \log \left (\frac {x}{1+x}\right )-2 x^3 \log \left (\frac {x}{1+x}\right )+x^2 \log ^2\left (\frac {x}{1+x}\right )+x \left (1+2 \log \left (\frac {x}{1+x}\right )\right )+x^2 \left (1+3 \log \left (\frac {x}{1+x}\right )\right )-x \left (5+6 \log \left (\frac {x}{1+x}\right )\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(24)=48\).

Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x \left (1+\left (1+e^x\right )^2 x-2 \left (1+e^x\right ) x^2+x^3-2 x \left (-1-e^x+x\right ) \log \left (\frac {x}{1+x}\right )+x \log ^2\left (\frac {x}{1+x}\right )\right ) \]

[In]

Integrate[(1 + 5*x - 6*x^2 - 2*x^3 + 4*x^4 + E^(2*x)*(2*x + 4*x^2 + 2*x^3) + E^x*(6*x - 6*x^3 - 2*x^4) + (6*x
- 2*x^2 - 6*x^3 + E^x*(4*x + 6*x^2 + 2*x^3))*Log[x/(1 + x)] + (2*x + 2*x^2)*Log[x/(1 + x)]^2)/(1 + x),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(23)=46\).

Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.83

method	result	size
parallelrisch	\(x^{4}-2 \,{\mathrm e}^{x} x^{3}-2 x^{3} \ln \left (\frac {x}{1+x}\right )+{\mathrm e}^{2 x} x^{2}+2 \,{\mathrm e}^{x} \ln \left (\frac {x}{1+x}\right ) x^{2}+\ln \left (\frac {x}{1+x}\right )^{2} x^{2}-2 x^{3}+2 \,{\mathrm e}^{x} x^{2}+2 x^{2} \ln \left (\frac {x}{1+x}\right )-\frac {1}{2}+x^{2}+x\)	\(92\)
risch	\(\text {Expression too large to display}\)	\(909\)

[In]

int(((2*x^2+2*x)*ln(x/(1+x))^2+((2*x^3+6*x^2+4*x)*exp(x)-6*x^3-2*x^2+6*x)*ln(x/(1+x))+(2*x^3+4*x^2+2*x)*exp(x)
^2+(-2*x^4-6*x^3+6*x)*exp(x)+4*x^4-2*x^3-6*x^2+5*x+1)/(1+x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (25) = 50\).

Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.08 \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x^{4} + x^{2} \log \left (\frac {x}{x + 1}\right )^{2} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + x^{2} - 2 \, {\left (x^{3} - x^{2}\right )} e^{x} - 2 \, {\left (x^{3} - x^{2} e^{x} - x^{2}\right )} \log \left (\frac {x}{x + 1}\right ) + x \]

[In]

integrate(((2*x^2+2*x)*log(x/(1+x))^2+((2*x^3+6*x^2+4*x)*exp(x)-6*x^3-2*x^2+6*x)*log(x/(1+x))+(2*x^3+4*x^2+2*x
)*exp(x)^2+(-2*x^4-6*x^3+6*x)*exp(x)+4*x^4-2*x^3-6*x^2+5*x+1)/(1+x),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (19) = 38\).

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x^{4} - 2 x^{3} + x^{2} e^{2 x} + x^{2} \log {\left (\frac {x}{x + 1} \right )}^{2} + x^{2} + x + \left (- 2 x^{3} + 2 x^{2}\right ) \log {\left (\frac {x}{x + 1} \right )} + \left (- 2 x^{3} + 2 x^{2} \log {\left (\frac {x}{x + 1} \right )} + 2 x^{2}\right ) e^{x} \]

[In]

integrate(((2*x**2+2*x)*ln(x/(1+x))**2+((2*x**3+6*x**2+4*x)*exp(x)-6*x**3-2*x**2+6*x)*ln(x/(1+x))+(2*x**3+4*x*
*2+2*x)*exp(x)**2+(-2*x**4-6*x**3+6*x)*exp(x)+4*x**4-2*x**3-6*x**2+5*x+1)/(1+x),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (25) = 50\).

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.50 \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x^{4} + x^{2} \log \left (x + 1\right )^{2} + x^{2} \log \left (x\right )^{2} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + x^{2} - 2 \, {\left (x^{3} - x^{2} \log \left (x\right ) - x^{2}\right )} e^{x} + 2 \, {\left (x^{3} - x^{2} e^{x} - x^{2} \log \left (x\right ) - x^{2} + 2\right )} \log \left (x + 1\right ) - 2 \, {\left (x^{3} - x^{2}\right )} \log \left (x\right ) + x - 4 \, \log \left (x + 1\right ) \]

[In]

integrate(((2*x^2+2*x)*log(x/(1+x))^2+((2*x^3+6*x^2+4*x)*exp(x)-6*x^3-2*x^2+6*x)*log(x/(1+x))+(2*x^3+4*x^2+2*x
)*exp(x)^2+(-2*x^4-6*x^3+6*x)*exp(x)+4*x^4-2*x^3-6*x^2+5*x+1)/(1+x),x, algorithm="maxima")

[Out]

x^4 + x^2*log(x + 1)^2 + x^2*log(x)^2 - 2*x^3 + x^2*e^(2*x) + x^2 - 2*(x^3 - x^2*log(x) - x^2)*e^x + 2*(x^3 -
x^2*e^x - x^2*log(x) - x^2 + 2)*log(x + 1) - 2*(x^3 - x^2)*log(x) + x - 4*log(x + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (25) = 50\).

Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.75 \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x^{4} - 2 \, x^{3} e^{x} - 2 \, x^{3} \log \left (\frac {x}{x + 1}\right ) + 2 \, x^{2} e^{x} \log \left (\frac {x}{x + 1}\right ) + x^{2} \log \left (\frac {x}{x + 1}\right )^{2} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{x} + 2 \, x^{2} \log \left (\frac {x}{x + 1}\right ) + x^{2} + x \]

[In]

integrate(((2*x^2+2*x)*log(x/(1+x))^2+((2*x^3+6*x^2+4*x)*exp(x)-6*x^3-2*x^2+6*x)*log(x/(1+x))+(2*x^3+4*x^2+2*x
)*exp(x)^2+(-2*x^4-6*x^3+6*x)*exp(x)+4*x^4-2*x^3-6*x^2+5*x+1)/(1+x),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.17 \[ \int \frac {1+5 x-6 x^2-2 x^3+4 x^4+e^{2 x} \left (2 x+4 x^2+2 x^3\right )+e^x \left (6 x-6 x^3-2 x^4\right )+\left (6 x-2 x^2-6 x^3+e^x \left (4 x+6 x^2+2 x^3\right )\right ) \log \left (\frac {x}{1+x}\right )+\left (2 x+2 x^2\right ) \log ^2\left (\frac {x}{1+x}\right )}{1+x} \, dx=x+{\mathrm {e}}^x\,\left (2\,x^2-2\,x^3\right )+x^2\,{\ln \left (\frac {x}{x+1}\right )}^2+x^2\,{\mathrm {e}}^{2\,x}+\ln \left (\frac {x}{x+1}\right )\,\left (2\,x^2\,{\mathrm {e}}^x+2\,x^2-2\,x^3\right )+x^2-2\,x^3+x^4 \]

[In]

int((5*x + exp(2*x)*(2*x + 4*x^2 + 2*x^3) + log(x/(x + 1))*(6*x - 2*x^2 - 6*x^3 + exp(x)*(4*x + 6*x^2 + 2*x^3)
) + log(x/(x + 1))^2*(2*x + 2*x^2) - 6*x^2 - 2*x^3 + 4*x^4 - exp(x)*(6*x^3 - 6*x + 2*x^4) + 1)/(x + 1),x)

[Out]

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