\(\int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log (\frac {4-3 x+e x}{x})}{4 x^2-3 x^3+e x^3} \, dx\) [5994]

Optimal result

Integrand size = 60, antiderivative size = 18 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2 \log \left (-3+e+\frac {4}{x}\right ) (5+\log (x))}{x} \]

[Out]

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

Rubi [C] (verified)

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

Time = 0.87 (sec) , antiderivative size = 185, normalized size of antiderivative = 10.28, number of steps used = 29, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {6, 1607, 6820, 6874, 14, 46, 2504, 2436, 2332, 2380, 2341, 2379, 2438, 2423, 2525, 2458, 45, 2393, 2354} \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=\frac {1}{2} (3-e) \operatorname {PolyLog}\left (2,1-\frac {4}{(3-e) x}\right )+\frac {1}{2} (3-e) \operatorname {PolyLog}\left (2,\frac {4}{(3-e) x}\right )+\frac {5}{2} \left (-\frac {4}{x}-e+3\right ) \log \left (\frac {4}{x}+e-3\right )+\frac {1}{2} (3-e) \log \left (\frac {4}{(3-e) x}\right ) \log \left (\frac {4}{x}+e-3\right )+\frac {1}{2} \left (-\frac {4}{x}-e+3\right ) \log (x) \log \left (\frac {4}{x}+e-3\right )-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)+\frac {5}{2} (3-e) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x) \]

[In]

Int[(40 + 8*Log[x] + (32 - 24*x + 8*E*x + (8 - 6*x + 2*E*x)*Log[x])*Log[(4 - 3*x + E*x)/x])/(4*x^2 - 3*x^3 + E
*x^3),x]

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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 2380

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

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2423

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2436

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

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 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 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 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 {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2+(-3+e) x^3} \, dx \\ & = \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{x^2 (4+(-3+e) x)} \, dx \\ & = \int \frac {40+8 \log (x)+2 (4+(-3+e) x) \log \left (-3+e+\frac {4}{x}\right ) (4+\log (x))}{x^2 (4+(-3+e) x)} \, dx \\ & = \int \left (\frac {8 \left (5+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )\right )}{x^2 (4-(3-e) x)}+\frac {2 \left (4+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )\right ) \log (x)}{x^2 (4-(3-e) x)}\right ) \, dx \\ & = 2 \int \frac {\left (4+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )\right ) \log (x)}{x^2 (4-(3-e) x)} \, dx+8 \int \frac {5+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )}{x^2 (4-(3-e) x)} \, dx \\ & = 2 \int \frac {\left (\frac {4}{4+(-3+e) x}+\log \left (-3+e+\frac {4}{x}\right )\right ) \log (x)}{x^2} \, dx+8 \int \frac {\frac {5}{4+(-3+e) x}+\log \left (-3+e+\frac {4}{x}\right )}{x^2} \, dx \\ & = 2 \int \left (\frac {4 \log (x)}{x^2 (4-(3-e) x)}+\frac {\log \left (-3+e+\frac {4}{x}\right ) \log (x)}{x^2}\right ) \, dx+8 \int \left (\frac {5}{x^2 (4-(3-e) x)}+\frac {\log \left (-3+e+\frac {4}{x}\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {\log \left (-3+e+\frac {4}{x}\right ) \log (x)}{x^2} \, dx+8 \int \frac {\log \left (-3+e+\frac {4}{x}\right )}{x^2} \, dx+8 \int \frac {\log (x)}{x^2 (4+(-3+e) x)} \, dx+40 \int \frac {1}{x^2 (4-(3-e) x)} \, dx \\ & = \frac {2 \log (x)}{x}+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-2 \int \left (\frac {1}{x^2}+\frac {\left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )}{4 x}\right ) \, dx+2 \int \frac {\log (x)}{x^2} \, dx-8 \text {Subst}\left (\int \log (-3+e+4 x) \, dx,x,\frac {1}{x}\right )+40 \int \left (\frac {1}{4 x^2}+\frac {3-e}{16 x}+\frac {(3-e)^2}{16 (4-(3-e) x)}\right ) \, dx+(2 (3-e)) \int \frac {\log (x)}{x (4+(-3+e) x)} \, dx \\ & = -\frac {10}{x}+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} \int \frac {\left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )}{x} \, dx-2 \text {Subst}\left (\int \log (x) \, dx,x,-3+e+\frac {4}{x}\right )+\frac {1}{2} (3-e) \int \frac {\log \left (1+\frac {4}{(-3+e) x}\right )}{x} \, dx \\ & = -\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (\frac {4}{(3-e) x}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {(3-e-4 x) \log (-3+e+4 x)}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (\frac {4}{(3-e) x}\right )-\frac {1}{8} \text {Subst}\left (\int \frac {x \log (x)}{\frac {3-e}{4}+\frac {x}{4}} \, dx,x,-3+e+\frac {4}{x}\right ) \\ & = -\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (\frac {4}{(3-e) x}\right )-\frac {1}{8} \text {Subst}\left (\int \left (4 \log (x)-\frac {4 (-3+e) \log (x)}{-3+e-x}\right ) \, dx,x,-3+e+\frac {4}{x}\right ) \\ & = -\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (\frac {4}{(3-e) x}\right )-\frac {1}{2} \text {Subst}\left (\int \log (x) \, dx,x,-3+e+\frac {4}{x}\right )-\frac {1}{2} (3-e) \text {Subst}\left (\int \frac {\log (x)}{-3+e-x} \, dx,x,-3+e+\frac {4}{x}\right ) \\ & = \frac {5}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {1}{2} (3-e) \log \left (-3+e+\frac {4}{x}\right ) \log \left (\frac {4}{(3-e) x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (\frac {4}{(3-e) x}\right )-\frac {1}{2} (3-e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{-3+e}\right )}{x} \, dx,x,-3+e+\frac {4}{x}\right ) \\ & = \frac {5}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {1}{2} (3-e) \log \left (-3+e+\frac {4}{x}\right ) \log \left (\frac {4}{(3-e) x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {4}{(3-e) x}\right )+\frac {1}{2} (3-e) \text {Li}_2\left (\frac {4}{(3-e) x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {10 \log \left (-3+e+\frac {4}{x}\right )}{x}-\frac {2 \log \left (-3+e+\frac {4}{x}\right ) \log (x)}{x} \]

[In]

Integrate[(40 + 8*Log[x] + (32 - 24*x + 8*E*x + (8 - 6*x + 2*E*x)*Log[x])*Log[(4 - 3*x + E*x)/x])/(4*x^2 - 3*x
^3 + E*x^3),x]

[Out]

(-10*Log[-3 + E + 4/x])/x - (2*Log[-3 + E + 4/x]*Log[x])/x

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(19)=38\).

Time = 1.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 7.28

method	result	size
parallelrisch	\(-\frac {80 \,{\mathrm e} \ln \left (\frac {x \,{\mathrm e}+4-3 x}{{\mathrm e}-3}\right ) x -80 x \,{\mathrm e} \ln \left (x \right )-80 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) {\mathrm e} x -240 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{{\mathrm e}-3}\right ) x +240 x \ln \left (x \right )+240 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) x +32 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \ln \left (x \right )+160 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )}{16 x}\)	\(131\)
default	\(\frac {2+2 \left (-\frac {{\mathrm e}}{4}+\frac {3}{4}\right ) x \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \ln \left (x \right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )}{x}-\frac {10}{x}+40 \left (-\frac {{\mathrm e}}{16}+\frac {3}{16}\right ) \ln \left (x \right )-\frac {5 \left ({\mathrm e}-3\right )^{2} \ln \left (-x \,{\mathrm e}-4+3 x \right )}{2 \left (3-{\mathrm e}\right )}-\frac {2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {2 x \,{\mathrm e}-6 x +8}{x}\)	\(149\)
parts	\(\frac {2+2 \left (-\frac {{\mathrm e}}{4}+\frac {3}{4}\right ) x \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \ln \left (x \right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )}{x}-\frac {10}{x}+40 \left (-\frac {{\mathrm e}}{16}+\frac {3}{16}\right ) \ln \left (x \right )-\frac {5 \left ({\mathrm e}-3\right )^{2} \ln \left (-x \,{\mathrm e}-4+3 x \right )}{2 \left (3-{\mathrm e}\right )}-\frac {2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {2 x \,{\mathrm e}-6 x +8}{x}\)	\(149\)
risch	\(-\frac {2 \left (5+\ln \left (x \right )\right ) \ln \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right ) \ln \left (x \right )-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \ln \left (x \right )-i \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \ln \left (x \right )+i \pi \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3} \ln \left (x \right )+5 i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )-5 i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2}-5 i \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2}+5 i \pi \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3}+2 \ln \left (x \right )^{2}+10 \ln \left (x \right )}{x}\)	\(296\)

[In]

int((((2*x*exp(1)-6*x+8)*ln(x)+8*x*exp(1)-24*x+32)*ln((x*exp(1)+4-3*x)/x)+8*ln(x)+40)/(x^3*exp(1)-3*x^3+4*x^2)
,x,method=_RETURNVERBOSE)

[Out]

-1/16*(80*exp(1)*ln((x*exp(1)+4-3*x)/(exp(1)-3))*x-80*x*exp(1)*ln(x)-80*ln((x*exp(1)+4-3*x)/x)*exp(1)*x-240*ln
((x*exp(1)+4-3*x)/(exp(1)-3))*x+240*x*ln(x)+240*ln((x*exp(1)+4-3*x)/x)*x+32*ln((x*exp(1)+4-3*x)/x)*ln(x)+160*l
n((x*exp(1)+4-3*x)/x))/x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2 \, {\left (\log \left (x\right ) + 5\right )} \log \left (\frac {x e - 3 \, x + 4}{x}\right )}{x} \]

[In]

integrate((((2*x*exp(1)-6*x+8)*log(x)+8*x*exp(1)-24*x+32)*log((x*exp(1)+4-3*x)/x)+8*log(x)+40)/(x^3*exp(1)-3*x
^3+4*x^2),x, algorithm="fricas")

[Out]

-2*(log(x) + 5)*log((x*e - 3*x + 4)/x)/x

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=\frac {\left (- 2 \log {\left (x \right )} - 10\right ) \log {\left (\frac {- 3 x + e x + 4}{x} \right )}}{x} \]

[In]

integrate((((2*x*exp(1)-6*x+8)*ln(x)+8*x*exp(1)-24*x+32)*ln((x*exp(1)+4-3*x)/x)+8*ln(x)+40)/(x**3*exp(1)-3*x**
3+4*x**2),x)

[Out]

(-2*log(x) - 10)*log((-3*x + E*x + 4)/x)/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (19) = 38\).

Time = 0.24 (sec) , antiderivative size = 304, normalized size of antiderivative = 16.89 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-2 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \left (x\right )\right )} e \log \left (\frac {4}{x} + e - 3\right ) + {\left (\log \left (x {\left (e - 3\right )} + 4\right )^{2} - 2 \, \log \left (x {\left (e - 3\right )} + 4\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e + \frac {5}{2} \, {\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - 3 \, \log \left (x {\left (e - 3\right )} + 4\right )^{2} - \frac {5}{2} \, {\left (e - 3\right )} \log \left (x\right ) + 6 \, \log \left (x {\left (e - 3\right )} + 4\right ) \log \left (x\right ) - 3 \, \log \left (x\right )^{2} + 2 \, {\left ({\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (e - 3\right )} \log \left (x\right ) - \frac {4}{x}\right )} \log \left (\frac {4}{x} + e - 3\right ) + 6 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \left (x\right )\right )} \log \left (\frac {4}{x} + e - 3\right ) - \frac {x {\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right )^{2} + x {\left (e - 3\right )} \log \left (x\right )^{2} - 2 \, x {\left (e - 3\right )} \log \left (x\right ) - 2 \, {\left (x {\left (e - 3\right )} \log \left (x\right ) - x {\left (e - 3\right )}\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - 8}{x} - \frac {{\left (x {\left (e - 3\right )} + 4 \, \log \left (x\right ) + 4\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (x {\left (e - 3\right )} + 4\right )} \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - 4}{2 \, x} - \frac {10}{x} \]

[In]

integrate((((2*x*exp(1)-6*x+8)*log(x)+8*x*exp(1)-24*x+32)*log((x*exp(1)+4-3*x)/x)+8*log(x)+40)/(x^3*exp(1)-3*x
^3+4*x^2),x, algorithm="maxima")

[Out]

-2*(log(x*(e - 3) + 4) - log(x))*e*log(4/x + e - 3) + (log(x*(e - 3) + 4)^2 - 2*log(x*(e - 3) + 4)*log(x) + lo
g(x)^2)*e + 5/2*(e - 3)*log(x*(e - 3) + 4) - 3*log(x*(e - 3) + 4)^2 - 5/2*(e - 3)*log(x) + 6*log(x*(e - 3) + 4
)*log(x) - 3*log(x)^2 + 2*((e - 3)*log(x*(e - 3) + 4) - (e - 3)*log(x) - 4/x)*log(4/x + e - 3) + 6*(log(x*(e -
3) + 4) - log(x))*log(4/x + e - 3) - (x*(e - 3)*log(x*(e - 3) + 4)^2 + x*(e - 3)*log(x)^2 - 2*x*(e - 3)*log(x
) - 2*(x*(e - 3)*log(x) - x*(e - 3))*log(x*(e - 3) + 4) - 8)/x - 1/2*((x*(e - 3) + 4*log(x) + 4)*log(x*(e - 3)
+ 4) - (x*(e - 3) + 4)*log(x) - 4*log(x)^2 - 4)/x - 10/x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).

Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.28 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2 \, {\left (\log \left (x e - 3 \, x + 4\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 5 \, \log \left (x e - 3 \, x + 4\right ) - 5 \, \log \left (x\right )\right )}}{x} \]

[In]

integrate((((2*x*exp(1)-6*x+8)*log(x)+8*x*exp(1)-24*x+32)*log((x*exp(1)+4-3*x)/x)+8*log(x)+40)/(x^3*exp(1)-3*x
^3+4*x^2),x, algorithm="giac")

[Out]

-2*(log(x*e - 3*x + 4)*log(x) - log(x)^2 + 5*log(x*e - 3*x + 4) - 5*log(x))/x

Mupad [B] (verification not implemented)

Time = 12.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2\,\ln \left (\frac {x\,\mathrm {e}-3\,x+4}{x}\right )\,\left (\ln \left (x\right )+5\right )}{x} \]

[In]

int((8*log(x) + log((x*exp(1) - 3*x + 4)/x)*(8*x*exp(1) - 24*x + log(x)*(2*x*exp(1) - 6*x + 8) + 32) + 40)/(x^
3*exp(1) + 4*x^2 - 3*x^3),x)

[Out]

-(2*log((x*exp(1) - 3*x + 4)/x)*(log(x) + 5))/x