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

Optimal. Leaf size=18 \[ -\frac {2 \log \left (-3+e+\frac {4}{x}\right ) (5+\log (x))}{x} \]

________________________________________________________________________________________

Rubi [C]  time = 1.40, antiderivative size = 202, normalized size of antiderivative = 11.22, number of steps used = 31, number of rules used = 20, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6, 1593, 6688, 6742, 14, 44, 2454, 2389, 2295, 2351, 2304, 2301, 2316, 2315, 2376, 2475, 2411, 43, 2317, 2391} \begin {gather*} \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 (1-\frac {1}{4} (3-e) x\right )+\frac {1}{4} (3-e) \log ^2(x)+\frac {1}{2} \left (-\frac {4}{x}-e+3\right ) \log \left (\frac {4}{x}+e-3\right ) \log (x)+\frac {5}{2} (3-e) \log (x)+\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}{x}+e-3\right ) \log \left (\frac {4}{(3-e) x}\right )-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)-\frac {5}{2} (3-e) \log (4-(3-e) x) \end {gather*}

Antiderivative was successfully verified.

[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[x]^2)/4 - (5*(3 - E)*Log[4 - (3 - E)*x])/2 - ((3
- E)*Log[4/(3 - E)]*Log[4 - (3 - E)*x])/2 + ((3 - E)*PolyLog[2, 1 - 4/((3 - E)*x)])/2 + ((3 - E)*PolyLog[2, 1
- ((3 - E)*x)/4])/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 43

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 44

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] && L
tQ[m + n + 2, 0])

Rule 1593

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 2295

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

Rule 2301

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 2304

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 2315

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

Rule 2316

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 2317

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 2351

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 2376

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 2389

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 2391

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 2411

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 2454

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 2475

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 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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+8 \int \left (\frac {\log (x)}{4 x^2}+\frac {(3-e) \log (x)}{16 x}+\frac {(3-e)^2 \log (x)}{16 (4-(3-e) x)}\right ) \, dx-8 \operatorname {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\\ &=-\frac {8}{x}+\frac {5}{2} (3-e) \log (x)+\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)-\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 \int \frac {\log (x)}{x^2} \, dx-2 \operatorname {Subst}\left (\int \log (x) \, dx,x,-3+e+\frac {4}{x}\right )+\frac {1}{2} (3-e) \int \frac {\log (x)}{x} \, dx+\frac {1}{2} (3-e)^2 \int \frac {\log (x)}{4+(-3+e) 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}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {(3-e-4 x) \log (-3+e+4 x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} (3-e)^2 \int \frac {\log \left (\frac {1}{4} (3-e) x\right )}{4+(-3+e) 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}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{8} \operatorname {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}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{8} \operatorname {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}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \log (x) \, dx,x,-3+e+\frac {4}{x}\right )-\frac {1}{2} (3-e) \operatorname {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}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{2} (3-e) \operatorname {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}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \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 (1-\frac {1}{4} (3-e) x\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 5.04, size = 31, normalized size = 1.72 \begin {gather*} -\frac {10 \log \left (-3+e+\frac {4}{x}\right )}{x}-\frac {2 \log \left (-3+e+\frac {4}{x}\right ) \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (\log \relax (x) + 5\right )} \log \left (\frac {x e - 3 \, x + 4}{x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [B]  time = 0.18, size = 41, normalized size = 2.28 \begin {gather*} -\frac {2 \, {\left (\log \left (x e - 3 \, x + 4\right ) \log \relax (x) - \log \relax (x)^{2} + 5 \, \log \left (x e - 3 \, x + 4\right ) - 5 \, \log \relax (x)\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maple [C]  time = 0.43, size = 296, normalized size = 16.44

method	result	size
risch	\(-\frac {2 \left (5+\ln \relax (x )\right ) \ln \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {-i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \ln \relax (x )+i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \ln \relax (x )+i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3} \ln \relax (x )-i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \ln \relax (x )-5 i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2}+5 i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+5 i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3}-5 i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+2 \ln \relax (x )^{2}+10 \ln \relax (x )}{x}\)	\(296\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

-2*(5+ln(x))/x*ln(x*exp(1)+4-3*x)+(-I*Pi*csgn(I*(x*exp(1)+4-3*x))*csgn(I/x*(x*exp(1)+4-3*x))^2*ln(x)+I*Pi*csgn
(I*(x*exp(1)+4-3*x))*csgn(I/x*(x*exp(1)+4-3*x))*csgn(I/x)*ln(x)+I*Pi*csgn(I/x*(x*exp(1)+4-3*x))^3*ln(x)-I*Pi*c
sgn(I/x*(x*exp(1)+4-3*x))^2*csgn(I/x)*ln(x)-5*I*Pi*csgn(I*(x*exp(1)+4-3*x))*csgn(I/x*(x*exp(1)+4-3*x))^2+5*I*P
i*csgn(I*(x*exp(1)+4-3*x))*csgn(I/x*(x*exp(1)+4-3*x))*csgn(I/x)+5*I*Pi*csgn(I/x*(x*exp(1)+4-3*x))^3-5*I*Pi*csg
n(I/x*(x*exp(1)+4-3*x))^2*csgn(I/x)+2*ln(x)^2+10*ln(x))/x

________________________________________________________________________________________

maxima [B]  time = 0.39, size = 304, normalized size = 16.89 \begin {gather*} -2 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \relax (x)\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 \relax (x) + \log \relax (x)^{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 \relax (x) + 6 \, \log \left (x {\left (e - 3\right )} + 4\right ) \log \relax (x) - 3 \, \log \relax (x)^{2} + 2 \, {\left ({\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (e - 3\right )} \log \relax (x) - \frac {4}{x}\right )} \log \left (\frac {4}{x} + e - 3\right ) + 6 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \relax (x)\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 \relax (x)^{2} - 2 \, x {\left (e - 3\right )} \log \relax (x) - 2 \, {\left (x {\left (e - 3\right )} \log \relax (x) - 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 \relax (x) + 4\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (x {\left (e - 3\right )} + 4\right )} \log \relax (x) - 4 \, \log \relax (x)^{2} - 4}{2 \, x} - \frac {10}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 4.75, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2\,\ln \left (\frac {x\,\mathrm {e}-3\,x+4}{x}\right )\,\left (\ln \relax (x)+5\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [A]  time = 0.68, size = 22, normalized size = 1.22 \begin {gather*} \frac {\left (- 2 \log {\relax (x )} - 10\right ) \log {\left (\frac {- 3 x + e x + 4}{x} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________