\(\int \frac {(8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 (2 x+16 x^2+16 x^3)) \log (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3})+(-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} (-8 x+2 x^2)+e^5 (-2 x-16 x^2-12 x^3+4 x^4)) \log ^2(\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3})}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 (-1-8 x-6 x^2+2 x^3)} \, dx\) [5956]

Optimal result

Integrand size = 243, antiderivative size = 26 \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=x^2 \log ^2\left (\frac {4-x+\frac {1}{e^5+x+x^2}}{x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 134.73 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81, number of steps used = 103, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6874, 2608, 2603, 6860, 648, 632, 210, 642, 2125, 2106, 2104, 814, 2604, 2465, 2441, 2352, 2437, 2338, 2440, 2438, 2092, 2090, 719, 31, 1642, 2605} \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=x^2 \log ^2\left (\frac {-x^3+3 x^2+\left (4-e^5\right ) x+4 e^5+1}{x \left (x^2+x+e^5\right )}\right ) \]

[In]

Int[((8*E^10*x + 4*x^2 + 14*x^3 + 16*x^4 + 8*x^5 + E^5*(2*x + 16*x^2 + 16*x^3))*Log[(1 + E^5*(4 - x) + 4*x + 3
*x^2 - x^3)/(E^5*x + x^2 + x^3)] + (-2*x^2 - 10*x^3 - 14*x^4 - 4*x^5 + 2*x^6 + E^10*(-8*x + 2*x^2) + E^5*(-2*x
- 16*x^2 - 12*x^3 + 4*x^4))*Log[(1 + E^5*(4 - x) + 4*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)]^2)/(E^10*(-4 + x)
- x - 5*x^2 - 7*x^3 - 2*x^4 + x^5 + E^5*(-1 - 8*x - 6*x^2 + 2*x^3)),x]

[Out]

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

Rule 31

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 719

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2090

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27
*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)
*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]]
/; FreeQ[{a, b, d}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]

Rule 2092

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2104

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/
3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

Rule 2106

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 2125

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coef
f[Qn, x, n])), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, 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 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 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 2440

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

Rule 2441

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

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 2603

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[x*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2604

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

Rule 2605

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m +
1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))), x] - Dist[b*n*(p/(e*(m + 1))), Int[SimplifyIntegrand[(d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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 {2 x \left (-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )}+2 x \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )\right ) \, dx \\ & = 2 \int \frac {x \left (-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )} \, dx+2 \int x \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right ) \, dx \\ & = x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )-2 \int \frac {x \left (-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )} \, dx+2 \int \left (4 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+\frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2}+\frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx \\ & = x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+2 \int \frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2} \, dx+2 \int \frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-2 \int \left (4 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+\frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2}+\frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx+8 \int \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right ) \, dx \\ & = 8 x \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )-2 \int \frac {\left (-e^5-\left (1-2 e^5\right ) x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{e^5+x+x^2} \, dx-2 \int \frac {\left (-3 \left (1+4 e^5\right )-3 \left (5+3 e^5\right ) x-\left (17-2 e^5\right ) x^2\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+2 \int \left (\frac {\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1-i \sqrt {-1+4 e^5}+2 x}+\frac {\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+i \sqrt {-1+4 e^5}+2 x}\right ) \, dx+2 \int \left (\frac {3 \left (-1-4 e^5\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}+\frac {3 \left (-5-3 e^5\right ) x \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}+\frac {\left (-17+2 e^5\right ) x^2 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx-8 \int \frac {-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )} \, dx-8 \int \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right ) \, dx \\ & = x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )-2 \int \left (\frac {\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1-i \sqrt {-1+4 e^5}+2 x}+\frac {\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+i \sqrt {-1+4 e^5}+2 x}\right ) \, dx-2 \int \left (\frac {3 \left (-1-4 e^5\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}+\frac {3 \left (-5-3 e^5\right ) x \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}+\frac {\left (-17+2 e^5\right ) x^2 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx+8 \int \frac {-e^5 \left (1+4 e^5\right )-2 \left (1+4 e^5\right ) x-\left (7+8 e^5\right ) x^2-8 x^3-4 x^4}{\left (e^5+x+x^2\right ) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )} \, dx-8 \int \left (\frac {2 e^5+x}{e^5+x+x^2}+\frac {-3 \left (1+4 e^5\right )-2 \left (4-e^5\right ) x-3 x^2}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx-\left (2 \left (17-2 e^5\right )\right ) \int \frac {x^2 \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-\left (6 \left (5+3 e^5\right )\right ) \int \frac {x \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-\left (6 \left (1+4 e^5\right )\right ) \int \frac {\log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-\left (2 \left (1-2 e^5-i \sqrt {-1+4 e^5}\right )\right ) \int \frac {\log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1-i \sqrt {-1+4 e^5}+2 x} \, dx-\left (2 \left (1-2 e^5+i \sqrt {-1+4 e^5}\right )\right ) \int \frac {\log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+i \sqrt {-1+4 e^5}+2 x} \, dx \\ & = -\left (\left (1-2 e^5-i \sqrt {-1+4 e^5}\right ) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )\right )-\left (1-2 e^5+i \sqrt {-1+4 e^5}\right ) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right ) \log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )-8 \int \frac {2 e^5+x}{e^5+x+x^2} \, dx-8 \int \frac {-3 \left (1+4 e^5\right )-2 \left (4-e^5\right ) x-3 x^2}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+8 \int \left (\frac {2 e^5+x}{e^5+x+x^2}+\frac {-3 \left (1+4 e^5\right )-2 \left (4-e^5\right ) x-3 x^2}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx+\left (2 \left (1-2 e^5-i \sqrt {-1+4 e^5}\right )\right ) \int \frac {\log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1-i \sqrt {-1+4 e^5}+2 x} \, dx-\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \int \frac {x \left (e^5+x+x^2\right ) \left (\frac {4-e^5+6 x-3 x^2}{x \left (e^5+x+x^2\right )}-\frac {(1+2 x) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )}{x \left (e^5+x+x^2\right )^2}-\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x^2 \left (e^5+x+x^2\right )}\right ) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+\left (2 \left (1-2 e^5+i \sqrt {-1+4 e^5}\right )\right ) \int \frac {\log \left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )}{1+i \sqrt {-1+4 e^5}+2 x} \, dx-\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \int \frac {x \left (e^5+x+x^2\right ) \left (\frac {4-e^5+6 x-3 x^2}{x \left (e^5+x+x^2\right )}-\frac {(1+2 x) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )}{x \left (e^5+x+x^2\right )^2}-\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x^2 \left (e^5+x+x^2\right )}\right ) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx \\ & = -8 \log \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )+x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )+\frac {8}{3} \int \frac {3 \left (7+11 e^5\right )+6 \left (7-e^5\right ) x}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-4 \int \frac {1+2 x}{e^5+x+x^2} \, dx+8 \int \frac {2 e^5+x}{e^5+x+x^2} \, dx+8 \int \frac {-3 \left (1+4 e^5\right )-2 \left (4-e^5\right ) x-3 x^2}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+\left (4 \left (1-4 e^5\right )\right ) \int \frac {1}{e^5+x+x^2} \, dx+\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \int \frac {x \left (e^5+x+x^2\right ) \left (\frac {4-e^5+6 x-3 x^2}{x \left (e^5+x+x^2\right )}-\frac {(1+2 x) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )}{x \left (e^5+x+x^2\right )^2}-\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x^2 \left (e^5+x+x^2\right )}\right ) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \int \left (-\frac {\log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{x}+\frac {(-1-2 x) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{e^5+x+x^2}+\frac {\left (4-e^5+6 x-3 x^2\right ) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx+\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \int \frac {x \left (e^5+x+x^2\right ) \left (\frac {4-e^5+6 x-3 x^2}{x \left (e^5+x+x^2\right )}-\frac {(1+2 x) \left (1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3\right )}{x \left (e^5+x+x^2\right )^2}-\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x^2 \left (e^5+x+x^2\right )}\right ) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx-\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \int \left (-\frac {\log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{x}+\frac {(-1-2 x) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{e^5+x+x^2}+\frac {\left (4-e^5+6 x-3 x^2\right ) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx \\ & = -4 \log \left (e^5+x+x^2\right )+x^2 \log ^2\left (\frac {1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}{x \left (e^5+x+x^2\right )}\right )-\frac {8}{3} \int \frac {3 \left (7+11 e^5\right )+6 \left (7-e^5\right ) x}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+\frac {8}{3} \text {Subst}\left (\int \frac {\frac {1}{3} \left (18 \left (7-e^5\right )+9 \left (7+11 e^5\right )\right )+6 \left (7-e^5\right ) x}{7+3 e^5+\left (7-e^5\right ) x-x^3} \, dx,x,-1+x\right )+4 \int \frac {1+2 x}{e^5+x+x^2} \, dx-\left (4 \left (1-4 e^5\right )\right ) \int \frac {1}{e^5+x+x^2} \, dx-\left (8 \left (1-4 e^5\right )\right ) \text {Subst}\left (\int \frac {1}{1-4 e^5-x^2} \, dx,x,1+2 x\right )-\left (1-2 e^5-i \sqrt {-1+4 e^5}\right ) \int \frac {\log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{x} \, dx-\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \int \frac {(-1-2 x) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{e^5+x+x^2} \, dx-\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \int \frac {\left (4-e^5+6 x-3 x^2\right ) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+\left (-1+2 e^5-i \sqrt {-1+4 e^5}\right ) \int \left (-\frac {\log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{x}+\frac {(-1-2 x) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{e^5+x+x^2}+\frac {\left (4-e^5+6 x-3 x^2\right ) \log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx-\left (1-2 e^5+i \sqrt {-1+4 e^5}\right ) \int \frac {\log \left (1+i \sqrt {-1+4 e^5}+2 x\right )}{x} \, dx-\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \int \frac {(-1-2 x) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{e^5+x+x^2} \, dx-\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \int \frac {\left (4-e^5+6 x-3 x^2\right ) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3} \, dx+\left (-1+2 e^5+i \sqrt {-1+4 e^5}\right ) \int \left (-\frac {\log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{x}+\frac {(-1-2 x) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{e^5+x+x^2}+\frac {\left (4-e^5+6 x-3 x^2\right ) \log \left (1-i \sqrt {-1+4 e^5}+2 x\right )}{1+4 e^5+\left (4-e^5\right ) x+3 x^2-x^3}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 15.81 (sec) , antiderivative size = 146774, normalized size of antiderivative = 5645.15 \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=\text {Result too large to show} \]

[In]

Integrate[((8*E^10*x + 4*x^2 + 14*x^3 + 16*x^4 + 8*x^5 + E^5*(2*x + 16*x^2 + 16*x^3))*Log[(1 + E^5*(4 - x) + 4
*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)] + (-2*x^2 - 10*x^3 - 14*x^4 - 4*x^5 + 2*x^6 + E^10*(-8*x + 2*x^2) + E^5
*(-2*x - 16*x^2 - 12*x^3 + 4*x^4))*Log[(1 + E^5*(4 - x) + 4*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)]^2)/(E^10*(-4
+ x) - x - 5*x^2 - 7*x^3 - 2*x^4 + x^5 + E^5*(-1 - 8*x - 6*x^2 + 2*x^3)),x]

[Out]

Result too large to show

Maple [A] (verified)

Time = 3.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73

method	result	size
norman	\(x^{2} \ln \left (\frac {\left (-x +4\right ) {\mathrm e}^{5}-x^{3}+3 x^{2}+4 x +1}{x \,{\mathrm e}^{5}+x^{3}+x^{2}}\right )^{2}\)	\(45\)
risch	\(x^{2} \ln \left (\frac {\left (-x +4\right ) {\mathrm e}^{5}-x^{3}+3 x^{2}+4 x +1}{x \,{\mathrm e}^{5}+x^{3}+x^{2}}\right )^{2}\)	\(45\)
parallelrisch	\(-\frac {\left (-16 x^{2} \ln \left (\frac {\left (-x +4\right ) {\mathrm e}^{5}-x^{3}+3 x^{2}+4 x +1}{x \left (x^{2}+{\mathrm e}^{5}+x \right )}\right )^{2} {\mathrm e}^{20}-8 \,{\mathrm e}^{15} x^{2} \ln \left (\frac {\left (-x +4\right ) {\mathrm e}^{5}-x^{3}+3 x^{2}+4 x +1}{x \left (x^{2}+{\mathrm e}^{5}+x \right )}\right )^{2}-{\mathrm e}^{10} x^{2} \ln \left (\frac {\left (-x +4\right ) {\mathrm e}^{5}-x^{3}+3 x^{2}+4 x +1}{x \left (x^{2}+{\mathrm e}^{5}+x \right )}\right )^{2}\right ) {\mathrm e}^{-10}}{\left (4 \,{\mathrm e}^{5}+1\right )^{2}}\)	\(160\)

[In]

int((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*ln(((-x+4)*exp(5)
-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*x^2)*ln
(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2*x^4-7*x
^3-5*x^2-x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=x^{2} \log \left (-\frac {x^{3} - 3 \, x^{2} + {\left (x - 4\right )} e^{5} - 4 \, x - 1}{x^{3} + x^{2} + x e^{5}}\right )^{2} \]

[In]

integrate((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*log(((-x+4)
*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*
x^2)*log(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2
*x^4-7*x^3-5*x^2-x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=x^{2} \log {\left (\frac {- x^{3} + 3 x^{2} + 4 x + \left (4 - x\right ) e^{5} + 1}{x^{3} + x^{2} + x e^{5}} \right )}^{2} \]

[In]

integrate((((2*x**2-8*x)*exp(5)**2+(4*x**4-12*x**3-16*x**2-2*x)*exp(5)+2*x**6-4*x**5-14*x**4-10*x**3-2*x**2)*l
n(((-x+4)*exp(5)-x**3+3*x**2+4*x+1)/(x*exp(5)+x**3+x**2))**2+(8*x*exp(5)**2+(16*x**3+16*x**2+2*x)*exp(5)+8*x**
5+16*x**4+14*x**3+4*x**2)*ln(((-x+4)*exp(5)-x**3+3*x**2+4*x+1)/(x*exp(5)+x**3+x**2)))/((x-4)*exp(5)**2+(2*x**3
-6*x**2-8*x-1)*exp(5)+x**5-2*x**4-7*x**3-5*x**2-x),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.35 \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=x^{2} \log \left (-x^{3} + 3 \, x^{2} - x {\left (e^{5} - 4\right )} + 4 \, e^{5} + 1\right )^{2} + x^{2} \log \left (x^{2} + x + e^{5}\right )^{2} + 2 \, x^{2} \log \left (x^{2} + x + e^{5}\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 2 \, {\left (x^{2} \log \left (x^{2} + x + e^{5}\right ) + x^{2} \log \left (x\right )\right )} \log \left (-x^{3} + 3 \, x^{2} - x {\left (e^{5} - 4\right )} + 4 \, e^{5} + 1\right ) \]

[In]

integrate((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*log(((-x+4)
*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*
x^2)*log(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2
*x^4-7*x^3-5*x^2-x),x, algorithm="maxima")

[Out]

x^2*log(-x^3 + 3*x^2 - x*(e^5 - 4) + 4*e^5 + 1)^2 + x^2*log(x^2 + x + e^5)^2 + 2*x^2*log(x^2 + x + e^5)*log(x)
+ x^2*log(x)^2 - 2*(x^2*log(x^2 + x + e^5) + x^2*log(x))*log(-x^3 + 3*x^2 - x*(e^5 - 4) + 4*e^5 + 1)

Giac [A] (verification not implemented)

none

Time = 2.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=x^{2} \log \left (-\frac {x^{3} - 3 \, x^{2} + x e^{5} - 4 \, x - 4 \, e^{5} - 1}{x^{3} + x^{2} + x e^{5}}\right )^{2} \]

[In]

integrate((((2*x^2-8*x)*exp(5)^2+(4*x^4-12*x^3-16*x^2-2*x)*exp(5)+2*x^6-4*x^5-14*x^4-10*x^3-2*x^2)*log(((-x+4)
*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2))^2+(8*x*exp(5)^2+(16*x^3+16*x^2+2*x)*exp(5)+8*x^5+16*x^4+14*x^3+4*
x^2)*log(((-x+4)*exp(5)-x^3+3*x^2+4*x+1)/(x*exp(5)+x^3+x^2)))/((x-4)*exp(5)^2+(2*x^3-6*x^2-8*x-1)*exp(5)+x^5-2
*x^4-7*x^3-5*x^2-x),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.55 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx=x^2\,{\ln \left (\frac {4\,x-{\mathrm {e}}^5\,\left (x-4\right )+3\,x^2-x^3+1}{x^3+x^2+{\mathrm {e}}^5\,x}\right )}^2 \]

[In]

int((log((4*x - exp(5)*(x - 4) + 3*x^2 - x^3 + 1)/(x*exp(5) + x^2 + x^3))^2*(exp(10)*(8*x - 2*x^2) + exp(5)*(2
*x + 16*x^2 + 12*x^3 - 4*x^4) + 2*x^2 + 10*x^3 + 14*x^4 + 4*x^5 - 2*x^6) - log((4*x - exp(5)*(x - 4) + 3*x^2 -
x^3 + 1)/(x*exp(5) + x^2 + x^3))*(8*x*exp(10) + exp(5)*(2*x + 16*x^2 + 16*x^3) + 4*x^2 + 14*x^3 + 16*x^4 + 8*
x^5))/(x + exp(5)*(8*x + 6*x^2 - 2*x^3 + 1) - exp(10)*(x - 4) + 5*x^2 + 7*x^3 + 2*x^4 - x^5),x)

[Out]

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