∫ 3−3x−5x2+3x3+2x4+(−3+6x+3x2+6x3+6x4) log( 4x__ 1+x2 ) _____________________________________________________________________________54x2 −108x3 +36x4 −36x5 +6x6 +72x7 +24x8 dx

Optimal. Leaf size=30 \[ \frac {\log \left (\frac {4}{\frac {1}{x}+x}\right )}{6 x (3+x-x (4+2 x))} \]

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Rubi [B] time = 7.88, antiderivative size = 973, normalized size of antiderivative = 32.43, number of steps used = 171, number of rules used = 25, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {6741, 12, 6725, 975, 1074, 632, 31, 635, 203, 260, 638, 618, 206, 1018, 1065, 2528, 2525, 453, 2524, 2418, 2392, 2391, 2416, 2394, 2393} \begin {gather*} \frac {2 x+29}{748 \left (-2 x^2-3 x+3\right )}+\frac {4 \left (3+\sqrt {33}\right ) \tan ^{-1}(x)}{99 \left (29+3 \sqrt {33}\right )}-\frac {56 \tan ^{-1}(x)}{99 \left (29+3 \sqrt {33}\right )}+\frac {4 \left (3-\sqrt {33}\right ) \tan ^{-1}(x)}{99 \left (29-3 \sqrt {33}\right )}-\frac {56 \tan ^{-1}(x)}{99 \left (29-3 \sqrt {33}\right )}+\frac {19}{306} \tan ^{-1}(x)+\frac {58 \tanh ^{-1}\left (\frac {4 x+3}{\sqrt {33}}\right )}{1683 \sqrt {33}}+\frac {14 \log (x)}{99 \left (3+\sqrt {33}\right )}+\frac {14 \log (x)}{99 \left (3-\sqrt {33}\right )}+\frac {7 \log (x)}{198}-\frac {\left (33693+5087 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{686664}+\frac {5 \left (16335+4513 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{7553304}+\frac {\left (2783+343 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{114444}+\frac {\left (5445-37 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{1258884}-\frac {\left (363-41 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{104907}-\frac {\left (363+41 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{104907}+\frac {\left (5445+37 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{1258884}+\frac {\left (2783-343 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{114444}+\frac {5 \left (16335-4513 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{7553304}-\frac {\left (33693-5087 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{686664}+\frac {7 \left (13-3 \sqrt {33}\right ) \log \left (2 \left (93-19 \sqrt {33}\right ) x+3 \left (151-25 \sqrt {33}\right )\right )}{99 \left (93-19 \sqrt {33}\right )}-\frac {\left (69-11 \sqrt {33}\right ) \log \left (2 \left (93-19 \sqrt {33}\right ) x+3 \left (151-25 \sqrt {33}\right )\right )}{99 \left (93-19 \sqrt {33}\right )}-\frac {\left (69+11 \sqrt {33}\right ) \log \left (2 \left (93+19 \sqrt {33}\right ) x+3 \left (151+25 \sqrt {33}\right )\right )}{99 \left (93+19 \sqrt {33}\right )}+\frac {7 \left (13+3 \sqrt {33}\right ) \log \left (2 \left (93+19 \sqrt {33}\right ) x+3 \left (151+25 \sqrt {33}\right )\right )}{99 \left (93+19 \sqrt {33}\right )}+\frac {\log \left (\frac {4 x}{x^2+1}\right )}{18 x}+\frac {\left (3-\sqrt {33}\right ) \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x-\sqrt {33}+3\right )}-\frac {14 \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x-\sqrt {33}+3\right )}+\frac {\left (3+\sqrt {33}\right ) \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x+\sqrt {33}+3\right )}-\frac {14 \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x+\sqrt {33}+3\right )}+\frac {\left (7+\sqrt {33}\right ) \log \left (x^2+1\right )}{33 \left (29+3 \sqrt {33}\right )}-\frac {7 \left (3+\sqrt {33}\right ) \log \left (x^2+1\right )}{99 \left (29+3 \sqrt {33}\right )}+\frac {\left (7-\sqrt {33}\right ) \log \left (x^2+1\right )}{33 \left (29-3 \sqrt {33}\right )}-\frac {7 \left (3-\sqrt {33}\right ) \log \left (x^2+1\right )}{99 \left (29-3 \sqrt {33}\right )}-\frac {1}{68} \log \left (x^2+1\right )+\frac {26 x+3}{1122 \left (-2 x^2-3 x+3\right )}-\frac {5 (58 x+93)}{6732 \left (-2 x^2-3 x+3\right )}-\frac {62 x+151}{2244 \left (-2 x^2-3 x+3\right )}+\frac {302 x+639}{6732 \left (-2 x^2-3 x+3\right )} \end {gather*}

Int[(3 - 3*x - 5*x^2 + 3*x^3 + 2*x^4 + (-3 + 6*x + 3*x^2 + 6*x^3 + 6*x^4)*Log[(4*x)/(1 + x^2)])/(54*x^2 - 108*

x^3 + 36*x^4 - 36*x^5 + 6*x^6 + 72*x^7 + 24*x^8),x]

(29 + 2*x)/(748*(3 - 3*x - 2*x^2)) + (3 + 26*x)/(1122*(3 - 3*x - 2*x^2)) - (5*(93 + 58*x))/(6732*(3 - 3*x - 2*

x^2)) - (151 + 62*x)/(2244*(3 - 3*x - 2*x^2)) + (639 + 302*x)/(6732*(3 - 3*x - 2*x^2)) + (19*ArcTan[x])/306 -

(56*ArcTan[x])/(99*(29 - 3*Sqrt[33])) + (4*(3 - Sqrt[33])*ArcTan[x])/(99*(29 - 3*Sqrt[33])) - (56*ArcTan[x])/(

99*(29 + 3*Sqrt[33])) + (4*(3 + Sqrt[33])*ArcTan[x])/(99*(29 + 3*Sqrt[33])) + (58*ArcTanh[(3 + 4*x)/Sqrt[33]])

/(1683*Sqrt[33]) + (7*Log[x])/198 + (14*Log[x])/(99*(3 - Sqrt[33])) + (14*Log[x])/(99*(3 + Sqrt[33])) - ((363

- 41*Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/104907 + ((5445 - 37*Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/1258884 + ((27

83 + 343*Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/114444 + (5*(16335 + 4513*Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/75533

04 - ((33693 + 5087*Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/686664 - ((33693 - 5087*Sqrt[33])*Log[3 + Sqrt[33] + 4*

x])/686664 + (5*(16335 - 4513*Sqrt[33])*Log[3 + Sqrt[33] + 4*x])/7553304 + ((2783 - 343*Sqrt[33])*Log[3 + Sqrt

[33] + 4*x])/114444 + ((5445 + 37*Sqrt[33])*Log[3 + Sqrt[33] + 4*x])/1258884 - ((363 + 41*Sqrt[33])*Log[3 + Sq

rt[33] + 4*x])/104907 - ((69 - 11*Sqrt[33])*Log[3*(151 - 25*Sqrt[33]) + 2*(93 - 19*Sqrt[33])*x])/(99*(93 - 19*

Sqrt[33])) + (7*(13 - 3*Sqrt[33])*Log[3*(151 - 25*Sqrt[33]) + 2*(93 - 19*Sqrt[33])*x])/(99*(93 - 19*Sqrt[33]))

+ (7*(13 + 3*Sqrt[33])*Log[3*(151 + 25*Sqrt[33]) + 2*(93 + 19*Sqrt[33])*x])/(99*(93 + 19*Sqrt[33])) - ((69 +

11*Sqrt[33])*Log[3*(151 + 25*Sqrt[33]) + 2*(93 + 19*Sqrt[33])*x])/(99*(93 + 19*Sqrt[33])) + Log[(4*x)/(1 + x^2

)]/(18*x) - (14*Log[(4*x)/(1 + x^2)])/(99*(3 - Sqrt[33] + 4*x)) + ((3 - Sqrt[33])*Log[(4*x)/(1 + x^2)])/(99*(3

- Sqrt[33] + 4*x)) - (14*Log[(4*x)/(1 + x^2)])/(99*(3 + Sqrt[33] + 4*x)) + ((3 + Sqrt[33])*Log[(4*x)/(1 + x^2

)])/(99*(3 + Sqrt[33] + 4*x)) - Log[1 + x^2]/68 - (7*(3 - Sqrt[33])*Log[1 + x^2])/(99*(29 - 3*Sqrt[33])) + ((7

- Sqrt[33])*Log[1 + x^2])/(33*(29 - 3*Sqrt[33])) - (7*(3 + Sqrt[33])*Log[1 + x^2])/(99*(29 + 3*Sqrt[33])) + (

(7 + Sqrt[33])*Log[1 + x^2])/(33*(29 + 3*Sqrt[33]))

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m

+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In

t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||

GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]

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]

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[

(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*

x), 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*

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/

(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[-(a*c)]

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -

b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2

- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((b^3*f + b*c*(c*d

- 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1))/((b^2 - 4*a*c)*(

b^2*d*f + (c*d - a*f)^2)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x

+ c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[2*c*(b^2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*

f*(p + 1) - c*d*(p + 2)) + (2*f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(2*a*f))*(b*f*(

p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x]

&& NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) &

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp

[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(

2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)

*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f

*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*

(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((g*c)*(-(b*(c*d + a*f))) + (g*b - a*h)*(2*c^2*d + b^2*f - c*

(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(g*f

) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}

, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (C_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Si

mp[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b)*(2*c^2*d + b^2*f - c

*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f)) + C*(b^2*d - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c

*d - a*f)^2)*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p

+ 1)*(d + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(C*d + A*f) + 2*(A*c

*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b)*

(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f*

(p + 1)))*x - c*f*(b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]

/; FreeQ[{a, b, c, d, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0]

&& !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol]

:> With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[1/q, Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*b*B*f -

a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 +

b*B*d*f - A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x^2), x], x] /; NeQ[q, 0]]

/; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]

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

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[

b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

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

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]

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q

_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,

b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

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

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]

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]

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

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]

/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{6 x^2 \left (3-3 x-2 x^2\right )^2 \left (1+x^2\right )} \, dx\\ &=\frac {1}{6} \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{x^2 \left (3-3 x-2 x^2\right )^2 \left (1+x^2\right )} \, dx\\ &=\frac {1}{6} \int \left (-\frac {5}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2}+\frac {3}{x^2 \left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2}-\frac {3}{x \left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2}+\frac {3 x}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2}+\frac {2 x^2}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2}+\frac {3 \left (-1+2 x+2 x^2\right ) \log \left (\frac {4 x}{1+x^2}\right )}{x^2 \left (-3+3 x+2 x^2\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {x^2}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2} \, dx+\frac {1}{2} \int \frac {1}{x^2 \left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2} \, dx-\frac {1}{2} \int \frac {1}{x \left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2} \, dx+\frac {1}{2} \int \frac {x}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2} \, dx+\frac {1}{2} \int \frac {\left (-1+2 x+2 x^2\right ) \log \left (\frac {4 x}{1+x^2}\right )}{x^2 \left (-3+3 x+2 x^2\right )^2} \, dx-\frac {5}{6} \int \frac {1}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )^2} \, dx\\ &=\frac {29+2 x}{748 \left (3-3 x-2 x^2\right )}+\frac {3+26 x}{1122 \left (3-3 x-2 x^2\right )}-\frac {5 (93+58 x)}{6732 \left (3-3 x-2 x^2\right )}-\frac {\int \frac {-87-99 x+78 x^2}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )} \, dx}{3366}-\frac {\int \frac {-93+165 x+6 x^2}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )} \, dx}{2244}-\frac {5 \int \frac {-223-99 x-58 x^2}{\left (1+x^2\right ) \left (-3+3 x+2 x^2\right )} \, dx}{6732}+\frac {1}{2} \int \left (\frac {1}{9 x^2}+\frac {2}{9 x}+\frac {-8-15 x}{578 \left (1+x^2\right )}+\frac {113+50 x}{102 \left (-3+3 x+2 x^2\right )^2}+\frac {-4075-2042 x}{5202 \left (-3+3 x+2 x^2\right )}\right ) \, dx-\frac {1}{2} \int \left (\frac {1}{9 x}+\frac {15-8 x}{578 \left (1+x^2\right )}+\frac {75+38 x}{102 \left (-3+3 x+2 x^2\right )^2}-\frac {2 (447+253 x)}{2601 \left (-3+3 x+2 x^2\right )}\right ) \, dx+\frac {1}{2} \int \left (-\frac {\log \left (\frac {4 x}{1+x^2}\right )}{9 x^2}+\frac {(7+2 x) \log \left (\frac {4 x}{1+x^2}\right )}{3 \left (-3+3 x+2 x^2\right )^2}+\frac {2 \log \left (\frac {4 x}{1+x^2}\right )}{9 \left (-3+3 x+2 x^2\right )}\right ) \, dx\\ &=-\frac {1}{18 x}+\frac {29+2 x}{748 \left (3-3 x-2 x^2\right )}+\frac {3+26 x}{1122 \left (3-3 x-2 x^2\right )}-\frac {5 (93+58 x)}{6732 \left (3-3 x-2 x^2\right )}+\frac {\log (x)}{18}-\frac {\int \frac {528+990 x}{1+x^2} \, dx}{114444}-\frac {\int \frac {-1374-1980 x}{-3+3 x+2 x^2} \, dx}{114444}-\frac {\int \frac {990-528 x}{1+x^2} \, dx}{76296}-\frac {\int \frac {-192+1056 x}{-3+3 x+2 x^2} \, dx}{76296}-\frac {5 \int \frac {528+990 x}{1+x^2} \, dx}{228888}-\frac {5 \int \frac {-5998-1980 x}{-3+3 x+2 x^2} \, dx}{228888}+\frac {\int \frac {-4075-2042 x}{-3+3 x+2 x^2} \, dx}{10404}+\frac {\int \frac {447+253 x}{-3+3 x+2 x^2} \, dx}{2601}+\frac {\int \frac {-8-15 x}{1+x^2} \, dx}{1156}-\frac {\int \frac {15-8 x}{1+x^2} \, dx}{1156}-\frac {1}{204} \int \frac {75+38 x}{\left (-3+3 x+2 x^2\right )^2} \, dx+\frac {1}{204} \int \frac {113+50 x}{\left (-3+3 x+2 x^2\right )^2} \, dx-\frac {1}{18} \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{x^2} \, dx+\frac {1}{9} \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{-3+3 x+2 x^2} \, dx+\frac {1}{6} \int \frac {(7+2 x) \log \left (\frac {4 x}{1+x^2}\right )}{\left (-3+3 x+2 x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 1.02, size = 30, normalized size = 1.00 \begin {gather*} -\frac {\log \left (\frac {4 x}{1+x^2}\right )}{6 x \left (-3+3 x+2 x^2\right )} \end {gather*}

Integrate[(3 - 3*x - 5*x^2 + 3*x^3 + 2*x^4 + (-3 + 6*x + 3*x^2 + 6*x^3 + 6*x^4)*Log[(4*x)/(1 + x^2)])/(54*x^2

- 108*x^3 + 36*x^4 - 36*x^5 + 6*x^6 + 72*x^7 + 24*x^8),x]

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

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fricas [A] time = 0.65, size = 29, normalized size = 0.97 \begin {gather*} -\frac {\log \left (\frac {4 \, x}{x^{2} + 1}\right )}{6 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} \end {gather*}

integrate(((6*x^4+6*x^3+3*x^2+6*x-3)*log(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3*x+3)/(24*x^8+72*x^7+6*x^6-36*x^5+36*

-1/6*log(4*x/(x^2 + 1))/(2*x^3 + 3*x^2 - 3*x)

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giac [A] time = 0.37, size = 37, normalized size = 1.23 \begin {gather*} -\frac {1}{18} \, {\left (\frac {2 \, x + 3}{2 \, x^{2} + 3 \, x - 3} - \frac {1}{x}\right )} \log \left (\frac {4 \, x}{x^{2} + 1}\right ) \end {gather*}

integrate(((6*x^4+6*x^3+3*x^2+6*x-3)*log(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3*x+3)/(24*x^8+72*x^7+6*x^6-36*x^5+36*

-1/18*((2*x + 3)/(2*x^2 + 3*x - 3) - 1/x)*log(4*x/(x^2 + 1))

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method	result	size

norman	\(-\frac {\ln \left (\frac {4 x}{x^{2}+1}\right )}{6 x \left (2 x^{2}+3 x -3\right )}\)	\(29\)
risch	\(-\frac {\ln \left (\frac {4 x}{x^{2}+1}\right )}{6 x \left (2 x^{2}+3 x -3\right )}\)	\(29\)

int(((6*x^4+6*x^3+3*x^2+6*x-3)*ln(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3*x+3)/(24*x^8+72*x^7+6*x^6-36*x^5+36*x^4-108

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maxima [B] time = 2.19, size = 191, normalized size = 6.37 \begin {gather*} -\frac {350 \, x^{2} + 587 \, x - 374}{2244 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} + \frac {68 \, x^{2} + 3 \, {\left (6 \, x^{3} + 9 \, x^{2} - 9 \, x + 34\right )} \log \left (x^{2} + 1\right ) - 34 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x + 3\right )} \log \relax (x) + 102 \, x - 204 \, \log \relax (2) - 102}{612 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} + \frac {62 \, x + 151}{2244 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} + \frac {5 \, {\left (58 \, x + 93\right )}}{6732 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {26 \, x + 3}{1122 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {2 \, x + 29}{748 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {1}{68} \, \log \left (x^{2} + 1\right ) + \frac {1}{18} \, \log \relax (x) \end {gather*}

integrate(((6*x^4+6*x^3+3*x^2+6*x-3)*log(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3*x+3)/(24*x^8+72*x^7+6*x^6-36*x^5+36*

-1/2244*(350*x^2 + 587*x - 374)/(2*x^3 + 3*x^2 - 3*x) + 1/612*(68*x^2 + 3*(6*x^3 + 9*x^2 - 9*x + 34)*log(x^2 +

1) - 34*(2*x^3 + 3*x^2 - 3*x + 3)*log(x) + 102*x - 204*log(2) - 102)/(2*x^3 + 3*x^2 - 3*x) + 1/2244*(62*x + 1

51)/(2*x^2 + 3*x - 3) + 5/6732*(58*x + 93)/(2*x^2 + 3*x - 3) - 1/1122*(26*x + 3)/(2*x^2 + 3*x - 3) - 1/748*(2*

x + 29)/(2*x^2 + 3*x - 3) - 1/68*log(x^2 + 1) + 1/18*log(x)

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mupad [B] time = 1.26, size = 33, normalized size = 1.10 \begin {gather*} -\frac {2\,\ln \relax (2)-\ln \left (x^2+1\right )+\ln \relax (x)}{12\,\left (x^3+\frac {3\,x^2}{2}-\frac {3\,x}{2}\right )} \end {gather*}

int((log((4*x)/(x^2 + 1))*(6*x + 3*x^2 + 6*x^3 + 6*x^4 - 3) - 3*x - 5*x^2 + 3*x^3 + 2*x^4 + 3)/(54*x^2 - 108*x

^3 + 36*x^4 - 36*x^5 + 6*x^6 + 72*x^7 + 24*x^8),x)

-(2*log(2) - log(x^2 + 1) + log(x))/(12*((3*x^2)/2 - (3*x)/2 + x^3))

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sympy [A] time = 0.22, size = 24, normalized size = 0.80 \begin {gather*} - \frac {\log {\left (\frac {4 x}{x^{2} + 1} \right )}}{12 x^{3} + 18 x^{2} - 18 x} \end {gather*}

integrate(((6*x**4+6*x**3+3*x**2+6*x-3)*ln(4*x/(x**2+1))+2*x**4+3*x**3-5*x**2-3*x+3)/(24*x**8+72*x**7+6*x**6-3

-log(4*x/(x**2 + 1))/(12*x**3 + 18*x**2 - 18*x)

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3.14.83 \(\int \frac {3-3 x-5 x^2+3 x^3+2 x^4+(-3+6 x+3 x^2+6 x^3+6 x^4) \log (\frac {4 x}{1+x^2})}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx\)