\(\int \frac {3-3 x-5 x^2+3 x^3+2 x^4+(-3+6 x+3 x^2+6 x^3+6 x^4) \log (\frac {4 x}{1+x^2})}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx\) [1383]

Optimal result

Integrand size = 91, antiderivative size = 30 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=\frac {\log \left (\frac {4}{\frac {1}{x}+x}\right )}{6 x (3+x-x (4+2 x))} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(973\) vs. \(2(30)=60\).

Time = 4.05 (sec) , antiderivative size = 973, normalized size of antiderivative = 32.43, number of steps used = 171, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {6873, 12, 6857, 989, 1088, 646, 31, 649, 209, 266, 652, 632, 212, 1032, 1079, 2608, 2605, 464, 2604, 2465, 2439, 2438, 2463, 2441, 2440} \[ \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 )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=\frac {2 x+29}{748 \left (-2 x^2-3 x+3\right )}+\frac {4 \left (3+\sqrt {33}\right ) \arctan (x)}{99 \left (29+3 \sqrt {33}\right )}-\frac {56 \arctan (x)}{99 \left (29+3 \sqrt {33}\right )}+\frac {4 \left (3-\sqrt {33}\right ) \arctan (x)}{99 \left (29-3 \sqrt {33}\right )}-\frac {56 \arctan (x)}{99 \left (29-3 \sqrt {33}\right )}+\frac {19 \arctan (x)}{306}+\frac {58 \text {arctanh}\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 )} \]

[In]

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]

[Out]

(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]))

Rule 12

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

Rule 31

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

Rule 209

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

Rule 212

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

Rule 266

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

Rule 464

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]

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 646

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*
c]

Rule 649

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]

Rule 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 989

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]) &
&  !IGtQ[q, 0]

Rule 1032

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)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 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), 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
])

Rule 1079

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)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 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), 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]

Rule 1088

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]

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 2439

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]

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 2463

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]

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 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 6857

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]

Rule 6873

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

Rubi steps \begin{align*} \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 \\ & = -\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 {151+62 x}{2244 \left (3-3 x-2 x^2\right )}+\frac {639+302 x}{6732 \left (3-3 x-2 x^2\right )}+\frac {\log (x)}{18}+\frac {\log \left (\frac {4 x}{1+x^2}\right )}{18 x}-\frac {4}{867} \int \frac {1}{1+x^2} \, dx-\frac {2}{289} \int \frac {1}{1+x^2} \, dx+2 \left (\frac {2}{289} \int \frac {x}{1+x^2} \, dx\right )-\frac {5}{578} \int \frac {x}{1+x^2} \, dx-\frac {10}{867} \int \frac {1}{1+x^2} \, dx-2 \frac {15 \int \frac {1}{1+x^2} \, dx}{1156}-\frac {15 \int \frac {x}{1+x^2} \, dx}{1156}-\frac {25 \int \frac {x}{1+x^2} \, dx}{1156}+\frac {31 \int \frac {1}{-3+3 x+2 x^2} \, dx}{1122}-\frac {151 \int \frac {1}{-3+3 x+2 x^2} \, dx}{3366}-\frac {1}{18} \int \frac {1-x^2}{x^2 \left (1+x^2\right )} \, dx+\frac {1}{9} \int \left (-\frac {4 \log \left (\frac {4 x}{1+x^2}\right )}{\sqrt {33} \left (-3+\sqrt {33}-4 x\right )}-\frac {4 \log \left (\frac {4 x}{1+x^2}\right )}{\sqrt {33} \left (3+\sqrt {33}+4 x\right )}\right ) \, dx+\frac {1}{6} \int \left (\frac {7 \log \left (\frac {4 x}{1+x^2}\right )}{\left (-3+3 x+2 x^2\right )^2}+\frac {2 x \log \left (\frac {4 x}{1+x^2}\right )}{\left (-3+3 x+2 x^2\right )^2}\right ) \, dx+\frac {\left (5 \left (16335-4513 \sqrt {33}\right )\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {33}}{2}+2 x} \, dx}{3776652}+\frac {\left (2783-343 \sqrt {33}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {33}}{2}+2 x} \, dx}{57222}-\frac {\left (2 \left (363-41 \sqrt {33}\right )\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {33}}{2}+2 x} \, dx}{104907}-\frac {\left (-5445+37 \sqrt {33}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {33}}{2}+2 x} \, dx}{629442}+\frac {\left (5445+37 \sqrt {33}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {33}}{2}+2 x} \, dx}{629442}-\frac {\left (2 \left (363+41 \sqrt {33}\right )\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {33}}{2}+2 x} \, dx}{104907}+\frac {\left (2783+343 \sqrt {33}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {33}}{2}+2 x} \, dx}{57222}+\frac {\left (5 \left (16335+4513 \sqrt {33}\right )\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {33}}{2}+2 x} \, dx}{3776652}+\frac {\left (-33693+5087 \sqrt {33}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {33}}{2}+2 x} \, dx}{343332}-\frac {\left (33693+5087 \sqrt {33}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {33}}{2}+2 x} \, dx}{343332} \\ & = \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 {151+62 x}{2244 \left (3-3 x-2 x^2\right )}+\frac {639+302 x}{6732 \left (3-3 x-2 x^2\right )}-\frac {5 \arctan (x)}{102}+\frac {\log (x)}{18}-\frac {\left (363-41 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{104907}+\frac {\left (5445-37 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{1258884}+\frac {\left (2783+343 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{114444}+\frac {5 \left (16335+4513 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{7553304}-\frac {\left (33693+5087 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{686664}-\frac {\left (33693-5087 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{686664}+\frac {5 \left (16335-4513 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{7553304}+\frac {\left (2783-343 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{114444}+\frac {\left (5445+37 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{1258884}-\frac {\left (363+41 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{104907}+\frac {\log \left (\frac {4 x}{1+x^2}\right )}{18 x}-\frac {1}{68} \log \left (1+x^2\right )-\frac {31}{561} \text {Subst}\left (\int \frac {1}{33-x^2} \, dx,x,3+4 x\right )+\frac {151 \text {Subst}\left (\int \frac {1}{33-x^2} \, dx,x,3+4 x\right )}{1683}+\frac {1}{9} \int \frac {1}{1+x^2} \, dx+\frac {1}{3} \int \frac {x \log \left (\frac {4 x}{1+x^2}\right )}{\left (-3+3 x+2 x^2\right )^2} \, dx+\frac {7}{6} \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{\left (-3+3 x+2 x^2\right )^2} \, dx-\frac {4 \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{-3+\sqrt {33}-4 x} \, dx}{9 \sqrt {33}}-\frac {4 \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{3+\sqrt {33}+4 x} \, dx}{9 \sqrt {33}} \\ & = \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 {151+62 x}{2244 \left (3-3 x-2 x^2\right )}+\frac {639+302 x}{6732 \left (3-3 x-2 x^2\right )}+\frac {19 \arctan (x)}{306}+\frac {58 \text {arctanh}\left (\frac {3+4 x}{\sqrt {33}}\right )}{1683 \sqrt {33}}+\frac {\log (x)}{18}-\frac {\left (363-41 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{104907}+\frac {\left (5445-37 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{1258884}+\frac {\left (2783+343 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{114444}+\frac {5 \left (16335+4513 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{7553304}-\frac {\left (33693+5087 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{686664}-\frac {\left (33693-5087 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{686664}+\frac {5 \left (16335-4513 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{7553304}+\frac {\left (2783-343 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{114444}+\frac {\left (5445+37 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{1258884}-\frac {\left (363+41 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{104907}+\frac {\log \left (\frac {4 x}{1+x^2}\right )}{18 x}+\frac {\log \left (-3+\sqrt {33}-4 x\right ) \log \left (\frac {4 x}{1+x^2}\right )}{9 \sqrt {33}}-\frac {\log \left (3+\sqrt {33}+4 x\right ) \log \left (\frac {4 x}{1+x^2}\right )}{9 \sqrt {33}}-\frac {1}{68} \log \left (1+x^2\right )+\frac {1}{3} \int \left (\frac {4 \left (-3+\sqrt {33}\right ) \log \left (\frac {4 x}{1+x^2}\right )}{33 \left (-3+\sqrt {33}-4 x\right )^2}-\frac {4 \log \left (\frac {4 x}{1+x^2}\right )}{11 \sqrt {33} \left (-3+\sqrt {33}-4 x\right )}+\frac {4 \left (-3-\sqrt {33}\right ) \log \left (\frac {4 x}{1+x^2}\right )}{33 \left (3+\sqrt {33}+4 x\right )^2}-\frac {4 \log \left (\frac {4 x}{1+x^2}\right )}{11 \sqrt {33} \left (3+\sqrt {33}+4 x\right )}\right ) \, dx+\frac {7}{6} \int \left (\frac {16 \log \left (\frac {4 x}{1+x^2}\right )}{33 \left (-3+\sqrt {33}-4 x\right )^2}+\frac {16 \log \left (\frac {4 x}{1+x^2}\right )}{33 \sqrt {33} \left (-3+\sqrt {33}-4 x\right )}+\frac {16 \log \left (\frac {4 x}{1+x^2}\right )}{33 \left (3+\sqrt {33}+4 x\right )^2}+\frac {16 \log \left (\frac {4 x}{1+x^2}\right )}{33 \sqrt {33} \left (3+\sqrt {33}+4 x\right )}\right ) \, dx-\frac {\int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (-3+\sqrt {33}-4 x\right )}{4 x} \, dx}{9 \sqrt {33}}+\frac {\int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (3+\sqrt {33}+4 x\right )}{4 x} \, dx}{9 \sqrt {33}} \\ & = \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 {151+62 x}{2244 \left (3-3 x-2 x^2\right )}+\frac {639+302 x}{6732 \left (3-3 x-2 x^2\right )}+\frac {19 \arctan (x)}{306}+\frac {58 \text {arctanh}\left (\frac {3+4 x}{\sqrt {33}}\right )}{1683 \sqrt {33}}+\frac {\log (x)}{18}-\frac {\left (363-41 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{104907}+\frac {\left (5445-37 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{1258884}+\frac {\left (2783+343 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{114444}+\frac {5 \left (16335+4513 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{7553304}-\frac {\left (33693+5087 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{686664}-\frac {\left (33693-5087 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{686664}+\frac {5 \left (16335-4513 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{7553304}+\frac {\left (2783-343 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{114444}+\frac {\left (5445+37 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{1258884}-\frac {\left (363+41 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{104907}+\frac {\log \left (\frac {4 x}{1+x^2}\right )}{18 x}+\frac {\log \left (-3+\sqrt {33}-4 x\right ) \log \left (\frac {4 x}{1+x^2}\right )}{9 \sqrt {33}}-\frac {\log \left (3+\sqrt {33}+4 x\right ) \log \left (\frac {4 x}{1+x^2}\right )}{9 \sqrt {33}}-\frac {1}{68} \log \left (1+x^2\right )+\frac {56}{99} \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{\left (-3+\sqrt {33}-4 x\right )^2} \, dx+\frac {56}{99} \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{\left (3+\sqrt {33}+4 x\right )^2} \, dx-\frac {\int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (-3+\sqrt {33}-4 x\right )}{x} \, dx}{36 \sqrt {33}}+\frac {\int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (3+\sqrt {33}+4 x\right )}{x} \, dx}{36 \sqrt {33}}-\frac {4 \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{-3+\sqrt {33}-4 x} \, dx}{33 \sqrt {33}}-\frac {4 \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{3+\sqrt {33}+4 x} \, dx}{33 \sqrt {33}}+\frac {56 \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{-3+\sqrt {33}-4 x} \, dx}{99 \sqrt {33}}+\frac {56 \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{3+\sqrt {33}+4 x} \, dx}{99 \sqrt {33}}-\frac {1}{99} \left (4 \left (3-\sqrt {33}\right )\right ) \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{\left (-3+\sqrt {33}-4 x\right )^2} \, dx-\frac {1}{99} \left (4 \left (3+\sqrt {33}\right )\right ) \int \frac {\log \left (\frac {4 x}{1+x^2}\right )}{\left (3+\sqrt {33}+4 x\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 {151+62 x}{2244 \left (3-3 x-2 x^2\right )}+\frac {639+302 x}{6732 \left (3-3 x-2 x^2\right )}+\frac {19 \arctan (x)}{306}+\frac {58 \text {arctanh}\left (\frac {3+4 x}{\sqrt {33}}\right )}{1683 \sqrt {33}}+\frac {\log (x)}{18}-\frac {\left (363-41 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{104907}+\frac {\left (5445-37 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{1258884}+\frac {\left (2783+343 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{114444}+\frac {5 \left (16335+4513 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{7553304}-\frac {\left (33693+5087 \sqrt {33}\right ) \log \left (3-\sqrt {33}+4 x\right )}{686664}-\frac {\left (33693-5087 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{686664}+\frac {5 \left (16335-4513 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{7553304}+\frac {\left (2783-343 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{114444}+\frac {\left (5445+37 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{1258884}-\frac {\left (363+41 \sqrt {33}\right ) \log \left (3+\sqrt {33}+4 x\right )}{104907}+\frac {\log \left (\frac {4 x}{1+x^2}\right )}{18 x}-\frac {14 \log \left (\frac {4 x}{1+x^2}\right )}{99 \left (3-\sqrt {33}+4 x\right )}+\frac {\left (3-\sqrt {33}\right ) \log \left (\frac {4 x}{1+x^2}\right )}{99 \left (3-\sqrt {33}+4 x\right )}-\frac {14 \log \left (\frac {4 x}{1+x^2}\right )}{99 \left (3+\sqrt {33}+4 x\right )}+\frac {\left (3+\sqrt {33}\right ) \log \left (\frac {4 x}{1+x^2}\right )}{99 \left (3+\sqrt {33}+4 x\right )}-\frac {1}{68} \log \left (1+x^2\right )+\frac {14}{99} \int \frac {1-x^2}{x \left (3+\sqrt {33}+4 x\right ) \left (1+x^2\right )} \, dx-\frac {14}{99} \int \frac {-1+x^2}{x \left (3-\sqrt {33}+4 x\right ) \left (1+x^2\right )} \, dx-\frac {\int \left (\frac {4 \log \left (-3+\sqrt {33}-4 x\right )}{x}-\frac {8 x \log \left (-3+\sqrt {33}-4 x\right )}{1+x^2}\right ) \, dx}{36 \sqrt {33}}+\frac {\int \left (\frac {4 \log \left (3+\sqrt {33}+4 x\right )}{x}-\frac {8 x \log \left (3+\sqrt {33}+4 x\right )}{1+x^2}\right ) \, dx}{36 \sqrt {33}}-\frac {\int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (-3+\sqrt {33}-4 x\right )}{4 x} \, dx}{33 \sqrt {33}}+\frac {\int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (3+\sqrt {33}+4 x\right )}{4 x} \, dx}{33 \sqrt {33}}+\frac {14 \int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (-3+\sqrt {33}-4 x\right )}{4 x} \, dx}{99 \sqrt {33}}-\frac {14 \int \frac {\left (1+x^2\right ) \left (-\frac {8 x^2}{\left (1+x^2\right )^2}+\frac {4}{1+x^2}\right ) \log \left (3+\sqrt {33}+4 x\right )}{4 x} \, dx}{99 \sqrt {33}}-\frac {1}{99} \left (-3+\sqrt {33}\right ) \int \frac {-1+x^2}{x \left (3-\sqrt {33}+4 x\right ) \left (1+x^2\right )} \, dx-\frac {1}{99} \left (3+\sqrt {33}\right ) \int \frac {1-x^2}{x \left (3+\sqrt {33}+4 x\right ) \left (1+x^2\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {\log \left (\frac {4 x}{1+x^2}\right )}{6 x \left (-3+3 x+2 x^2\right )} \]

[In]

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]

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97

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

[In]

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
*x^3+54*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {\log \left (\frac {4 \, x}{x^{2} + 1}\right )}{6 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} \]

[In]

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*
x^4-108*x^3+54*x^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=- \frac {\log {\left (\frac {4 x}{x^{2} + 1} \right )}}{12 x^{3} + 18 x^{2} - 18 x} \]

[In]

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
6*x**5+36*x**4-108*x**3+54*x**2),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (28) = 56\).

Time = 1.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 6.37 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {350 \, x^{2} + 587 \, x - 374}{2244 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} + \frac {68 \, x^{2} + 3 \, {\left (6 \, x^{3} + 9 \, x^{2} - 9 \, x + 34\right )} \log \left (x^{2} + 1\right ) - 34 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x + 3\right )} \log \left (x\right ) + 102 \, x - 204 \, \log \left (2\right ) - 102}{612 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} + \frac {62 \, x + 151}{2244 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} + \frac {5 \, {\left (58 \, x + 93\right )}}{6732 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {26 \, x + 3}{1122 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {2 \, x + 29}{748 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {1}{68} \, \log \left (x^{2} + 1\right ) + \frac {1}{18} \, \log \left (x\right ) \]

[In]

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*
x^4-108*x^3+54*x^2),x, algorithm="maxima")

[Out]

-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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {1}{18} \, {\left (\frac {2 \, x + 3}{2 \, x^{2} + 3 \, x - 3} - \frac {1}{x}\right )} \log \left (\frac {4 \, x}{x^{2} + 1}\right ) \]

[In]

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*
x^4-108*x^3+54*x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {2\,\ln \left (2\right )-\ln \left (x^2+1\right )+\ln \left (x\right )}{12\,\left (x^3+\frac {3\,x^2}{2}-\frac {3\,x}{2}\right )} \]

[In]

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)

[Out]

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