3.40 \(\int x^3 (a+b \log (c x^n))^3 \log (d (\frac{1}{d}+f x^2)) \, dx\)

Optimal. Leaf size=591 \[ \frac{3 b^2 n^2 \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{16 d^2 f^2}+\frac{3 b^2 n^2 \text{PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}-\frac{3 b n \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{PolyLog}\left (2,-d f x^2\right )}{64 d^2 f^2}-\frac{3 b^3 n^3 \text{PolyLog}\left (3,-d f x^2\right )}{32 d^2 f^2}-\frac{3 b^3 n^3 \text{PolyLog}\left (4,-d f x^2\right )}{16 d^2 f^2}-\frac{3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{16 d^2 f^2}+\frac{1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{16} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b^3 n^3 \log \left (d f x^2+1\right )}{128 d^2 f^2}-\frac{45 b^3 n^3 x^2}{128 d f}-\frac{3}{128} b^3 n^3 x^4 \log \left (d f x^2+1\right )+\frac{3}{64} b^3 n^3 x^4 \]

[Out]

(-45*b^3*n^3*x^2)/(128*d*f) + (3*b^3*n^3*x^4)/64 + (21*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(32*d*f) - (9*b^2*n^2*x
^4*(a + b*Log[c*x^n]))/64 - (9*b*n*x^2*(a + b*Log[c*x^n])^2)/(16*d*f) + (3*b*n*x^4*(a + b*Log[c*x^n])^2)/16 +
(x^2*(a + b*Log[c*x^n])^3)/(4*d*f) - (x^4*(a + b*Log[c*x^n])^3)/8 + (3*b^3*n^3*Log[1 + d*f*x^2])/(128*d^2*f^2)
- (3*b^3*n^3*x^4*Log[1 + d*f*x^2])/128 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(32*d^2*f^2) + (3*b^
2*n^2*x^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/32 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(16*d^2*f^2)
- (3*b*n*x^4*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/16 - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(4*d^2*f^2)
+ (x^4*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/4 - (3*b^3*n^3*PolyLog[2, -(d*f*x^2)])/(64*d^2*f^2) + (3*b^2*n^2
*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*x^2)])/(16*d^2*f^2) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^2)])
/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[3, -(d*f*x^2)])/(32*d^2*f^2) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d*
f*x^2)])/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[4, -(d*f*x^2)])/(16*d^2*f^2)

________________________________________________________________________________________

Rubi [A]  time = 0.734313, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {2454, 2395, 43, 2377, 2305, 2304, 2374, 2383, 6589, 2376, 2391} \[ \frac{3 b^2 n^2 \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{16 d^2 f^2}+\frac{3 b^2 n^2 \text{PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}-\frac{3 b n \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{PolyLog}\left (2,-d f x^2\right )}{64 d^2 f^2}-\frac{3 b^3 n^3 \text{PolyLog}\left (3,-d f x^2\right )}{32 d^2 f^2}-\frac{3 b^3 n^3 \text{PolyLog}\left (4,-d f x^2\right )}{16 d^2 f^2}-\frac{3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac{3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{16 d^2 f^2}+\frac{1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{16} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b^3 n^3 \log \left (d f x^2+1\right )}{128 d^2 f^2}-\frac{45 b^3 n^3 x^2}{128 d f}-\frac{3}{128} b^3 n^3 x^4 \log \left (d f x^2+1\right )+\frac{3}{64} b^3 n^3 x^4 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)],x]

[Out]

(-45*b^3*n^3*x^2)/(128*d*f) + (3*b^3*n^3*x^4)/64 + (21*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(32*d*f) - (9*b^2*n^2*x
^4*(a + b*Log[c*x^n]))/64 - (9*b*n*x^2*(a + b*Log[c*x^n])^2)/(16*d*f) + (3*b*n*x^4*(a + b*Log[c*x^n])^2)/16 +
(x^2*(a + b*Log[c*x^n])^3)/(4*d*f) - (x^4*(a + b*Log[c*x^n])^3)/8 + (3*b^3*n^3*Log[1 + d*f*x^2])/(128*d^2*f^2)
- (3*b^3*n^3*x^4*Log[1 + d*f*x^2])/128 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(32*d^2*f^2) + (3*b^
2*n^2*x^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/32 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(16*d^2*f^2)
- (3*b*n*x^4*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/16 - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(4*d^2*f^2)
+ (x^4*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/4 - (3*b^3*n^3*PolyLog[2, -(d*f*x^2)])/(64*d^2*f^2) + (3*b^2*n^2
*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*x^2)])/(16*d^2*f^2) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^2)])
/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[3, -(d*f*x^2)])/(32*d^2*f^2) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d*
f*x^2)])/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[4, -(d*f*x^2)])/(16*d^2*f^2)

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 2395

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

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 2377

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[
Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] &&
RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q,
0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1]))

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

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

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 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]

Rubi steps

\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac{1}{d}+f x^2\right )\right ) \, dx &=\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(3 b n) \int \left (\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac{1}{4} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )\right ) \, dx\\ &=\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac{1}{8} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{1}{4} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right ) \, dx+\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x} \, dx}{4 d^2 f^2}-\frac{(3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 d f}\\ &=-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac{1}{16} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{1}{2} \left (3 b^2 n^2\right ) \int \left (\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac{1}{8} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac{1}{4} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )\right ) \, dx+\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{x} \, dx}{4 d^2 f^2}+\frac{\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 d f}\\ &=-\frac{3 b^3 n^3 x^2}{16 d f}+\frac{3}{256} b^3 n^3 x^4+\frac{3 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}-\frac{3}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{1}{16} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{1}{8} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right ) \, dx-\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{8 d^2 f^2}+\frac{\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 d f}-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_3\left (-d f x^2\right )}{x} \, dx}{8 d^2 f^2}\\ &=-\frac{9 b^3 n^3 x^2}{32 d f}+\frac{3}{128} b^3 n^3 x^4+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac{1}{8} \left (3 b^3 n^3\right ) \int \left (\frac{x}{4 d f}-\frac{x^3}{8}-\frac{\log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac{1}{4} x^3 \log \left (1+d f x^2\right )\right ) \, dx-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_2\left (-d f x^2\right )}{x} \, dx}{16 d^2 f^2}\\ &=-\frac{21 b^3 n^3 x^2}{64 d f}+\frac{9}{256} b^3 n^3 x^4+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac{1}{32} \left (3 b^3 n^3\right ) \int x^3 \log \left (1+d f x^2\right ) \, dx+\frac{\left (3 b^3 n^3\right ) \int \frac{\log \left (1+d f x^2\right )}{x} \, dx}{32 d^2 f^2}\\ &=-\frac{21 b^3 n^3 x^2}{64 d f}+\frac{9}{256} b^3 n^3 x^4+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{3 b^3 n^3 \text{Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac{1}{64} \left (3 b^3 n^3\right ) \operatorname{Subst}\left (\int x \log (1+d f x) \, dx,x,x^2\right )\\ &=-\frac{21 b^3 n^3 x^2}{64 d f}+\frac{9}{256} b^3 n^3 x^4+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{3 b^3 n^3 \text{Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_4\left (-d f x^2\right )}{16 d^2 f^2}+\frac{1}{128} \left (3 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+d f x} \, dx,x,x^2\right )\\ &=-\frac{21 b^3 n^3 x^2}{64 d f}+\frac{9}{256} b^3 n^3 x^4+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{3 b^3 n^3 \text{Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_4\left (-d f x^2\right )}{16 d^2 f^2}+\frac{1}{128} \left (3 b^3 d f n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{d^2 f^2}+\frac{x}{d f}+\frac{1}{d^2 f^2 (1+d f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{45 b^3 n^3 x^2}{128 d f}+\frac{3}{64} b^3 n^3 x^4+\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac{9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^3 n^3 \log \left (1+d f x^2\right )}{128 d^2 f^2}-\frac{3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac{3 b^3 n^3 \text{Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{3 b^3 n^3 \text{Li}_4\left (-d f x^2\right )}{16 d^2 f^2}\\ \end{align*}

Mathematica [C]  time = 1.0407, size = 1234, normalized size = 2.09 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)],x]

[Out]

-(-2*d*f*x^2*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*b*(
-(n*Log[x]) + Log[c*x^n]) + 12*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 24
*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n])^3) + d^2*f^2*x^4*(32*a^3 - 24*a^2*b*n
+ 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*b^3
*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 24*b^3*n*(-(n*Log[x]) + Log[c*x^n])^
2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 2*d^2*f^2*x^4*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 12*
b*(8*a^2 - 4*a*b*n + b^2*n^2)*Log[c*x^n] - 24*b^2*(-4*a + b*n)*Log[c*x^n]^2 + 32*b^3*Log[c*x^n]^3)*Log[1 + d*f
*x^2] + 2*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*b*(-(n
*Log[x]) + Log[c*x^n]) + 12*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 24*b^
3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n])^3)*Log[1 + d*f*x^2] + 24*b*n*(8*a^2 - 4*a
*b*n + b^2*n^2 + 4*b^2*n*(n*Log[x] - Log[c*x^n]) + 16*a*b*(-(n*Log[x]) + Log[c*x^n]) + 8*b^2*(-(n*Log[x]) + Lo
g[c*x^n])^2)*((d*f*x^2)/2 - (d^2*f^2*x^4)/8 - d*f*x^2*Log[x] + (d^2*f^2*x^4*Log[x])/2 + Log[x]*Log[1 - I*Sqrt[
d]*Sqrt[f]*x] + Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d
]*Sqrt[f]*x]) - 96*b^2*n^2*(4*a - b*n - 4*b*n*Log[x] + 4*b*Log[c*x^n])*((d*f*x^2*(1 - 2*Log[x] + 2*Log[x]^2))/
4 - (d^2*f^2*x^4*(1 - 4*Log[x] + 8*Log[x]^2))/32 - (Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x])/2 - (Log[x]^2*Log[1
+ I*Sqrt[d]*Sqrt[f]*x])/2 - Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x
] + PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] + PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]) + b^3*n^3*(-16*d*f*x^2*(-3 + 6*Log[x
] - 6*Log[x]^2 + 4*Log[x]^3) + d^2*f^2*x^4*(-3 + 12*Log[x] - 24*Log[x]^2 + 32*Log[x]^3) + 64*(Log[x]^3*Log[1 +
I*Sqrt[d]*Sqrt[f]*x] + 3*Log[x]^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]*PolyLog[3, (-I)*Sqrt[d]*Sqrt[
f]*x] + 6*PolyLog[4, (-I)*Sqrt[d]*Sqrt[f]*x]) + 64*(Log[x]^3*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + 3*Log[x]^2*PolyLog
[2, I*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x] + 6*PolyLog[4, I*Sqrt[d]*Sqrt[f]*x])))/(25
6*d^2*f^2)

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Maple [F]  time = 0.153, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2)),x)

[Out]

int(x^3*(a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{128} \,{\left (32 \, b^{3} x^{4} \log \left (x^{n}\right )^{3} - 24 \,{\left (b^{3}{\left (n - 4 \, \log \left (c\right )\right )} - 4 \, a b^{2}\right )} x^{4} \log \left (x^{n}\right )^{2} + 12 \,{\left ({\left (n^{2} - 4 \, n \log \left (c\right ) + 8 \, \log \left (c\right )^{2}\right )} b^{3} - 4 \, a b^{2}{\left (n - 4 \, \log \left (c\right )\right )} + 8 \, a^{2} b\right )} x^{4} \log \left (x^{n}\right ) +{\left (12 \,{\left (n^{2} - 4 \, n \log \left (c\right ) + 8 \, \log \left (c\right )^{2}\right )} a b^{2} -{\left (3 \, n^{3} - 12 \, n^{2} \log \left (c\right ) + 24 \, n \log \left (c\right )^{2} - 32 \, \log \left (c\right )^{3}\right )} b^{3} - 24 \, a^{2} b{\left (n - 4 \, \log \left (c\right )\right )} + 32 \, a^{3}\right )} x^{4}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac{32 \, b^{3} d f x^{5} \log \left (x^{n}\right )^{3} + 24 \,{\left (4 \, a b^{2} d f -{\left (d f n - 4 \, d f \log \left (c\right )\right )} b^{3}\right )} x^{5} \log \left (x^{n}\right )^{2} + 12 \,{\left (8 \, a^{2} b d f - 4 \,{\left (d f n - 4 \, d f \log \left (c\right )\right )} a b^{2} +{\left (d f n^{2} - 4 \, d f n \log \left (c\right ) + 8 \, d f \log \left (c\right )^{2}\right )} b^{3}\right )} x^{5} \log \left (x^{n}\right ) +{\left (32 \, a^{3} d f - 24 \,{\left (d f n - 4 \, d f \log \left (c\right )\right )} a^{2} b + 12 \,{\left (d f n^{2} - 4 \, d f n \log \left (c\right ) + 8 \, d f \log \left (c\right )^{2}\right )} a b^{2} -{\left (3 \, d f n^{3} - 12 \, d f n^{2} \log \left (c\right ) + 24 \, d f n \log \left (c\right )^{2} - 32 \, d f \log \left (c\right )^{3}\right )} b^{3}\right )} x^{5}}{64 \,{\left (d f x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="maxima")

[Out]

1/128*(32*b^3*x^4*log(x^n)^3 - 24*(b^3*(n - 4*log(c)) - 4*a*b^2)*x^4*log(x^n)^2 + 12*((n^2 - 4*n*log(c) + 8*lo
g(c)^2)*b^3 - 4*a*b^2*(n - 4*log(c)) + 8*a^2*b)*x^4*log(x^n) + (12*(n^2 - 4*n*log(c) + 8*log(c)^2)*a*b^2 - (3*
n^3 - 12*n^2*log(c) + 24*n*log(c)^2 - 32*log(c)^3)*b^3 - 24*a^2*b*(n - 4*log(c)) + 32*a^3)*x^4)*log(d*f*x^2 +
1) - integrate(1/64*(32*b^3*d*f*x^5*log(x^n)^3 + 24*(4*a*b^2*d*f - (d*f*n - 4*d*f*log(c))*b^3)*x^5*log(x^n)^2
+ 12*(8*a^2*b*d*f - 4*(d*f*n - 4*d*f*log(c))*a*b^2 + (d*f*n^2 - 4*d*f*n*log(c) + 8*d*f*log(c)^2)*b^3)*x^5*log(
x^n) + (32*a^3*d*f - 24*(d*f*n - 4*d*f*log(c))*a^2*b + 12*(d*f*n^2 - 4*d*f*n*log(c) + 8*d*f*log(c)^2)*a*b^2 -
(3*d*f*n^3 - 12*d*f*n^2*log(c) + 24*d*f*n*log(c)^2 - 32*d*f*log(c)^3)*b^3)*x^5)/(d*f*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{3} x^{3} \log \left (d f x^{2} + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="fricas")

[Out]

integral(b^3*x^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*x^3*log(d*f*x^2 + 1)*log(c*x^n)^2 + 3*a^2*b*x^3*log(d
*f*x^2 + 1)*log(c*x^n) + a^3*x^3*log(d*f*x^2 + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*ln(c*x**n))**3*ln(d*(1/d+f*x**2)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3} \log \left ({\left (f x^{2} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^3*x^3*log((f*x^2 + 1/d)*d), x)