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

Optimal. Leaf size=367 \[ -\frac{b n \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{b^2 n^2 \text{PolyLog}\left (2,-d f x^2\right )}{16 d^2 f^2}+\frac{b^2 n^2 \text{PolyLog}\left (3,-d f x^2\right )}{8 d^2 f^2}-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{b^2 n^2 \log \left (d f x^2+1\right )}{32 d^2 f^2}+\frac{7 b^2 n^2 x^2}{32 d f}+\frac{1}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right )-\frac{3}{64} b^2 n^2 x^4 \]

[Out]

(7*b^2*n^2*x^2)/(32*d*f) - (3*b^2*n^2*x^4)/64 - (3*b*n*x^2*(a + b*Log[c*x^n]))/(8*d*f) + (b*n*x^4*(a + b*Log[c
*x^n]))/8 + (x^2*(a + b*Log[c*x^n])^2)/(4*d*f) - (x^4*(a + b*Log[c*x^n])^2)/8 - (b^2*n^2*Log[1 + d*f*x^2])/(32
*d^2*f^2) + (b^2*n^2*x^4*Log[1 + d*f*x^2])/32 + (b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(8*d^2*f^2) - (b*n*x
^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/8 - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*d^2*f^2) + (x^4*(a + b*
Log[c*x^n])^2*Log[1 + d*f*x^2])/4 + (b^2*n^2*PolyLog[2, -(d*f*x^2)])/(16*d^2*f^2) - (b*n*(a + b*Log[c*x^n])*Po
lyLog[2, -(d*f*x^2)])/(4*d^2*f^2) + (b^2*n^2*PolyLog[3, -(d*f*x^2)])/(8*d^2*f^2)

________________________________________________________________________________________

Rubi [A]  time = 0.364539, antiderivative size = 367, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {2454, 2395, 43, 2377, 2304, 2374, 6589, 2376, 2391} \[ -\frac{b n \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac{b^2 n^2 \text{PolyLog}\left (2,-d f x^2\right )}{16 d^2 f^2}+\frac{b^2 n^2 \text{PolyLog}\left (3,-d f x^2\right )}{8 d^2 f^2}-\frac{\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac{b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac{1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{b^2 n^2 \log \left (d f x^2+1\right )}{32 d^2 f^2}+\frac{7 b^2 n^2 x^2}{32 d f}+\frac{1}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right )-\frac{3}{64} b^2 n^2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*b^2*n^2*x^2)/(32*d*f) - (3*b^2*n^2*x^4)/64 - (3*b*n*x^2*(a + b*Log[c*x^n]))/(8*d*f) + (b*n*x^4*(a + b*Log[c
*x^n]))/8 + (x^2*(a + b*Log[c*x^n])^2)/(4*d*f) - (x^4*(a + b*Log[c*x^n])^2)/8 - (b^2*n^2*Log[1 + d*f*x^2])/(32
*d^2*f^2) + (b^2*n^2*x^4*Log[1 + d*f*x^2])/32 + (b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(8*d^2*f^2) - (b*n*x
^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/8 - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*d^2*f^2) + (x^4*(a + b*
Log[c*x^n])^2*Log[1 + d*f*x^2])/4 + (b^2*n^2*PolyLog[2, -(d*f*x^2)])/(16*d^2*f^2) - (b*n*(a + b*Log[c*x^n])*Po
lyLog[2, -(d*f*x^2)])/(4*d^2*f^2) + (b^2*n^2*PolyLog[3, -(d*f*x^2)])/(8*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 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 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 )^2 \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 )^2}{4 d f}-\frac{1}{8} x^4 \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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-(2 b n) \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{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^4 \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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{1}{4} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{1}{2} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right ) \, dx+\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{2 d^2 f^2}-\frac{(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d f}\\ &=\frac{b^2 n^2 x^2}{8 d f}-\frac{1}{64} b^2 n^2 x^4-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac{1}{2} \left (b^2 n^2\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 (b^2 n^2\right ) \int \frac{\text{Li}_2\left (-d f x^2\right )}{x} \, dx}{4 d^2 f^2}\\ &=\frac{3 b^2 n^2 x^2}{16 d f}-\frac{1}{32} b^2 n^2 x^4-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}+\frac{1}{8} \left (b^2 n^2\right ) \int x^3 \log \left (1+d f x^2\right ) \, dx-\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+d f x^2\right )}{x} \, dx}{8 d^2 f^2}\\ &=\frac{3 b^2 n^2 x^2}{16 d f}-\frac{1}{32} b^2 n^2 x^4-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{b^2 n^2 \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}+\frac{1}{16} \left (b^2 n^2\right ) \operatorname{Subst}\left (\int x \log (1+d f x) \, dx,x,x^2\right )\\ &=\frac{3 b^2 n^2 x^2}{16 d f}-\frac{1}{32} b^2 n^2 x^4-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{32} b^2 n^2 x^4 \log \left (1+d f x^2\right )+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{b^2 n^2 \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{1}{32} \left (b^2 d f n^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+d f x} \, dx,x,x^2\right )\\ &=\frac{3 b^2 n^2 x^2}{16 d f}-\frac{1}{32} b^2 n^2 x^4-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{32} b^2 n^2 x^4 \log \left (1+d f x^2\right )+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{b^2 n^2 \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac{1}{32} \left (b^2 d f n^2\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{7 b^2 n^2 x^2}{32 d f}-\frac{3}{64} b^2 n^2 x^4-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}+\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac{1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b^2 n^2 \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac{1}{32} b^2 n^2 x^4 \log \left (1+d f x^2\right )+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{8 d^2 f^2}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{b^2 n^2 \text{Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{4 d^2 f^2}+\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{8 d^2 f^2}\\ \end{align*}

Mathematica [C]  time = 0.337078, size = 654, normalized size = 1.78 \[ \frac{b n \left (8 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+8 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-d^2 f^2 x^4+4 d^2 f^2 x^4 \log (x)+4 d f x^2-8 d f x^2 \log (x)+8 \log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )+8 \log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right ) \left (-4 a-4 b \log \left (c x^n\right )+4 b n \log (x)+b n\right )+32 b^2 n^2 \left (\text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )+\text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )-\log (x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )-\log (x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )-\frac{1}{32} d^2 f^2 x^4 \left (8 \log ^2(x)-4 \log (x)+1\right )+\frac{1}{4} d f x^2 \left (2 \log ^2(x)-2 \log (x)+1\right )-\frac{1}{2} \log ^2(x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )-\frac{1}{2} \log ^2(x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )+2 d^2 f^2 x^4 \log \left (d f x^2+1\right ) \left (8 a^2-4 b (b n-4 a) \log \left (c x^n\right )-4 a b n+8 b^2 \log ^2\left (c x^n\right )+b^2 n^2\right )-d^2 f^2 x^4 \left (8 a^2+16 a b \left (\log \left (c x^n\right )-n \log (x)\right )-4 a b n+8 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )+2 d f x^2 \left (8 a^2+16 a b \left (\log \left (c x^n\right )-n \log (x)\right )-4 a b n+8 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )-2 \log \left (d f x^2+1\right ) \left (8 a^2+16 a b \left (\log \left (c x^n\right )-n \log (x)\right )-4 a b n+8 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )}{64 d^2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*f*x^2*(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]) + Log[c*x^n])^2) - d^2*f^2*x^4*(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]) + Log[c*x^n])^2) + 2*d^2*f^2*x^4*(8*a^2 - 4*a*b*n +
b^2*n^2 - 4*b*(-4*a + b*n)*Log[c*x^n] + 8*b^2*Log[c*x^n]^2)*Log[1 + d*f*x^2] - 2*(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]) + Log[c*x^n])^2)*Log[
1 + d*f*x^2] + b*n*(-4*a + b*n + 4*b*n*Log[x] - 4*b*Log[c*x^n])*(4*d*f*x^2 - d^2*f^2*x^4 - 8*d*f*x^2*Log[x] +
4*d^2*f^2*x^4*Log[x] + 8*Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + 8*Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 8*PolyL
og[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 8*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) + 32*b^2*n^2*((d*f*x^2*(1 - 2*Log[x] + 2*Lo
g[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]))/(64*d^2*f^2)

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\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))^2*ln(d*(1/d+f*x^2)),x)

[Out]

int(x^3*(a+b*ln(c*x^n))^2*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}{32} \,{\left (8 \, b^{2} x^{4} \log \left (x^{n}\right )^{2} - 4 \,{\left (b^{2}{\left (n - 4 \, \log \left (c\right )\right )} - 4 \, a b\right )} x^{4} \log \left (x^{n}\right ) +{\left ({\left (n^{2} - 4 \, n \log \left (c\right ) + 8 \, \log \left (c\right )^{2}\right )} b^{2} - 4 \, a b{\left (n - 4 \, \log \left (c\right )\right )} + 8 \, a^{2}\right )} x^{4}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac{8 \, b^{2} d f x^{5} \log \left (x^{n}\right )^{2} + 4 \,{\left (4 \, a b d f -{\left (d f n - 4 \, d f \log \left (c\right )\right )} b^{2}\right )} x^{5} \log \left (x^{n}\right ) +{\left (8 \, a^{2} d f - 4 \,{\left (d f n - 4 \, d f \log \left (c\right )\right )} a b +{\left (d f n^{2} - 4 \, d f n \log \left (c\right ) + 8 \, d f \log \left (c\right )^{2}\right )} b^{2}\right )} x^{5}}{16 \,{\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))^2*log(d*(1/d+f*x^2)),x, algorithm="maxima")

[Out]

1/32*(8*b^2*x^4*log(x^n)^2 - 4*(b^2*(n - 4*log(c)) - 4*a*b)*x^4*log(x^n) + ((n^2 - 4*n*log(c) + 8*log(c)^2)*b^
2 - 4*a*b*(n - 4*log(c)) + 8*a^2)*x^4)*log(d*f*x^2 + 1) - integrate(1/16*(8*b^2*d*f*x^5*log(x^n)^2 + 4*(4*a*b*
d*f - (d*f*n - 4*d*f*log(c))*b^2)*x^5*log(x^n) + (8*a^2*d*f - 4*(d*f*n - 4*d*f*log(c))*a*b + (d*f*n^2 - 4*d*f*
n*log(c) + 8*d*f*log(c)^2)*b^2)*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^{2} x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 2 \, a b x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{2} 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))^2*log(d*(1/d+f*x^2)),x, algorithm="fricas")

[Out]

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

[Out]

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