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

Optimal. Leaf size=241 \[ \frac{b n \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d f}-\frac{b^2 n^2 \text{PolyLog}\left (2,-d f x^2\right )}{4 d f}-\frac{b^2 n^2 \text{PolyLog}\left (3,-d f x^2\right )}{4 d f}-\frac{b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d f}+\frac{\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d f}+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b^2 n^2 \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{4 d f}-\frac{3}{4} b^2 n^2 x^2 \]

[Out]

(-3*b^2*n^2*x^2)/4 + b*n*x^2*(a + b*Log[c*x^n]) - (x^2*(a + b*Log[c*x^n])^2)/2 + (b^2*n^2*(1 + d*f*x^2)*Log[1
+ d*f*x^2])/(4*d*f) - (b*n*(1 + d*f*x^2)*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(2*d*f) + ((1 + d*f*x^2)*(a + b*
Log[c*x^n])^2*Log[1 + d*f*x^2])/(2*d*f) - (b^2*n^2*PolyLog[2, -(d*f*x^2)])/(4*d*f) + (b*n*(a + b*Log[c*x^n])*P
olyLog[2, -(d*f*x^2)])/(2*d*f) - (b^2*n^2*PolyLog[3, -(d*f*x^2)])/(4*d*f)

________________________________________________________________________________________

Rubi [A]  time = 0.505307, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 16, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {2454, 2389, 2295, 2377, 2304, 14, 2351, 2301, 6742, 2374, 6589, 2376, 2475, 2411, 43, 2315} \[ \frac{b n \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d f}-\frac{b^2 n^2 \text{PolyLog}\left (2,-d f x^2\right )}{4 d f}-\frac{b^2 n^2 \text{PolyLog}\left (3,-d f x^2\right )}{4 d f}-\frac{b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d f}+\frac{\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d f}+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b^2 n^2 \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{4 d f}-\frac{3}{4} b^2 n^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^2*n^2*x^2)/4 + b*n*x^2*(a + b*Log[c*x^n]) - (x^2*(a + b*Log[c*x^n])^2)/2 + (b^2*n^2*(1 + d*f*x^2)*Log[1
+ d*f*x^2])/(4*d*f) - (b*n*(1 + d*f*x^2)*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(2*d*f) + ((1 + d*f*x^2)*(a + b*
Log[c*x^n])^2*Log[1 + d*f*x^2])/(2*d*f) - (b^2*n^2*PolyLog[2, -(d*f*x^2)])/(4*d*f) + (b*n*(a + b*Log[c*x^n])*P
olyLog[2, -(d*f*x^2)])/(2*d*f) - (b^2*n^2*PolyLog[3, -(d*f*x^2)])/(4*d*f)

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 2389

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 6742

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

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 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int x \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{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}-(2 b n) \int \left (-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f x}\right ) \, dx\\ &=-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}+(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{(b n) \int \frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{d f}\\ &=-\frac{1}{4} b^2 n^2 x^2+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}-\frac{(b n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}+d f x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )\right ) \, dx}{d f}\\ &=-\frac{1}{4} b^2 n^2 x^2+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}-(b n) \int x \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}{d f}\\ &=-\frac{1}{4} b^2 n^2 x^2+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{2 d f}+\left (b^2 n^2\right ) \int \left (-\frac{x}{2}+\frac{\left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{2 d f x}\right ) \, dx-\frac{\left (b^2 n^2\right ) \int \frac{\text{Li}_2\left (-d f x^2\right )}{x} \, dx}{2 d f}\\ &=-\frac{1}{2} b^2 n^2 x^2+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{2 d f}-\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{4 d f}+\frac{\left (b^2 n^2\right ) \int \frac{\left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{x} \, dx}{2 d f}\\ &=-\frac{1}{2} b^2 n^2 x^2+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{2 d f}-\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{4 d f}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{(1+d f x) \log (1+d f x)}{x} \, dx,x,x^2\right )}{4 d f}\\ &=-\frac{1}{2} b^2 n^2 x^2+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{2 d f}-\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{4 d f}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \log (x)}{-\frac{1}{d f}+\frac{x}{d f}} \, dx,x,1+d f x^2\right )}{4 d^2 f^2}\\ &=-\frac{1}{2} b^2 n^2 x^2+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{2 d f}-\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{4 d f}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (d f \log (x)+\frac{d f \log (x)}{-1+x}\right ) \, dx,x,1+d f x^2\right )}{4 d^2 f^2}\\ &=-\frac{1}{2} b^2 n^2 x^2+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{2 d f}-\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{4 d f}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+d f x^2\right )}{4 d f}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-1+x} \, dx,x,1+d f x^2\right )}{4 d f}\\ &=-\frac{3}{4} b^2 n^2 x^2+b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b^2 n^2 \left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac{b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 d f}+\frac{\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 d f}-\frac{b^2 n^2 \text{Li}_2\left (-d f x^2\right )}{4 d f}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )}{2 d f}-\frac{b^2 n^2 \text{Li}_3\left (-d f x^2\right )}{4 d f}\\ \end{align*}

Mathematica [C]  time = 0.256498, size = 519, normalized size = 2.15 \[ \frac{2 b n \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\frac{1}{2} d f x^2-d f x^2 \log (x)+\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right ) \left (2 a+2 b \log \left (c x^n\right )-2 b n \log (x)-b n\right )-b^2 n^2 \left (4 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )+4 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )-4 \log (x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )-4 \log (x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+d f x^2+2 d f x^2 \log ^2(x)-2 d f x^2 \log (x)-2 \log ^2(x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )-2 \log ^2(x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )+d f x^2 \log \left (d f x^2+1\right ) \left (2 a^2-2 b (b n-2 a) \log \left (c x^n\right )-2 a b n+2 b^2 \log ^2\left (c x^n\right )+b^2 n^2\right )-d f x^2 \left (2 a^2+4 a b \left (\log \left (c x^n\right )-n \log (x)\right )-2 a b n+2 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )+\log \left (d f x^2+1\right ) \left (2 a^2+4 a b \left (\log \left (c x^n\right )-n \log (x)\right )-2 a b n+2 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+b^2 n^2\right )}{4 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(d*f*x^2*(2*a^2 - 2*a*b*n + b^2*n^2 + 2*b^2*n*(n*Log[x] - Log[c*x^n]) + 4*a*b*(-(n*Log[x]) + Log[c*x^n]) + 2
*b^2*(-(n*Log[x]) + Log[c*x^n])^2)) + d*f*x^2*(2*a^2 - 2*a*b*n + b^2*n^2 - 2*b*(-2*a + b*n)*Log[c*x^n] + 2*b^2
*Log[c*x^n]^2)*Log[1 + d*f*x^2] + (2*a^2 - 2*a*b*n + b^2*n^2 + 2*b^2*n*(n*Log[x] - Log[c*x^n]) + 4*a*b*(-(n*Lo
g[x]) + Log[c*x^n]) + 2*b^2*(-(n*Log[x]) + Log[c*x^n])^2)*Log[1 + d*f*x^2] + 2*b*n*(2*a - b*n - 2*b*n*Log[x] +
2*b*Log[c*x^n])*((d*f*x^2)/2 - d*f*x^2*Log[x] + 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]) - b^2*n^2*(d*f*x^2 - 2*d*
f*x^2*Log[x] + 2*d*f*x^2*Log[x]^2 - 2*Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] - 2*Log[x]^2*Log[1 + I*Sqrt[d]*Sqr
t[f]*x] - 4*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - 4*Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + 4*PolyLog[3
, (-I)*Sqrt[d]*Sqrt[f]*x] + 4*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]))/(4*d*f)

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Maple [F]  time = 0.208, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^2)),x)

[Out]

int(x*(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}{4} \,{\left (2 \, b^{2} x^{2} \log \left (x^{n}\right )^{2} - 2 \,{\left (b^{2}{\left (n - 2 \, \log \left (c\right )\right )} - 2 \, a b\right )} x^{2} \log \left (x^{n}\right ) +{\left ({\left (n^{2} - 2 \, n \log \left (c\right ) + 2 \, \log \left (c\right )^{2}\right )} b^{2} - 2 \, a b{\left (n - 2 \, \log \left (c\right )\right )} + 2 \, a^{2}\right )} x^{2}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac{2 \, b^{2} d f x^{3} \log \left (x^{n}\right )^{2} + 2 \,{\left (2 \, a b d f -{\left (d f n - 2 \, d f \log \left (c\right )\right )} b^{2}\right )} x^{3} \log \left (x^{n}\right ) +{\left (2 \, a^{2} d f - 2 \,{\left (d f n - 2 \, d f \log \left (c\right )\right )} a b +{\left (d f n^{2} - 2 \, d f n \log \left (c\right ) + 2 \, d f \log \left (c\right )^{2}\right )} b^{2}\right )} x^{3}}{2 \,{\left (d f x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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