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

Optimal. Leaf size=938 \[ 36 n^3 x b^3-36 n^2 x \log \left (c x^n\right ) b^3+\frac {12 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right ) b^3}{\sqrt {d} \sqrt {f}}-6 n^3 x \log \left (d f x^2+1\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right ) b^3-\frac {6 i n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} \sqrt {f}}+\frac {6 i n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} \sqrt {f}}+\frac {6 n^3 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}-\frac {6 n^3 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}+\frac {6 n^3 \text {Li}_4\left (-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}-\frac {6 n^3 \text {Li}_4\left (\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}-24 a n^2 x b^2-12 n^2 (a-b n) x b^2+\frac {12 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) b^2}{\sqrt {d} \sqrt {f}}+6 a n^2 x \log \left (d f x^2+1\right ) b^2-\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}+\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}-\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}+\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}+12 n x \left (a+b \log \left (c x^n\right )\right )^2 b+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right ) b}{\sqrt {-d} \sqrt {f}}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt {-d} \sqrt {f} x+1\right ) b}{\sqrt {-d} \sqrt {f}}-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) b+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right ) b}{\sqrt {-d} \sqrt {f}}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right ) b}{\sqrt {-d} \sqrt {f}}-2 x \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-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\sqrt {-d} \sqrt {f} x+1\right )}{\sqrt {-d} \sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d f x^2+1\right ) \]

[Out]

-3*b*n*(a+b*ln(c*x^n))^2*ln(1+x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-x*
(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+3*b*n*(a+b*ln(c*x^n))^2*polylog(2,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(
1/2)+6*I*b^3*n^3*polylog(2,I*x*d^(1/2)*f^(1/2))/d^(1/2)/f^(1/2)+6*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,x*(-d)^(1/
2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-3*b*n*(a+b*ln(c*x^n))^2*polylog(2,x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^
2*n^2*(a+b*ln(c*x^n))*polylog(3,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+6*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,
x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+12*b^2*n^2*(-b*n+a)*arctan(x*d^(1/2)*f^(1/2))/d^(1/2)/f^(1/2)+12*b^3*
n^2*arctan(x*d^(1/2)*f^(1/2))*ln(c*x^n)/d^(1/2)/f^(1/2)-6*I*b^3*n^3*polylog(2,-I*x*d^(1/2)*f^(1/2))/d^(1/2)/f^
(1/2)-12*b^2*n^2*(-b*n+a)*x-36*b^3*n^2*x*ln(c*x^n)+12*b*n*x*(a+b*ln(c*x^n))^2-24*a*b^2*n^2*x+6*b^3*n^3*polylog
(3,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^3*n^3*polylog(3,x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+6*b^
3*n^3*polylog(4,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^3*n^3*polylog(4,x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)
/f^(1/2)-2*x*(a+b*ln(c*x^n))^3+36*b^3*n^3*x+x*(a+b*ln(c*x^n))^3*ln(d*f*x^2+1)+6*a*b^2*n^2*x*ln(d*f*x^2+1)+6*b^
3*n^2*x*ln(c*x^n)*ln(d*f*x^2+1)-3*b*n*x*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)-6*b^3*n^3*x*ln(d*f*x^2+1)-(a+b*ln(c*x^
n))^3*ln(1-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+(a+b*ln(c*x^n))^3*ln(1+x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^
(1/2)+3*b*n*(a+b*ln(c*x^n))^2*ln(1-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.55, antiderivative size = 938, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {2296, 2295, 2371, 6, 321, 203, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589, 2383} \[ 36 n^3 x b^3-36 n^2 x \log \left (c x^n\right ) b^3+\frac {12 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right ) b^3}{\sqrt {d} \sqrt {f}}-6 n^3 x \log \left (d f x^2+1\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right ) b^3-\frac {6 i n^3 \text {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} \sqrt {f}}+\frac {6 i n^3 \text {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} \sqrt {f}}+\frac {6 n^3 \text {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}-\frac {6 n^3 \text {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}+\frac {6 n^3 \text {PolyLog}\left (4,-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}-\frac {6 n^3 \text {PolyLog}\left (4,\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} \sqrt {f}}-24 a n^2 x b^2-12 n^2 (a-b n) x b^2+\frac {12 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) b^2}{\sqrt {d} \sqrt {f}}+6 a n^2 x \log \left (d f x^2+1\right ) b^2-\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}+\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}-\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}+\frac {6 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} \sqrt {f}}+12 n x \left (a+b \log \left (c x^n\right )\right )^2 b+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right ) b}{\sqrt {-d} \sqrt {f}}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt {-d} \sqrt {f} x+1\right ) b}{\sqrt {-d} \sqrt {f}}-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) b+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) b}{\sqrt {-d} \sqrt {f}}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) b}{\sqrt {-d} \sqrt {f}}-2 x \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-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\sqrt {-d} \sqrt {f} x+1\right )}{\sqrt {-d} \sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d f x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-24*a*b^2*n^2*x + 36*b^3*n^3*x - 12*b^2*n^2*(a - b*n)*x + (12*b^2*n^2*(a - b*n)*ArcTan[Sqrt[d]*Sqrt[f]*x])/(Sq
rt[d]*Sqrt[f]) - 36*b^3*n^2*x*Log[c*x^n] + (12*b^3*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x]*Log[c*x^n])/(Sqrt[d]*Sqrt[f])
+ 12*b*n*x*(a + b*Log[c*x^n])^2 - 2*x*(a + b*Log[c*x^n])^3 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqr
t[f]*x])/(Sqrt[-d]*Sqrt[f]) - ((a + b*Log[c*x^n])^3*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (3*b*n*(
a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + ((a + b*Log[c*x^n])^3*Log[1 + Sqrt[-d]*S
qrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + 6*a*b^2*n^2*x*Log[1 + d*f*x^2] - 6*b^3*n^3*x*Log[1 + d*f*x^2] + 6*b^3*n^2*x*Lo
g[c*x^n]*Log[1 + d*f*x^2] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] + x*(a + b*Log[c*x^n])^3*Log[1 + d*f
*x^2] - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) + (3*b*n*(a + b*Lo
g[c*x^n])^2*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, S
qrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]
*Sqrt[f]) - ((6*I)*b^3*n^3*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) + ((6*I)*b^3*n^3*PolyLog[2, I
*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) + (6*b^3*n^3*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) - (6
*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) - (6*b^3*n^3*PolyLog[3, Sqrt
[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*
Sqrt[f]) + (6*b^3*n^3*PolyLog[4, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) - (6*b^3*n^3*PolyLog[4, Sqrt[-d]*S
qrt[f]*x])/(Sqrt[-d]*Sqrt[f])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 203

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, 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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2317

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

Rule 2324

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

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 2353

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

Rule 2371

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

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 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]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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]

Rubi steps

\begin {align*} \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 f) \int \left (\frac {6 a b^2 d n^2 x^2}{1+d f x^2}-\frac {6 b^3 d n^3 x^2}{1+d f x^2}+\frac {6 b^3 d n^2 x^2 \log \left (c x^n\right )}{1+d f x^2}-\frac {3 b d n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 f) \int \left (\frac {d \left (6 a b^2 n^2-6 b^3 n^3\right ) x^2}{1+d f x^2}+\frac {6 b^3 d n^2 x^2 \log \left (c x^n\right )}{1+d f x^2}-\frac {3 b d n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 d f) \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2} \, dx+(6 b d f n) \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx-\left (12 b^3 d f n^2\right ) \int \frac {x^2 \log \left (c x^n\right )}{1+d f x^2} \, dx-\left (12 b^2 d f n^2 (a-b n)\right ) \int \frac {x^2}{1+d f x^2} \, dx\\ &=-12 b^2 n^2 (a-b n) x+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(2 d f) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{d f \left (1+d f x^2\right )}\right ) \, dx+(6 b d f n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d f \left (1+d f x^2\right )}\right ) \, dx-\left (12 b^3 d f n^2\right ) \int \left (\frac {\log \left (c x^n\right )}{d f}-\frac {\log \left (c x^n\right )}{d f \left (1+d f x^2\right )}\right ) \, dx+\left (12 b^2 n^2 (a-b n)\right ) \int \frac {1}{1+d f x^2} \, dx\\ &=-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-2 \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx+2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1+d f x^2} \, dx+(6 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(6 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx-\left (12 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx+\left (12 b^3 n^2\right ) \int \frac {\log \left (c x^n\right )}{1+d f x^2} \, dx\\ &=12 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-12 b^3 n^2 x \log \left (c x^n\right )+\frac {12 b^3 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right )}{\sqrt {d} \sqrt {f}}+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+2 \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 \left (1-\sqrt {-d} \sqrt {f} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 \left (1+\sqrt {-d} \sqrt {f} x\right )}\right ) \, dx+(6 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(6 b n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1-\sqrt {-d} \sqrt {f} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 \left (1+\sqrt {-d} \sqrt {f} x\right )}\right ) \, dx-\left (12 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (12 b^3 n^3\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f} x} \, dx\\ &=-12 a b^2 n^2 x+12 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-12 b^3 n^2 x \log \left (c x^n\right )+\frac {12 b^3 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right )}{\sqrt {d} \sqrt {f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1-\sqrt {-d} \sqrt {f} x} \, dx-(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+\sqrt {-d} \sqrt {f} x} \, dx-\left (12 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (12 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac {\left (12 b^3 n^3\right ) \int \frac {\tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {d} \sqrt {f}}+\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1-\sqrt {-d} \sqrt {f} x} \, dx+\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1+\sqrt {-d} \sqrt {f} x} \, dx\\ &=-24 a b^2 n^2 x+24 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-24 b^3 n^2 x \log \left (c x^n\right )+\frac {12 b^3 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right )}{\sqrt {d} \sqrt {f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \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-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}-\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}-\left (12 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}+\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}-\frac {\left (6 i b^3 n^3\right ) \int \frac {\log \left (1-i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {d} \sqrt {f}}+\frac {\left (6 i b^3 n^3\right ) \int \frac {\log \left (1+i \sqrt {d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {d} \sqrt {f}}\\ &=-24 a b^2 n^2 x+36 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-36 b^3 n^2 x \log \left (c x^n\right )+\frac {12 b^3 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right )}{\sqrt {d} \sqrt {f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \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-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 i b^3 n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {6 i b^3 n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}+\frac {\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}+\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}\\ &=-24 a b^2 n^2 x+36 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-36 b^3 n^2 x \log \left (c x^n\right )+\frac {12 b^3 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right )}{\sqrt {d} \sqrt {f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \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-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 i b^3 n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {6 i b^3 n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {6 b^3 n^3 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^3 n^3 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}-\frac {\left (6 b^3 n^3\right ) \int \frac {\text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{x} \, dx}{\sqrt {-d} \sqrt {f}}\\ &=-24 a b^2 n^2 x+36 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-36 b^3 n^2 x \log \left (c x^n\right )+\frac {12 b^3 n^2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right )}{\sqrt {d} \sqrt {f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \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-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 i b^3 n^3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {6 i b^3 n^3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {6 b^3 n^3 \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^3 n^3 \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^3 n^3 \text {Li}_4\left (-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^3 n^3 \text {Li}_4\left (\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}\\ \end {align*}

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Mathematica [A]  time = 0.72, size = 1027, normalized size = 1.09 \[ \frac {2 b^3 \left (-\sqrt {d} \sqrt {f} x \left (\log ^3(x)-3 \log ^2(x)+6 \log (x)-6\right )-\frac {1}{2} i \left (\log \left (i \sqrt {d} \sqrt {f} x+1\right ) \log ^3(x)+3 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)-6 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right ) \log (x)+6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )\right )+\frac {1}{2} i \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right ) \log ^3(x)+3 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)-6 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right ) \log (x)+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right )\right ) n^3-6 b^2 \left (a-b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\sqrt {d} \sqrt {f} x \left (\log ^2(x)-2 \log (x)+2\right )+\frac {1}{2} i \left (\log \left (i \sqrt {d} \sqrt {f} x+1\right ) \log ^2(x)+2 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right ) \log (x)-2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)+2 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right ) \log (x)-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )\right ) n^2+3 b \left (a^2-2 b n a+2 b \left (\log \left (c x^n\right )-n \log (x)\right ) a+2 b^2 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 )\right ) \left (-2 \sqrt {d} \sqrt {f} x (\log (x)-1)-i \left (\log (x) \log \left (i \sqrt {d} \sqrt {f} x+1\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )\right )+i \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )\right ) n-2 \sqrt {d} \sqrt {f} x \left (a^3-3 b n a^2+3 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+3 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right ) a-6 b^3 n^3+b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3-3 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )+2 \tan ^{-1}\left (\sqrt {d} \sqrt {f} x\right ) \left (a^3-3 b n a^2+3 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+3 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right ) a-6 b^3 n^3+b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3-3 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )+\sqrt {d} \sqrt {f} x \left (a^3-3 b n a^2+6 b^2 n^2 a-6 b^3 n^3+b^3 \log ^3\left (c x^n\right )+3 b^2 (a-b n) \log ^2\left (c x^n\right )+3 b \left (a^2-2 b n a+2 b^2 n^2\right ) \log \left (c x^n\right )\right ) \log \left (d f x^2+1\right )}{\sqrt {d} \sqrt {f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d]*Sqrt[f]*x*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 6*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 3*a^2*b
*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 3*
b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a^3 - 3*
a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 6*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6
*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 3*b^3*n*(-(n*Log[x]) + Log[c*x^n]
)^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) + Sqrt[d]*Sqrt[f]*x*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 3*b*(
a^2 - 2*a*b*n + 2*b^2*n^2)*Log[c*x^n] + 3*b^2*(a - b*n)*Log[c*x^n]^2 + b^3*Log[c*x^n]^3)*Log[1 + d*f*x^2] + 3*
b*n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b^2*n*(n*Log[x] - Log[c*x^n]) + 2*a*b*(-(n*Log[x]) + Log[c*x^n]) + b^2*(-(n
*Log[x]) + Log[c*x^n])^2)*(-2*Sqrt[d]*Sqrt[f]*x*(-1 + Log[x]) - I*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyL
og[2, (-I)*Sqrt[d]*Sqrt[f]*x]) + I*(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])) -
6*b^2*n^2*(a - b*n - b*n*Log[x] + b*Log[c*x^n])*(Sqrt[d]*Sqrt[f]*x*(2 - 2*Log[x] + Log[x]^2) + (I/2)*(Log[x]^2
*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f
]*x]) - (I/2)*(Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3,
I*Sqrt[d]*Sqrt[f]*x])) + 2*b^3*n^3*(-(Sqrt[d]*Sqrt[f]*x*(-6 + 6*Log[x] - 3*Log[x]^2 + Log[x]^3)) - (I/2)*(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]) + (I/2)*(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]*S
qrt[f]*x])))/(Sqrt[d]*Sqrt[f])

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{3} \log \left (d f x^{2} + 1\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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*log((f*x^2 + 1/d)*d), x)

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maple [F]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (b^{3} x \log \left (x^{n}\right )^{3} - 3 \, {\left (b^{3} {\left (n - \log \relax (c)\right )} - a b^{2}\right )} x \log \left (x^{n}\right )^{2} + 3 \, {\left ({\left (2 \, n^{2} - 2 \, n \log \relax (c) + \log \relax (c)^{2}\right )} b^{3} - 2 \, a b^{2} {\left (n - \log \relax (c)\right )} + a^{2} b\right )} x \log \left (x^{n}\right ) + {\left (3 \, {\left (2 \, n^{2} - 2 \, n \log \relax (c) + \log \relax (c)^{2}\right )} a b^{2} - {\left (6 \, n^{3} - 6 \, n^{2} \log \relax (c) + 3 \, n \log \relax (c)^{2} - \log \relax (c)^{3}\right )} b^{3} - 3 \, a^{2} b {\left (n - \log \relax (c)\right )} + a^{3}\right )} x\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {2 \, {\left (b^{3} d f x^{2} \log \left (x^{n}\right )^{3} + 3 \, {\left (a b^{2} d f - {\left (d f n - d f \log \relax (c)\right )} b^{3}\right )} x^{2} \log \left (x^{n}\right )^{2} + 3 \, {\left (a^{2} b d f - 2 \, {\left (d f n - d f \log \relax (c)\right )} a b^{2} + {\left (2 \, d f n^{2} - 2 \, d f n \log \relax (c) + d f \log \relax (c)^{2}\right )} b^{3}\right )} x^{2} \log \left (x^{n}\right ) + {\left (a^{3} d f - 3 \, {\left (d f n - d f \log \relax (c)\right )} a^{2} b + 3 \, {\left (2 \, d f n^{2} - 2 \, d f n \log \relax (c) + d f \log \relax (c)^{2}\right )} a b^{2} - {\left (6 \, d f n^{3} - 6 \, d f n^{2} \log \relax (c) + 3 \, d f n \log \relax (c)^{2} - d f \log \relax (c)^{3}\right )} b^{3}\right )} x^{2}\right )}}{d f x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*x*log(x^n)^3 - 3*(b^3*(n - log(c)) - a*b^2)*x*log(x^n)^2 + 3*((2*n^2 - 2*n*log(c) + log(c)^2)*b^3 - 2*a*b
^2*(n - log(c)) + a^2*b)*x*log(x^n) + (3*(2*n^2 - 2*n*log(c) + log(c)^2)*a*b^2 - (6*n^3 - 6*n^2*log(c) + 3*n*l
og(c)^2 - log(c)^3)*b^3 - 3*a^2*b*(n - log(c)) + a^3)*x)*log(d*f*x^2 + 1) - integrate(2*(b^3*d*f*x^2*log(x^n)^
3 + 3*(a*b^2*d*f - (d*f*n - d*f*log(c))*b^3)*x^2*log(x^n)^2 + 3*(a^2*b*d*f - 2*(d*f*n - d*f*log(c))*a*b^2 + (2
*d*f*n^2 - 2*d*f*n*log(c) + d*f*log(c)^2)*b^3)*x^2*log(x^n) + (a^3*d*f - 3*(d*f*n - d*f*log(c))*a^2*b + 3*(2*d
*f*n^2 - 2*d*f*n*log(c) + d*f*log(c)^2)*a*b^2 - (6*d*f*n^3 - 6*d*f*n^2*log(c) + 3*d*f*n*log(c)^2 - d*f*log(c)^
3)*b^3)*x^2)/(d*f*x^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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