3.2.12 \(\int (a+b \log (c x^n))^3 \log (d (e+f x^2)^m) \, dx\) [112]

Optimal. Leaf size=977 \[ -24 a b^2 m n^2 x+36 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac {12 b^2 \sqrt {e} m n^2 (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-36 b^3 m n^2 x \log \left (c x^n\right )+\frac {12 b^3 \sqrt {e} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}+12 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {3 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 i b^3 \sqrt {e} m n^3 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {6 i b^3 \sqrt {e} m n^3 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {6 b^3 \sqrt {-e} m n^3 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 b^3 \sqrt {-e} m n^3 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {6 b^2 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {6 b^3 \sqrt {-e} m n^3 \text {Li}_4\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {6 b^3 \sqrt {-e} m n^3 \text {Li}_4\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.00, antiderivative size = 977, 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 = {2333, 2332, 2418, 6, 327, 211, 2393, 2361, 12, 4940, 2438, 2395, 2367, 2354, 2421, 6724, 2430} \begin {gather*} 36 m n^3 x b^3-36 m n^2 x \log \left (c x^n\right ) b^3+\frac {12 \sqrt {e} m n^2 \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right ) b^3}{\sqrt {f}}-6 n^3 x \log \left (d \left (f x^2+e\right )^m\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) b^3-\frac {6 i \sqrt {e} m n^3 \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{\sqrt {f}}+\frac {6 i \sqrt {e} m n^3 \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{\sqrt {f}}+\frac {6 \sqrt {-e} m n^3 \text {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{\sqrt {f}}-\frac {6 \sqrt {-e} m n^3 \text {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{\sqrt {f}}+\frac {6 \sqrt {-e} m n^3 \text {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{\sqrt {f}}-\frac {6 \sqrt {-e} m n^3 \text {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{\sqrt {f}}-24 a m n^2 x b^2-12 m n^2 (a-b n) x b^2+\frac {12 \sqrt {e} m n^2 (a-b n) \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) b^2}{\sqrt {f}}+6 a n^2 x \log \left (d \left (f x^2+e\right )^m\right ) b^2-\frac {6 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{\sqrt {f}}+\frac {6 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{\sqrt {f}}-\frac {6 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{\sqrt {f}}+\frac {6 \sqrt {-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^2}{\sqrt {f}}+12 m n x \left (a+b \log \left (c x^n\right )\right )^2 b+\frac {3 \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{\sqrt {f}}-\frac {3 \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) b}{\sqrt {f}}-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) b+\frac {3 \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{\sqrt {f}}-\frac {3 \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^3-\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-24*a*b^2*m*n^2*x + 36*b^3*m*n^3*x - 12*b^2*m*n^2*(a - b*n)*x + (12*b^2*Sqrt[e]*m*n^2*(a - b*n)*ArcTan[(Sqrt[f
]*x)/Sqrt[e]])/Sqrt[f] - 36*b^3*m*n^2*x*Log[c*x^n] + (12*b^3*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x
^n])/Sqrt[f] + 12*b*m*n*x*(a + b*Log[c*x^n])^2 - 2*m*x*(a + b*Log[c*x^n])^3 + (3*b*Sqrt[-e]*m*n*(a + b*Log[c*x
^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - (Sqrt[-e]*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]]
)/Sqrt[f] - (3*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (Sqrt[-e]*m*(a + b
*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + 6*a*b^2*n^2*x*Log[d*(e + f*x^2)^m] - 6*b^3*n^3*x*Log[d
*(e + f*x^2)^m] + 6*b^3*n^2*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)
^m] + x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m] - (6*b^2*Sqrt[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqr
t[f]*x)/Sqrt[-e])])/Sqrt[f] + (3*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt
[f] + (6*b^2*Sqrt[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - (3*b*Sqrt[-e]*m*n*(
a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - ((6*I)*b^3*Sqrt[e]*m*n^3*PolyLog[2, ((-I)*Sqrt
[f]*x)/Sqrt[e]])/Sqrt[f] + ((6*I)*b^3*Sqrt[e]*m*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] + (6*b^3*Sqrt[-
e]*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (6*b^2*Sqrt[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(
(Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (6*b^3*Sqrt[-e]*m*n^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (6*b^2*Sqr
t[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (6*b^3*Sqrt[-e]*m*n^3*PolyLog[4, -(
(Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (6*b^3*Sqrt[-e]*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/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 211

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

Rule 327

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 2332

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

Rule 2333

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 2354

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 2361

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 2367

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 2393

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 2395

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 2418

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2438

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

Rule 4940

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 6724

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2302\) vs. \(2(977)=1954\).
time = 0.44, size = 2302, normalized size = 2.36 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*Sqrt[f]*m*x + 12*a^2*b*Sqrt[f]*m*n*x - 36*a*b^2*Sqrt[f]*m*n^2*x + 48*b^3*Sqrt[f]*m*n^3*x + 2*a^3*Sqrt[
e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a^2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*Sqrt[e]*m*n^2*Ar
cTan[(Sqrt[f]*x)/Sqrt[e]] - 12*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a^2*b*Sqrt[e]*m*n*ArcTan[(Sqr
t[f]*x)/Sqrt[e]]*Log[x] + 12*a*b^2*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 12*b^3*Sqrt[e]*m*n^3*Arc
Tan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 6*a*b^2*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 6*b^3*Sqrt[e]*m
*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 2*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 - 6*a^2*b
*Sqrt[f]*m*x*Log[c*x^n] + 24*a*b^2*Sqrt[f]*m*n*x*Log[c*x^n] - 36*b^3*Sqrt[f]*m*n^2*x*Log[c*x^n] + 6*a^2*b*Sqrt
[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*a*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 1
2*b^3*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*a*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*
Log[x]*Log[c*x^n] + 12*b^3*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 6*b^3*Sqrt[e]*m*n^2*A
rcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] - 6*a*b^2*Sqrt[f]*m*x*Log[c*x^n]^2 + 12*b^3*Sqrt[f]*m*n*x*Log[c
*x^n]^2 + 6*a*b^2*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - 6*b^3*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sq
rt[e]]*Log[c*x^n]^2 - 6*b^3*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n]^2 - 2*b^3*Sqrt[f]*m*x*Lo
g[c*x^n]^3 + 2*b^3*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^3 + (3*I)*a^2*b*Sqrt[e]*m*n*Log[x]*Log[1 -
(I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*a*b^2*Sqrt[e]*m*n^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[e]*
m*n^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*a*b^2*Sqrt[e]*m*n^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]
] + (3*I)*b^3*Sqrt[e]*m*n^3*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + I*b^3*Sqrt[e]*m*n^3*Log[x]^3*Log[1 - (I*
Sqrt[f]*x)/Sqrt[e]] + (6*I)*a*b^2*Sqrt[e]*m*n*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3*Sqr
t[e]*m*n^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*b^3*Sqrt[e]*m*n^2*Log[x]^2*Log[c*x^n]*Log[
1 - (I*Sqrt[f]*x)/Sqrt[e]] + (3*I)*b^3*Sqrt[e]*m*n*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*
a^2*b*Sqrt[e]*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*a*b^2*Sqrt[e]*m*n^2*Log[x]*Log[1 + (I*Sqrt[f]*
x)/Sqrt[e]] - (6*I)*b^3*Sqrt[e]*m*n^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (3*I)*a*b^2*Sqrt[e]*m*n^2*Log[x]
^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*b^3*Sqrt[e]*m*n^3*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - I*b^3*Sq
rt[e]*m*n^3*Log[x]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*a*b^2*Sqrt[e]*m*n*Log[x]*Log[c*x^n]*Log[1 + (I*Sqr
t[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[e]*m*n^2*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (3*I)*b^3*Sqrt[e
]*m*n^2*Log[x]^2*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*b^3*Sqrt[e]*m*n*Log[x]*Log[c*x^n]^2*Log[1 +
(I*Sqrt[f]*x)/Sqrt[e]] + a^3*Sqrt[f]*x*Log[d*(e + f*x^2)^m] - 3*a^2*b*Sqrt[f]*n*x*Log[d*(e + f*x^2)^m] + 6*a*
b^2*Sqrt[f]*n^2*x*Log[d*(e + f*x^2)^m] - 6*b^3*Sqrt[f]*n^3*x*Log[d*(e + f*x^2)^m] + 3*a^2*b*Sqrt[f]*x*Log[c*x^
n]*Log[d*(e + f*x^2)^m] - 6*a*b^2*Sqrt[f]*n*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 6*b^3*Sqrt[f]*n^2*x*Log[c*x^n]
*Log[d*(e + f*x^2)^m] + 3*a*b^2*Sqrt[f]*x*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - 3*b^3*Sqrt[f]*n*x*Log[c*x^n]^2*L
og[d*(e + f*x^2)^m] + b^3*Sqrt[f]*x*Log[c*x^n]^3*Log[d*(e + f*x^2)^m] - (3*I)*b*Sqrt[e]*m*n*(a^2 - 2*a*b*n + 2
*b^2*n^2 + 2*b*(a - b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (3*I)*b*Sqrt[e]
*m*n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b*(a - b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e
]] + (6*I)*a*b^2*Sqrt[e]*m*n^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3*Sqrt[e]*m*n^3*PolyLog[3, ((-I)
*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[e]*m*n^2*Log[c*x^n]*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (6*I)*a*b^2*S
qrt[e]*m*n^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[e]*m*n^3*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - (
6*I)*b^3*Sqrt[e]*m*n^2*Log[c*x^n]*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^3*Sqrt[e]*m*n^3*PolyLog[4, ((-I)
*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[e]*m*n^3*PolyLog[4, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f]

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*m*x*log(x^n)^3 - 3*((m*n - m*log(c))*b^3 - a*b^2*m)*x*log(x^n)^2 - 3*(2*(m*n - m*log(c))*a*b^2 - (2*m*n^2
- 2*m*n*log(c) + m*log(c)^2)*b^3 - a^2*b*m)*x*log(x^n) - (3*(m*n - m*log(c))*a^2*b - 3*(2*m*n^2 - 2*m*n*log(c
) + m*log(c)^2)*a*b^2 + (6*m*n^3 - 6*m*n^2*log(c) + 3*m*n*log(c)^2 - m*log(c)^3)*b^3 - a^3*m)*x)*log(f*x^2 + e
) + integrate(-(((2*f*m - f*log(d))*b^3*x^2 - b^3*e*log(d))*log(x^n)^3 + ((2*f*m - f*log(d))*a^3 - 3*(2*f*m*n
- (2*f*m - f*log(d))*log(c))*a^2*b + 3*(4*f*m*n^2 - 4*f*m*n*log(c) + (2*f*m - f*log(d))*log(c)^2)*a*b^2 - (12*
f*m*n^3 - 12*f*m*n^2*log(c) + 6*f*m*n*log(c)^2 - (2*f*m - f*log(d))*log(c)^3)*b^3)*x^2 + 3*(((2*f*m - f*log(d)
)*a*b^2 - (2*f*m*n - (2*f*m - f*log(d))*log(c))*b^3)*x^2 - (b^3*log(c)*log(d) + a*b^2*log(d))*e)*log(x^n)^2 -
(b^3*log(c)^3*log(d) + 3*a*b^2*log(c)^2*log(d) + 3*a^2*b*log(c)*log(d) + a^3*log(d))*e + 3*(((2*f*m - f*log(d)
)*a^2*b - 2*(2*f*m*n - (2*f*m - f*log(d))*log(c))*a*b^2 + (4*f*m*n^2 - 4*f*m*n*log(c) + (2*f*m - f*log(d))*log
(c)^2)*b^3)*x^2 - (b^3*log(c)^2*log(d) + 2*a*b^2*log(c)*log(d) + a^2*b*log(d))*e)*log(x^n))/(f*x^2 + e), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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