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

Optimal. Leaf size=519 \[ \frac{2 b n \text{PolyLog}\left (2,-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{-d} \sqrt{f}}-\frac{2 b n \text{PolyLog}\left (2,\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{-d} \sqrt{f}}+\frac{2 i b^2 n^2 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{2 i b^2 n^2 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{\log \left (1-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d} \sqrt{f}}+\frac{\log \left (\sqrt{-d} \sqrt{f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d} \sqrt{f}}+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d f x^2+1\right )-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+4 a b n x+4 b n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d f x^2+1\right )-\frac{4 b^2 n \log \left (c x^n\right ) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+8 b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d f x^2+1\right )-8 b^2 n^2 x \]

[Out]

4*a*b*n*x - 8*b^2*n^2*x + 4*b*n*(a - b*n)*x - (4*b*n*(a - b*n)*ArcTan[Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) +
8*b^2*n*x*Log[c*x^n] - (4*b^2*n*ArcTan[Sqrt[d]*Sqrt[f]*x]*Log[c*x^n])/(Sqrt[d]*Sqrt[f]) - 2*x*(a + b*Log[c*x^n
])^2 - ((a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + ((a + b*Log[c*x^n])^2*Log[1 + S
qrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - 2*a*b*n*x*Log[1 + d*f*x^2] + 2*b^2*n^2*x*Log[1 + d*f*x^2] - 2*b^2*n*x
*Log[c*x^n]*Log[1 + d*f*x^2] + x*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2,
-(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-
d]*Sqrt[f]) + ((2*I)*b^2*n^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - ((2*I)*b^2*n^2*PolyLog[2,
I*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - (2*b^2*n^2*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) +
(2*b^2*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f])

________________________________________________________________________________________

Rubi [A]  time = 0.80112, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {2296, 2295, 2371, 6, 321, 203, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589} \[ \frac{2 b n \text{PolyLog}\left (2,-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{-d} \sqrt{f}}-\frac{2 b n \text{PolyLog}\left (2,\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{-d} \sqrt{f}}+\frac{2 i b^2 n^2 \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{2 i b^2 n^2 \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-\frac{\log \left (1-\sqrt{-d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d} \sqrt{f}}+\frac{\log \left (\sqrt{-d} \sqrt{f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d} \sqrt{f}}+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d f x^2+1\right )-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+4 a b n x+4 b n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d f x^2+1\right )-\frac{4 b^2 n \log \left (c x^n\right ) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+8 b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d f x^2+1\right )-8 b^2 n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*b*n*x - 8*b^2*n^2*x + 4*b*n*(a - b*n)*x - (4*b*n*(a - b*n)*ArcTan[Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) +
8*b^2*n*x*Log[c*x^n] - (4*b^2*n*ArcTan[Sqrt[d]*Sqrt[f]*x]*Log[c*x^n])/(Sqrt[d]*Sqrt[f]) - 2*x*(a + b*Log[c*x^n
])^2 - ((a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + ((a + b*Log[c*x^n])^2*Log[1 + S
qrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - 2*a*b*n*x*Log[1 + d*f*x^2] + 2*b^2*n^2*x*Log[1 + d*f*x^2] - 2*b^2*n*x
*Log[c*x^n]*Log[1 + d*f*x^2] + x*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2,
-(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-
d]*Sqrt[f]) + ((2*I)*b^2*n^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - ((2*I)*b^2*n^2*PolyLog[2,
I*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - (2*b^2*n^2*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*Sqrt[f]) +
(2*b^2*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f])

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 2295

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

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 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 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 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 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 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 12

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

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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac{1}{d}+f x^2\right )\right ) \, dx &=-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-(2 f) \int \left (-\frac{2 a b d n x^2}{1+d f x^2}+\frac{2 b^2 d n^2 x^2}{1+d f x^2}-\frac{2 b^2 d n x^2 \log \left (c x^n\right )}{1+d f x^2}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}\right ) \, dx\\ &=-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-(2 f) \int \left (\frac{d \left (-2 a b n+2 b^2 n^2\right ) x^2}{1+d f x^2}-\frac{2 b^2 d n x^2 \log \left (c x^n\right )}{1+d f x^2}+\frac{d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2}\right ) \, dx\\ &=-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \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 )^2}{1+d f x^2} \, dx+\left (4 b^2 d f n\right ) \int \frac{x^2 \log \left (c x^n\right )}{1+d f x^2} \, dx+(4 b d f n (a-b n)) \int \frac{x^2}{1+d f x^2} \, dx\\ &=4 b n (a-b n) x-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-(2 d f) \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 (4 b^2 d f n\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-(4 b n (a-b n)) \int \frac{1}{1+d f x^2} \, dx\\ &=4 b n (a-b n) x-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-2 \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx+2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+d f x^2} \, dx+\left (4 b^2 n\right ) \int \log \left (c x^n\right ) \, dx-\left (4 b^2 n\right ) \int \frac{\log \left (c x^n\right )}{1+d f x^2} \, dx\\ &=-4 b^2 n^2 x+4 b n (a-b n) x-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+4 b^2 n x \log \left (c x^n\right )-\frac{4 b^2 n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+2 \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+(4 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (4 b^2 n^2\right ) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f} x} \, dx\\ &=4 a b n x-4 b^2 n^2 x+4 b n (a-b n) x-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+4 b^2 n x \log \left (c x^n\right )-\frac{4 b^2 n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\left (4 b^2 n\right ) \int \log \left (c x^n\right ) \, dx+\frac{\left (4 b^2 n^2\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 )^2}{1-\sqrt{-d} \sqrt{f} x} \, dx+\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+\sqrt{-d} \sqrt{f} x} \, dx\\ &=4 a b n x-8 b^2 n^2 x+4 b n (a-b n) x-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+8 b^2 n x \log \left (c x^n\right )-\frac{4 b^2 n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )^2-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{(2 b n) \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{(2 b n) \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 (2 i b^2 n^2\right ) \int \frac{\log \left (1-i \sqrt{d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{d} \sqrt{f}}-\frac{\left (2 i b^2 n^2\right ) \int \frac{\log \left (1+i \sqrt{d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{d} \sqrt{f}}\\ &=4 a b n x-8 b^2 n^2 x+4 b n (a-b n) x-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+8 b^2 n x \log \left (c x^n\right )-\frac{4 b^2 n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )^2-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{2 b n \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{2 b n \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{2 i b^2 n^2 \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{2 i b^2 n^2 \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}+\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (\sqrt{-d} \sqrt{f} x\right )}{x} \, dx}{\sqrt{-d} \sqrt{f}}\\ &=4 a b n x-8 b^2 n^2 x+4 b n (a-b n) x-\frac{4 b n (a-b n) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}+8 b^2 n x \log \left (c x^n\right )-\frac{4 b^2 n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \log \left (c x^n\right )}{\sqrt{d} \sqrt{f}}-2 x \left (a+b \log \left (c x^n\right )\right )^2-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}-2 a b n x \log \left (1+d f x^2\right )+2 b^2 n^2 x \log \left (1+d f x^2\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac{2 b n \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{2 b n \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{2 i b^2 n^2 \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{2 i b^2 n^2 \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{\sqrt{d} \sqrt{f}}-\frac{2 b^2 n^2 \text{Li}_3\left (-\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}+\frac{2 b^2 n^2 \text{Li}_3\left (\sqrt{-d} \sqrt{f} x\right )}{\sqrt{-d} \sqrt{f}}\\ \end{align*}

Mathematica [A]  time = 0.320161, size = 544, normalized size = 1.05 \[ \frac{2 b n \left (-i \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )+i \left (\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )\right )-2 \sqrt{d} \sqrt{f} x (\log (x)-1)\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)-b n\right )-2 b^2 n^2 \left (\frac{1}{2} i \left (-2 \text{PolyLog}\left (3,-i \sqrt{d} \sqrt{f} x\right )+2 \log (x) \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\log ^2(x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )-\frac{1}{2} i \left (-2 \text{PolyLog}\left (3,i \sqrt{d} \sqrt{f} x\right )+2 \log (x) \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log ^2(x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )\right )+\sqrt{d} \sqrt{f} x \left (\log ^2(x)-2 \log (x)+2\right )\right )+\sqrt{d} \sqrt{f} x \log \left (d f x^2+1\right ) \left (a^2+2 b (a-b n) \log \left (c x^n\right )-2 a b n+b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )-2 \sqrt{d} \sqrt{f} x \left (a^2+2 a b \left (\log \left (c x^n\right )-n \log (x)\right )-2 a b n+b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+2 b^2 n^2\right )+2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a^2+2 a b \left (\log \left (c x^n\right )-n \log (x)\right )-2 a b n+b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+2 b^2 n^2\right )}{\sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d]*Sqrt[f]*x*(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*ArcTan[Sqrt[d]*Sqrt[f]*x]*(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) + Sqrt[d]*Sqrt
[f]*x*(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)*Log[1 + d*f*x^2] + 2*b*n*(a -
b*n - b*n*Log[x] + b*Log[c*x^n])*(-2*Sqrt[d]*Sqrt[f]*x*(-1 + Log[x]) - I*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x]
+ PolyLog[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])) - 2*b^2*n^2*(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])))/(Sqrt[
d]*Sqrt[f])

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Maple [F]  time = 0.156, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^2)),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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