3.105 \(\int (a+b \log (c x^n))^2 \log (d (e+f x^2)^m) \, dx\)

Optimal. Leaf size=546 \[ \frac{2 b \sqrt{-e} m n \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{f}}-\frac{2 b \sqrt{-e} m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{f}}+\frac{2 i b^2 \sqrt{e} m n^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-\frac{2 i b^2 \sqrt{e} m n^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-\frac{2 b^2 \sqrt{-e} m n^2 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{2 b^2 \sqrt{-e} m n^2 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{\sqrt{-e} m \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{f}}+\frac{\sqrt{-e} m \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac{4 b \sqrt{e} m n (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+4 a b m n x+4 b m n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{4 b^2 \sqrt{e} m n \log \left (c x^n\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+8 b^2 m n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-8 b^2 m n^2 x \]

[Out]

4*a*b*m*n*x - 8*b^2*m*n^2*x + 4*b*m*n*(a - b*n)*x - (4*b*Sqrt[e]*m*n*(a - b*n)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sq
rt[f] + 8*b^2*m*n*x*Log[c*x^n] - (4*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n])/Sqrt[f] - 2*m*x*(a
+ b*Log[c*x^n])^2 - (Sqrt[-e]*m*(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])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - 2*a*b*n*x*Log[d*(e + f*x^2)^m] + 2*b^2*n^2*x*Log[d*
(e + f*x^2)^m] - 2*b^2*n*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] + (2*
b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (2*b*Sqrt[-e]*m*n*(a + b*Log[
c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + ((2*I)*b^2*Sqrt[e]*m*n^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[
e]])/Sqrt[f] - ((2*I)*b^2*Sqrt[e]*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] - (2*b^2*Sqrt[-e]*m*n^2*Pol
yLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] + (2*b^2*Sqrt[-e]*m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f]

________________________________________________________________________________________

Rubi [A]  time = 0.806212, antiderivative size = 546, 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, 205, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589} \[ \frac{2 b \sqrt{-e} m n \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{f}}-\frac{2 b \sqrt{-e} m n \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{f}}+\frac{2 i b^2 \sqrt{e} m n^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-\frac{2 i b^2 \sqrt{e} m n^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-\frac{2 b^2 \sqrt{-e} m n^2 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{2 b^2 \sqrt{-e} m n^2 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{\sqrt{-e} m \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{f}}+\frac{\sqrt{-e} m \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac{4 b \sqrt{e} m n (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+4 a b m n x+4 b m n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac{4 b^2 \sqrt{e} m n \log \left (c x^n\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+8 b^2 m n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-8 b^2 m n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*b*m*n*x - 8*b^2*m*n^2*x + 4*b*m*n*(a - b*n)*x - (4*b*Sqrt[e]*m*n*(a - b*n)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sq
rt[f] + 8*b^2*m*n*x*Log[c*x^n] - (4*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n])/Sqrt[f] - 2*m*x*(a
+ b*Log[c*x^n])^2 - (Sqrt[-e]*m*(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])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - 2*a*b*n*x*Log[d*(e + f*x^2)^m] + 2*b^2*n^2*x*Log[d*
(e + f*x^2)^m] - 2*b^2*n*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] + (2*
b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (2*b*Sqrt[-e]*m*n*(a + b*Log[
c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + ((2*I)*b^2*Sqrt[e]*m*n^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[
e]])/Sqrt[f] - ((2*I)*b^2*Sqrt[e]*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] - (2*b^2*Sqrt[-e]*m*n^2*Pol
yLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] + (2*b^2*Sqrt[-e]*m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/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 205

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

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

Mathematica [A]  time = 0.313669, size = 993, normalized size = 1.82 \[ \frac{-2 \sqrt{f} m x a^2+2 \sqrt{e} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) a^2+\sqrt{f} x \log \left (d \left (f x^2+e\right )^m\right ) a^2+8 b \sqrt{f} m n x a-4 b \sqrt{e} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) a-4 b \sqrt{e} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x) a-4 b \sqrt{f} m x \log \left (c x^n\right ) a+4 b \sqrt{e} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right ) a+2 i b \sqrt{e} m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) a-2 i b \sqrt{e} m n \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) a-2 b \sqrt{f} n x \log \left (d \left (f x^2+e\right )^m\right ) a+2 b \sqrt{f} x \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) a+2 b^2 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log ^2(x)-2 b^2 \sqrt{f} m x \log ^2\left (c x^n\right )+2 b^2 \sqrt{e} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log ^2\left (c x^n\right )-12 b^2 \sqrt{f} m n^2 x+4 b^2 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )+4 b^2 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x)+8 b^2 \sqrt{f} m n x \log \left (c x^n\right )-4 b^2 \sqrt{e} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )-4 b^2 \sqrt{e} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log (x) \log \left (c x^n\right )-i b^2 \sqrt{e} m n^2 \log ^2(x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 i b^2 \sqrt{e} m n^2 \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 i b^2 \sqrt{e} m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+i b^2 \sqrt{e} m n^2 \log ^2(x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )+2 i b^2 \sqrt{e} m n^2 \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )-2 i b^2 \sqrt{e} m n \log (x) \log \left (c x^n\right ) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )+b^2 \sqrt{f} x \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+2 b^2 \sqrt{f} n^2 x \log \left (d \left (f x^2+e\right )^m\right )-2 b^2 \sqrt{f} n x \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )-2 i b \sqrt{e} m n \left (a-b n+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 i b \sqrt{e} m n \left (a-b n+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 i b^2 \sqrt{e} m n^2 \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 i b^2 \sqrt{e} m n^2 \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*Sqrt[f]*m*x + 8*a*b*Sqrt[f]*m*n*x - 12*b^2*Sqrt[f]*m*n^2*x + 2*a^2*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e
]] - 4*a*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 4*b^2*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*a*b*S
qrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 4*b^2*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 2*b^2
*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 4*a*b*Sqrt[f]*m*x*Log[c*x^n] + 8*b^2*Sqrt[f]*m*n*x*Log[c
*x^n] + 4*a*b*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 4*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]
*Log[c*x^n] - 4*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 2*b^2*Sqrt[f]*m*x*Log[c*x^n]^2
+ 2*b^2*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + (2*I)*a*b*Sqrt[e]*m*n*Log[x]*Log[1 - (I*Sqrt[f]*
x)/Sqrt[e]] - (2*I)*b^2*Sqrt[e]*m*n^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - I*b^2*Sqrt[e]*m*n^2*Log[x]^2*Log
[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (2*I)*b^2*Sqrt[e]*m*n*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (2*I)*a
*b*Sqrt[e]*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (2*I)*b^2*Sqrt[e]*m*n^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sq
rt[e]] + I*b^2*Sqrt[e]*m*n^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (2*I)*b^2*Sqrt[e]*m*n*Log[x]*Log[c*x^n]
*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + a^2*Sqrt[f]*x*Log[d*(e + f*x^2)^m] - 2*a*b*Sqrt[f]*n*x*Log[d*(e + f*x^2)^m]
+ 2*b^2*Sqrt[f]*n^2*x*Log[d*(e + f*x^2)^m] + 2*a*b*Sqrt[f]*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 2*b^2*Sqrt[f]*n
*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] + b^2*Sqrt[f]*x*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - (2*I)*b*Sqrt[e]*m*n*(a
- b*n + b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (2*I)*b*Sqrt[e]*m*n*(a - b*n + b*Log[c*x^n])*Poly
Log[2, (I*Sqrt[f]*x)/Sqrt[e]] + (2*I)*b^2*Sqrt[e]*m*n^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (2*I)*b^2*Sqrt[
e]*m*n^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f]

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Maple [F]  time = 5.022, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \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*(f*x^2+e)^m),x)

[Out]

int((a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m),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*(f*x^2+e)^m),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 ({\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\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*(f*x^2+e)^m),x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*log((f*x^2 + e)^m*d), 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*(f*x**2+e)**m),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} + e\right )}^{m} 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*(f*x^2+e)^m),x, algorithm="giac")

[Out]

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