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

Optimal. Leaf size=546 \[ x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b \sqrt {-e} m n \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-\frac {2 b \sqrt {-e} m n \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-\frac {\sqrt {-e} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {f}}+\frac {\sqrt {-e} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac {4 b \sqrt {e} m n (a-b n) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+4 a b m n x+4 b m n x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {4 b^2 \sqrt {e} m n \log \left (c x^n\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+8 b^2 m n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 i b^2 \sqrt {e} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 b^2 \sqrt {-e} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 b^2 \sqrt {-e} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-8 b^2 m n^2 x \]

[Out]

4*a*b*m*n*x-8*b^2*m*n^2*x+4*b*m*n*(-b*n+a)*x+8*b^2*m*n*x*ln(c*x^n)-2*m*x*(a+b*ln(c*x^n))^2-2*a*b*n*x*ln(d*(f*x
^2+e)^m)+2*b^2*n^2*x*ln(d*(f*x^2+e)^m)-2*b^2*n*x*ln(c*x^n)*ln(d*(f*x^2+e)^m)+x*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e
)^m)-m*(a+b*ln(c*x^n))^2*ln(1-x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)+m*(a+b*ln(c*x^n))^2*ln(1+x*f^(1/2)/(-e)
^(1/2))*(-e)^(1/2)/f^(1/2)+2*b*m*n*(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)-2*b*m*n
*(a+b*ln(c*x^n))*polylog(2,x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)-2*b^2*m*n^2*polylog(3,-x*f^(1/2)/(-e)^(1/2
))*(-e)^(1/2)/f^(1/2)+2*b^2*m*n^2*polylog(3,x*f^(1/2)/(-e)^(1/2))*(-e)^(1/2)/f^(1/2)-4*b*m*n*(-b*n+a)*arctan(x
*f^(1/2)/e^(1/2))*e^(1/2)/f^(1/2)-4*b^2*m*n*arctan(x*f^(1/2)/e^(1/2))*ln(c*x^n)*e^(1/2)/f^(1/2)+2*I*b^2*m*n^2*
polylog(2,-I*x*f^(1/2)/e^(1/2))*e^(1/2)/f^(1/2)-2*I*b^2*m*n^2*polylog(2,I*x*f^(1/2)/e^(1/2))*e^(1/2)/f^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.81, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, 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 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 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 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 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 )^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*}

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Mathematica [A]  time = 0.34, 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 {Li}_2\left (-\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 {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {e} m n^2 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {e} m n^2 \text {Li}_3\left (\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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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \]

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

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)

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maple [F]  time = 130.68, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (b^{2} m x \log \left (x^{n}\right )^{2} - 2 \, {\left ({\left (m n - m \log \relax (c)\right )} b^{2} - a b m\right )} x \log \left (x^{n}\right ) - {\left (2 \, {\left (m n - m \log \relax (c)\right )} a b - {\left (2 \, m n^{2} - 2 \, m n \log \relax (c) + m \log \relax (c)^{2}\right )} b^{2} - a^{2} m\right )} x\right )} \log \left (f x^{2} + e\right ) + \int \frac {b^{2} e \log \relax (c)^{2} \log \relax (d) + 2 \, a b e \log \relax (c) \log \relax (d) + a^{2} e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a^{2} - 2 \, {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (4 \, f m n^{2} - 4 \, f m n \log \relax (c) + {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{2} - {\left ({\left (2 \, f m - f \log \relax (d)\right )} b^{2} x^{2} - b^{2} e \log \relax (d)\right )} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} e \log \relax (c) \log \relax (d) + a b e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a b - {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{2}\right )} \log \left (x^{n}\right )}{f x^{2} + e}\,{d x} \]

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]

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

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

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))^2,x)

[Out]

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

[Out]

Timed out

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