3.2.5 \(\int (a+b \log (c x^n))^2 \log (d (e+f x^2)^m) \, dx\) [105]

3.2.5.1 Optimal result

Integrand size = 25, antiderivative size = 546 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=4 a b m n x-8 b^2 m n^2 x+4 b m n (a-b n) x-\frac {4 b \sqrt {e} m n (a-b n) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+8 b^2 m n x \log \left (c x^n\right )-\frac {4 b^2 \sqrt {e} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^2-\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {\sqrt {-e} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}-\frac {2 b \sqrt {-e} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 i b^2 \sqrt {e} m n^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 i b^2 \sqrt {e} m n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-\frac {2 b^2 \sqrt {-e} m n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}}+\frac {2 b^2 \sqrt {-e} m n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {f}} \]

output

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)
 

3.2.5.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 993, normalized size of antiderivative = 1.82 \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\frac {-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 \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-4 a b \sqrt {e} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+4 b^2 \sqrt {e} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-4 a b \sqrt {e} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+4 b^2 \sqrt {e} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+2 b^2 \sqrt {e} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)-4 a b \sqrt {f} m x \log \left (c x^n\right )+8 b^2 \sqrt {f} m n x \log \left (c x^n\right )+4 a b \sqrt {e} m \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-4 b^2 \sqrt {e} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-4 b^2 \sqrt {e} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) \log \left (c x^n\right )-2 b^2 \sqrt {f} m x \log ^2\left (c x^n\right )+2 b^2 \sqrt {e} m \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )+2 i a b \sqrt {e} m n \log (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 )-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 \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i a b \sqrt {e} m n \log (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 )+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 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+a^2 \sqrt {f} x \log \left (d \left (e+f x^2\right )^m\right )-2 a b \sqrt {f} n x \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 \sqrt {f} n^2 x \log \left (d \left (e+f x^2\right )^m\right )+2 a b \sqrt {f} x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 \sqrt {f} n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+b^2 \sqrt {f} x \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 i b \sqrt {e} m n \left (a-b n+b \log \left (c x^n\right )\right ) \operatorname {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 ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {e} m n^2 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {e} m n^2 \operatorname {PolyLog}\left (3,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}} \]

input

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

output

(-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*Sqr 
t[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[(Sq 
rt[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*S 
qrt[f]*x)/Sqrt[e]] - (2*I)*a*b*Sqrt[e]*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sq 
rt[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*Sqr 
t[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...
 

3.2.5.3 Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2818, 6, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2818

\(\displaystyle -2 f m \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2 x^2}{f x^2+e}-\frac {2 b^2 n \log \left (c x^n\right ) x^2}{f x^2+e}+\frac {2 b^2 n^2 x^2}{f x^2+e}-\frac {2 a b n x^2}{f x^2+e}\right )dx+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )\)

\(\Big \downarrow \) 6

\(\displaystyle -2 f m \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2 x^2}{f x^2+e}-\frac {2 b^2 n \log \left (c x^n\right ) x^2}{f x^2+e}+\frac {\left (2 b^2 n^2-2 a b n\right ) x^2}{f x^2+e}\right )dx+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 f m \left (\frac {2 b \sqrt {e} n (a-b n) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{f^{3/2}}-\frac {b \sqrt {-e} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^{3/2}}+\frac {b \sqrt {-e} n \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^{3/2}}+\frac {\sqrt {-e} \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^{3/2}}-\frac {\sqrt {-e} \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {2 a b n x}{f}-\frac {2 b n x (a-b n)}{f}+\frac {2 b^2 \sqrt {e} n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )}{f^{3/2}}-\frac {4 b^2 n x \log \left (c x^n\right )}{f}-\frac {i b^2 \sqrt {e} n^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{f^{3/2}}+\frac {i b^2 \sqrt {e} n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{f^{3/2}}+\frac {b^2 \sqrt {-e} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}-\frac {b^2 \sqrt {-e} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {4 b^2 n^2 x}{f}\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-2 a b n x \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )\)

input

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

output

-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*f*m*((-2*a*b*n*x)/f + (4*b^2*n^2*x)/f - (2*b*n*(a - b*n)*x)/f 
 + (2*b*Sqrt[e]*n*(a - b*n)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/f^(3/2) - (4*b^2* 
n*x*Log[c*x^n])/f + (2*b^2*Sqrt[e]*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n 
])/f^(3/2) + (x*(a + b*Log[c*x^n])^2)/f + (Sqrt[-e]*(a + b*Log[c*x^n])^2*L 
og[1 - (Sqrt[f]*x)/Sqrt[-e]])/(2*f^(3/2)) - (Sqrt[-e]*(a + b*Log[c*x^n])^2 
*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(2*f^(3/2)) - (b*Sqrt[-e]*n*(a + b*Log[c*x 
^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (b*Sqrt[-e]*n*(a + b*L 
og[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) - (I*b^2*Sqrt[e]*n^2* 
PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (I*b^2*Sqrt[e]*n^2*PolyLog 
[2, (I*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (b^2*Sqrt[-e]*n^2*PolyLog[3, -((Sqrt 
[f]*x)/Sqrt[-e])])/f^(3/2) - (b^2*Sqrt[-e]*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt 
[-e]])/f^(3/2))
 

3.2.5.3.1 Defintions of rubi rules used

rule 6

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2818

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]}, 
Simp[Log[d*(e + f*x^m)^r]   u, x] - Simp[f*m*r   Int[x^(m - 1)/(e + f*x^m) 
  u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && Inte 
gerQ[m]
 

3.2.5.4 Maple [F]

\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )d x\]

input

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

output

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

3.2.5.5 Fricas [F]

\[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]

input

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

output

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)
 

3.2.5.6 Sympy [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.2.5.7 Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.2.5.8 Giac [F]

\[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]

input

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

output

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

3.2.5.9 Mupad [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int \ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]

input

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

output

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