\(\int \frac {(a+b \log (c x^n))^3 \log (d (\frac {1}{d}+f x^2))}{x^2} \, dx\) [51]

Optimal result

Integrand size = 28, antiderivative size = 747 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=12 b^3 \sqrt {d} \sqrt {f} n^3 \arctan \left (\sqrt {d} \sqrt {f} x\right )+12 b^2 \sqrt {d} \sqrt {f} n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+6 b \sqrt {d} \sqrt {f} n \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 \sqrt {d} \sqrt {f} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {6 b^3 n^3 \log \left (1+d f x^2\right )}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{x}-6 b^3 \sqrt {-d} \sqrt {f} n^3 \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )-3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )+3 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )+6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )-6 b^3 \sqrt {-d} \sqrt {f} n^3 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )-6 b^2 \sqrt {-d} \sqrt {f} n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )-6 b^3 \sqrt {-d} \sqrt {f} n^3 \operatorname {PolyLog}\left (4,-\sqrt {-d} \sqrt {f} x\right )+6 b^3 \sqrt {-d} \sqrt {f} n^3 \operatorname {PolyLog}\left (4,\sqrt {-d} \sqrt {f} x\right ) \] Output:

12*b^3*d^(1/2)*f^(1/2)*n^3*arctan(d^(1/2)*f^(1/2)*x)+12*b^2*d^(1/2)*f^(1/2 
)*n^2*arctan(d^(1/2)*f^(1/2)*x)*(a+b*ln(c*x^n))+6*b*d^(1/2)*f^(1/2)*n*arct 
an(d^(1/2)*f^(1/2)*x)*(a+b*ln(c*x^n))^2+2*d^(1/2)*f^(1/2)*arctan(d^(1/2)*f 
^(1/2)*x)*(a+b*ln(c*x^n))^3-6*b^3*n^3*ln(d*f*x^2+1)/x-6*b^2*n^2*(a+b*ln(c* 
x^n))*ln(d*f*x^2+1)/x-3*b*n*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)/x-(a+b*ln(c*x^ 
n))^3*ln(d*f*x^2+1)/x-6*b^3*(-d)^(1/2)*f^(1/2)*n^3*polylog(2,-(-d)^(1/2)*f 
^(1/2)*x)-6*b^2*(-d)^(1/2)*f^(1/2)*n^2*(a+b*ln(c*x^n))*polylog(2,-(-d)^(1/ 
2)*f^(1/2)*x)-3*b*(-d)^(1/2)*f^(1/2)*n*(a+b*ln(c*x^n))^2*polylog(2,-(-d)^( 
1/2)*f^(1/2)*x)+6*b^3*(-d)^(1/2)*f^(1/2)*n^3*polylog(2,(-d)^(1/2)*f^(1/2)* 
x)+6*b^2*(-d)^(1/2)*f^(1/2)*n^2*(a+b*ln(c*x^n))*polylog(2,(-d)^(1/2)*f^(1/ 
2)*x)+3*b*(-d)^(1/2)*f^(1/2)*n*(a+b*ln(c*x^n))^2*polylog(2,(-d)^(1/2)*f^(1 
/2)*x)+6*b^3*(-d)^(1/2)*f^(1/2)*n^3*polylog(3,-(-d)^(1/2)*f^(1/2)*x)+6*b^2 
*(-d)^(1/2)*f^(1/2)*n^2*(a+b*ln(c*x^n))*polylog(3,-(-d)^(1/2)*f^(1/2)*x)-6 
*b^3*(-d)^(1/2)*f^(1/2)*n^3*polylog(3,(-d)^(1/2)*f^(1/2)*x)-6*b^2*(-d)^(1/ 
2)*f^(1/2)*n^2*(a+b*ln(c*x^n))*polylog(3,(-d)^(1/2)*f^(1/2)*x)-6*b^3*(-d)^ 
(1/2)*f^(1/2)*n^3*polylog(4,-(-d)^(1/2)*f^(1/2)*x)+6*b^3*(-d)^(1/2)*f^(1/2 
)*n^3*polylog(4,(-d)^(1/2)*f^(1/2)*x)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)])/x^2,x]
 

Output:

2*Sqrt[d]*Sqrt[f]*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 
 + 6*b^3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) 
 + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x 
]) + Log[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x 
]) + Log[c*x^n])^3) - ((a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*b*(a 
^2 + 2*a*b*n + 2*b^2*n^2)*Log[c*x^n] + 3*b^2*(a + b*n)*Log[c*x^n]^2 + b^3* 
Log[c*x^n]^3)*Log[1 + d*f*x^2])/x + (3*I)*b*Sqrt[d]*Sqrt[f]*n*(a^2 + 2*a*b 
*n + 2*b^2*n^2 + 2*a*b*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + 
 Log[c*x^n]) + b^2*(-(n*Log[x]) + Log[c*x^n])^2)*(Log[x]*(Log[1 - I*Sqrt[d 
]*Sqrt[f]*x] - Log[1 + I*Sqrt[d]*Sqrt[f]*x]) - PolyLog[2, (-I)*Sqrt[d]*Sqr 
t[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) + (6*I)*b^2*Sqrt[d]*Sqrt[f]*n^2 
*(a + b*n - b*n*Log[x] + b*Log[c*x^n])*((Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f 
]*x])/2 - (Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f]*x])/2 - Log[x]*PolyLog[2, (- 
I)*Sqrt[d]*Sqrt[f]*x] + Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + PolyLog[3 
, (-I)*Sqrt[d]*Sqrt[f]*x] - PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]) + I*b^3*Sqrt[ 
d]*Sqrt[f]*n^3*(Log[x]^3*Log[1 - I*Sqrt[d]*Sqrt[f]*x] - Log[x]^3*Log[1 + I 
*Sqrt[d]*Sqrt[f]*x] - 3*Log[x]^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 3*Lo 
g[x]^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + 6*Log[x]*PolyLog[3, (-I)*Sqrt[d]* 
Sqrt[f]*x] - 6*Log[x]*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x] - 6*PolyLog[4, (-I)* 
Sqrt[d]*Sqrt[f]*x] + 6*PolyLog[4, I*Sqrt[d]*Sqrt[f]*x])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.41 (sec) , antiderivative size = 866, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 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.

\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx\)

\(\Big \downarrow \) 2825

\(\displaystyle -2 f \int \left (-\frac {6 b^3 d n^3}{d f x^2+1}-\frac {6 b^2 d \left (a+b \log \left (c x^n\right )\right ) n^2}{d f x^2+1}-\frac {3 b d \left (a+b \log \left (c x^n\right )\right )^2 n}{d f x^2+1}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{d f x^2+1}\right )dx-\frac {6 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {6 b^3 n^3 \log \left (d f x^2+1\right )}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 b^3 \log \left (d f x^2+1\right ) n^3}{x}-\frac {6 b^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d f x^2+1\right ) n^2}{x}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) n}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d f x^2+1\right )}{x}-2 f \left (-\frac {6 b^3 \sqrt {d} \arctan \left (\sqrt {d} \sqrt {f} x\right ) n^3}{\sqrt {f}}+\frac {3 i b^3 \sqrt {d} \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right ) n^3}{\sqrt {f}}-\frac {3 i b^3 \sqrt {d} \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right ) n^3}{\sqrt {f}}-\frac {3 b^3 \sqrt {-d} \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) n^3}{\sqrt {f}}+\frac {3 b^3 \sqrt {-d} \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) n^3}{\sqrt {f}}+\frac {3 b^3 \sqrt {-d} \operatorname {PolyLog}\left (4,-\sqrt {-d} \sqrt {f} x\right ) n^3}{\sqrt {f}}-\frac {3 b^3 \sqrt {-d} \operatorname {PolyLog}\left (4,\sqrt {-d} \sqrt {f} x\right ) n^3}{\sqrt {f}}-\frac {6 b^2 \sqrt {d} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right ) n^2}{\sqrt {f}}+\frac {3 b^2 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) n^2}{\sqrt {f}}-\frac {3 b^2 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) n^2}{\sqrt {f}}-\frac {3 b^2 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) n^2}{\sqrt {f}}+\frac {3 b^2 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) n^2}{\sqrt {f}}-\frac {3 b \sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right ) n}{2 \sqrt {f}}+\frac {3 b \sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt {-d} \sqrt {f} x+1\right ) n}{2 \sqrt {f}}+\frac {3 b \sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) n}{2 \sqrt {f}}-\frac {3 b \sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) n}{2 \sqrt {f}}-\frac {\sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{2 \sqrt {f}}+\frac {\sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\sqrt {-d} \sqrt {f} x+1\right )}{2 \sqrt {f}}\right )\)

Input:

Int[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)])/x^2,x]
 

Output:

(-6*b^3*n^3*Log[1 + d*f*x^2])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d* 
f*x^2])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/x - ((a + b*Log[ 
c*x^n])^3*Log[1 + d*f*x^2])/x - 2*f*((-6*b^3*Sqrt[d]*n^3*ArcTan[Sqrt[d]*Sq 
rt[f]*x])/Sqrt[f] - (6*b^2*Sqrt[d]*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Lo 
g[c*x^n]))/Sqrt[f] - (3*b*Sqrt[-d]*n*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d] 
*Sqrt[f]*x])/(2*Sqrt[f]) - (Sqrt[-d]*(a + b*Log[c*x^n])^3*Log[1 - Sqrt[-d] 
*Sqrt[f]*x])/(2*Sqrt[f]) + (3*b*Sqrt[-d]*n*(a + b*Log[c*x^n])^2*Log[1 + Sq 
rt[-d]*Sqrt[f]*x])/(2*Sqrt[f]) + (Sqrt[-d]*(a + b*Log[c*x^n])^3*Log[1 + Sq 
rt[-d]*Sqrt[f]*x])/(2*Sqrt[f]) + (3*b^2*Sqrt[-d]*n^2*(a + b*Log[c*x^n])*Po 
lyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/Sqrt[f] + (3*b*Sqrt[-d]*n*(a + b*Log[c*x^ 
n])^2*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(2*Sqrt[f]) - (3*b^2*Sqrt[-d]*n^2 
*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/Sqrt[f] - (3*b*Sqrt[-d 
]*n*(a + b*Log[c*x^n])^2*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[f]) + ((3 
*I)*b^3*Sqrt[d]*n^3*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/Sqrt[f] - ((3*I)*b 
^3*Sqrt[d]*n^3*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/Sqrt[f] - (3*b^3*Sqrt[-d]* 
n^3*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/Sqrt[f] - (3*b^2*Sqrt[-d]*n^2*(a + 
b*Log[c*x^n])*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/Sqrt[f] + (3*b^3*Sqrt[-d] 
*n^3*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/Sqrt[f] + (3*b^2*Sqrt[-d]*n^2*(a + b* 
Log[c*x^n])*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/Sqrt[f] + (3*b^3*Sqrt[-d]*n^3* 
PolyLog[4, -(Sqrt[-d]*Sqrt[f]*x)])/Sqrt[f] - (3*b^3*Sqrt[-d]*n^3*PolyLo...
 

Defintions of rubi rules used

rule 2009

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

rule 2825

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* 
(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, g, r, m 
, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )}{x^{2}}d x\]

Input:

int((a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2))/x^2,x)
 

Output:

int((a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2))/x^2,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^2,x, algorithm="fricas")
 

Output:

integral((b^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*log(d*f*x^2 + 1)*log 
(c*x^n)^2 + 3*a^2*b*log(d*f*x^2 + 1)*log(c*x^n) + a^3*log(d*f*x^2 + 1))/x^ 
2, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))**3*ln(d*(1/d+f*x**2))/x**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^2,x, algorithm="maxima")
 

Output:

-(b^3*log(x^n)^3 + 3*(2*n^2 + 2*n*log(c) + log(c)^2)*a*b^2 + (6*n^3 + 6*n^ 
2*log(c) + 3*n*log(c)^2 + log(c)^3)*b^3 + 3*a^2*b*(n + log(c)) + a^3 + 3*( 
b^3*(n + log(c)) + a*b^2)*log(x^n)^2 + 3*((2*n^2 + 2*n*log(c) + log(c)^2)* 
b^3 + 2*a*b^2*(n + log(c)) + a^2*b)*log(x^n))*log(d*f*x^2 + 1)/x + integra 
te(2*(b^3*d*f*log(x^n)^3 + a^3*d*f + 3*(d*f*n + d*f*log(c))*a^2*b + 3*(2*d 
*f*n^2 + 2*d*f*n*log(c) + d*f*log(c)^2)*a*b^2 + (6*d*f*n^3 + 6*d*f*n^2*log 
(c) + 3*d*f*n*log(c)^2 + d*f*log(c)^3)*b^3 + 3*(a*b^2*d*f + (d*f*n + d*f*l 
og(c))*b^3)*log(x^n)^2 + 3*(a^2*b*d*f + 2*(d*f*n + d*f*log(c))*a*b^2 + (2* 
d*f*n^2 + 2*d*f*n*log(c) + d*f*log(c)^2)*b^3)*log(x^n))/(d*f*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^2,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^3*log((f*x^2 + 1/d)*d)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \] Input:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3)/x^2,x)
 

Output:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3)/x^2, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx =\text {Too large to display} \] Input:

int((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^2,x)
 

Output:

(2*sqrt(f)*sqrt(d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*a**3*x + 6*sqrt(f)*sqrt 
(d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*a**2*b*n*x + 12*sqrt(f)*sqrt(d)*atan(( 
d*f*x)/(sqrt(f)*sqrt(d)))*a*b**2*n**2*x + 12*sqrt(f)*sqrt(d)*atan((d*f*x)/ 
(sqrt(f)*sqrt(d)))*b**3*n**3*x - 2*int(log(x**n*c)**3/(d*f*x**4 + x**2),x) 
*b**3*x - 6*int(log(x**n*c)**2/(d*f*x**4 + x**2),x)*a*b**2*x - 6*int(log(x 
**n*c)**2/(d*f*x**4 + x**2),x)*b**3*n*x - 6*int(log(x**n*c)/(d*f*x**4 + x* 
*2),x)*a**2*b*x - 12*int(log(x**n*c)/(d*f*x**4 + x**2),x)*a*b**2*n*x - 12* 
int(log(x**n*c)/(d*f*x**4 + x**2),x)*b**3*n**2*x - log(d*f*x**2 + 1)*log(x 
**n*c)**3*b**3 - 3*log(d*f*x**2 + 1)*log(x**n*c)**2*a*b**2 - 3*log(d*f*x** 
2 + 1)*log(x**n*c)**2*b**3*n - 3*log(d*f*x**2 + 1)*log(x**n*c)*a**2*b - 6* 
log(d*f*x**2 + 1)*log(x**n*c)*a*b**2*n - 6*log(d*f*x**2 + 1)*log(x**n*c)*b 
**3*n**2 - log(d*f*x**2 + 1)*a**3 - 3*log(d*f*x**2 + 1)*a**2*b*n - 6*log(d 
*f*x**2 + 1)*a*b**2*n**2 - 6*log(d*f*x**2 + 1)*b**3*n**3 - 2*log(x**n*c)** 
3*b**3 - 6*log(x**n*c)**2*a*b**2 - 12*log(x**n*c)**2*b**3*n - 6*log(x**n*c 
)*a**2*b - 24*log(x**n*c)*a*b**2*n - 36*log(x**n*c)*b**3*n**2 - 6*a**2*b*n 
 - 24*a*b**2*n**2 - 36*b**3*n**3)/x