\(\int \frac {(a+b \log (c x^n))^2 \log (d (\frac {1}{d}+f x^2))}{x} \, dx\) [40]

Optimal result

Integrand size = 28, antiderivative size = 70 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^2\right )-\frac {1}{4} b^2 n^2 \operatorname {PolyLog}\left (4,-d f x^2\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 484, normalized size of antiderivative = 6.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\frac {1}{3} \left (\log (x) \left (b^2 n^2 \log ^2(x)-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \log \left (1+d f x^2\right )-3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )+3 b n \left (-a+b n \log (x)-b \log \left (c x^n\right )\right ) \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )-b^2 n^2 \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,-i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,i \sqrt {d} \sqrt {f} x\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2821, 2830, 7143}

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.

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*((a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^2)]) + b*n*(((a + b*Log[c*x^ 
n])*PolyLog[3, -(d*f*x^2)])/2 - (b*n*PolyLog[4, -(d*f*x^2)])/4)
 

Defintions of rubi rules used

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[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 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 15.66 (sec) , antiderivative size = 825, normalized size of antiderivative = 11.79

method	result	size
risch	\(\text {Expression too large to display}\)	\(825\)

Input:

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

Output:

ln(d*f*x^2+1)*ln(x)^3*b^2*n^2-ln(x)^3*ln(1+x*(-d*f)^(1/2))*b^2*n^2-ln(x)^3 
*ln(1-x*(-d*f)^(1/2))*b^2*n^2-2*ln(d*f*x^2+1)*ln(x)^2*ln(x^n)*b^2*n-ln(x)^ 
2*dilog(1+x*(-d*f)^(1/2))*b^2*n^2-ln(x)^2*dilog(1-x*(-d*f)^(1/2))*b^2*n^2+ 
2*ln(x)^2*ln(1+x*(-d*f)^(1/2))*ln(x^n)*b^2*n+2*ln(x)^2*ln(1-x*(-d*f)^(1/2) 
)*ln(x^n)*b^2*n+ln(d*f*x^2+1)*ln(x)*ln(x^n)^2*b^2+2*ln(x)*dilog(1+x*(-d*f) 
^(1/2))*ln(x^n)*b^2*n+2*ln(x)*dilog(1-x*(-d*f)^(1/2))*ln(x^n)*b^2*n-ln(x)* 
ln(1+x*(-d*f)^(1/2))*ln(x^n)^2*b^2-ln(x)*ln(1-x*(-d*f)^(1/2))*ln(x^n)^2*b^ 
2-dilog(1+x*(-d*f)^(1/2))*ln(x^n)^2*b^2-dilog(1-x*(-d*f)^(1/2))*ln(x^n)^2* 
b^2+1/2*ln(x)^2*polylog(2,-d*f*x^2)*b^2*n^2-1/4*b^2*n^2*polylog(4,-d*f*x^2 
)-ln(x)*ln(x^n)*polylog(2,-d*f*x^2)*b^2*n+1/2*ln(x^n)*polylog(3,-d*f*x^2)* 
b^2*n+(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n) 
*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln( 
c)+2*a)*b*((ln(x^n)-n*ln(x))*(ln(x)*ln(d*f*x^2+1)-2*d*f*(1/2*ln(x)*(ln(1+x 
*(-d*f)^(1/2))+ln(1-x*(-d*f)^(1/2)))/d/f+1/2*(dilog(1+x*(-d*f)^(1/2))+dilo 
g(1-x*(-d*f)^(1/2)))/d/f))+n*(-1/2*ln(x)*polylog(2,-d*f*x^2)+1/4*polylog(3 
,-d*f*x^2)))+1/4*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*cs 
gn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I 
*c)+2*b*ln(c)+2*a)^2*(ln(x)*ln(d*f*x^2+1)-2*d*f*(1/2*ln(x)*(ln(1+x*(-d*f)^ 
(1/2))+ln(1-x*(-d*f)^(1/2)))/d/f+1/2*(dilog(1+x*(-d*f)^(1/2))+dilog(1-x*(- 
d*f)^(1/2)))/d/f))
 

Fricas [F]

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

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

Output:

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, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\left (\int \frac {\mathrm {log}\left (d f \,x^{2}+1\right )}{d f \,x^{3}+x}d x \right ) a^{2}+\left (\int \frac {\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right )^{2}}{x}d x \right ) b^{2}+2 \left (\int \frac {\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right )}{x}d x \right ) a b +\frac {\mathrm {log}\left (d f \,x^{2}+1\right )^{2} a^{2}}{4} \] Input:

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

Output:

(4*int(log(d*f*x**2 + 1)/(d*f*x**3 + x),x)*a**2 + 4*int((log(d*f*x**2 + 1) 
*log(x**n*c)**2)/x,x)*b**2 + 8*int((log(d*f*x**2 + 1)*log(x**n*c))/x,x)*a* 
b + log(d*f*x**2 + 1)**2*a**2)/4