3.66 \(\int \frac{(a+b \log (c x^n))^2 \log (d (\frac{1}{d}+f x^m))}{x} \, dx\)
Optimal. Leaf size=73 \[ \frac{2 b n \text{PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac{2 b^2 n^2 \text{PolyLog}\left (4,-d f x^m\right )}{m^3} \]
[Out]
-(((a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^m)])/m) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*x^m)])/m^2 - (
2*b^2*n^2*PolyLog[4, -(d*f*x^m)])/m^3
________________________________________________________________________________________
Rubi [A] time = 0.0712155, antiderivative size = 73, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used =
{2374, 2383, 6589} \[ \frac{2 b n \text{PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac{2 b^2 n^2 \text{PolyLog}\left (4,-d f x^m\right )}{m^3} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*Log[c*x^n])^2*Log[d*(d^(-1) + f*x^m)])/x,x]
[Out]
-(((a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^m)])/m) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*x^m)])/m^2 - (
2*b^2*n^2*PolyLog[4, -(d*f*x^m)])/m^3
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 2383
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(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 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 \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac{1}{d}+f x^m\right )\right )}{x} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^m\right )}{m}+\frac{(2 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^m\right )}{x} \, dx}{m}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^m\right )}{m}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^m\right )}{m^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_3\left (-d f x^m\right )}{x} \, dx}{m^2}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-d f x^m\right )}{m}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-d f x^m\right )}{m^2}-\frac{2 b^2 n^2 \text{Li}_4\left (-d f x^m\right )}{m^3}\\ \end{align*}
Mathematica [B] time = 0.233456, size = 526, normalized size = 7.21 \[ \frac{b n \log (x) \text{PolyLog}\left (2,-\frac{x^{-m}}{d f}\right ) \left (2 \left (a+b \log \left (c x^n\right )\right )-b n \log (x)\right )}{m}+\frac{\text{PolyLog}\left (2,d f x^m+1\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2}{m}+\frac{2 a b n \text{PolyLog}\left (3,-\frac{x^{-m}}{d f}\right )}{m^2}+\frac{2 b^2 n \log \left (c x^n\right ) \text{PolyLog}\left (3,-\frac{x^{-m}}{d f}\right )}{m^2}+\frac{2 b^2 n^2 \text{PolyLog}\left (4,-\frac{x^{-m}}{d f}\right )}{m^3}+\frac{a^2 \log \left (-d f x^m\right ) \log \left (d f x^m+1\right )}{m}+\frac{2 a b \log \left (c x^n\right ) \log \left (-d f x^m\right ) \log \left (d f x^m+1\right )}{m}-a b n \log ^2(x) \log \left (\frac{x^{-m}}{d f}+1\right )+a b n \log ^2(x) \log \left (d f x^m+1\right )-\frac{2 a b n \log (x) \log \left (-d f x^m\right ) \log \left (d f x^m+1\right )}{m}-\frac{1}{3} a b m n \log ^3(x)-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (\frac{x^{-m}}{d f}+1\right )+b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d f x^m+1\right )+\frac{b^2 \log ^2\left (c x^n\right ) \log \left (-d f x^m\right ) \log \left (d f x^m+1\right )}{m}-\frac{2 b^2 n \log (x) \log \left (c x^n\right ) \log \left (-d f x^m\right ) \log \left (d f x^m+1\right )}{m}-\frac{1}{3} b^2 m n \log ^3(x) \log \left (c x^n\right )+\frac{2}{3} b^2 n^2 \log ^3(x) \log \left (\frac{x^{-m}}{d f}+1\right )-\frac{2}{3} b^2 n^2 \log ^3(x) \log \left (d f x^m+1\right )+\frac{b^2 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log \left (d f x^m+1\right )}{m}+\frac{1}{4} b^2 m n^2 \log ^4(x) \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*Log[c*x^n])^2*Log[d*(d^(-1) + f*x^m)])/x,x]
[Out]
-(a*b*m*n*Log[x]^3)/3 + (b^2*m*n^2*Log[x]^4)/4 - (b^2*m*n*Log[x]^3*Log[c*x^n])/3 - a*b*n*Log[x]^2*Log[1 + 1/(d
*f*x^m)] + (2*b^2*n^2*Log[x]^3*Log[1 + 1/(d*f*x^m)])/3 - b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + 1/(d*f*x^m)] + a*b*
n*Log[x]^2*Log[1 + d*f*x^m] - (2*b^2*n^2*Log[x]^3*Log[1 + d*f*x^m])/3 + (a^2*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])
/m - (2*a*b*n*Log[x]*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m + (b^2*n^2*Log[x]^2*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])
/m + b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + d*f*x^m] + (2*a*b*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*f*x^m])/m - (2*b
^2*n*Log[x]*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*f*x^m])/m + (b^2*Log[-(d*f*x^m)]*Log[c*x^n]^2*Log[1 + d*f*x^m
])/m + (b*n*Log[x]*(-(b*n*Log[x]) + 2*(a + b*Log[c*x^n]))*PolyLog[2, -(1/(d*f*x^m))])/m + ((a - b*n*Log[x] + b
*Log[c*x^n])^2*PolyLog[2, 1 + d*f*x^m])/m + (2*a*b*n*PolyLog[3, -(1/(d*f*x^m))])/m^2 + (2*b^2*n*Log[c*x^n]*Pol
yLog[3, -(1/(d*f*x^m))])/m^2 + (2*b^2*n^2*PolyLog[4, -(1/(d*f*x^m))])/m^3
________________________________________________________________________________________
Maple [C] time = 0.089, size = 2578, normalized size = 35.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^m))/x,x)
[Out]
-I*n/m*ln(x)*polylog(2,-d*f*x^m)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*n/m^2*polylog(3,-d*f*x^m)*b^2*Pi*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*n*ln(d*(1/d+f*x^m))*ln(x)^2*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*n/
m*ln(x)*polylog(2,-d*f*x^m)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I/m*dilog(d*f*x^m+1)*n*ln(x)*b^2*Pi*csg
n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/m*dilog(d*f*x^m+1)*a^2+2*b^2/m*dilog(d*f*x^m+1)*ln(x)*ln(x^n)*n-1/2/m*dilog
(d*f*x^m+1)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+1/4/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-1
/2/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+1/4/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*c)^2*csgn(I*
c*x^n)^4-2*b^2*n^2*polylog(4,-d*f*x^m)/m^3+n*ln(x)^2*ln(d*f*x^m+1)*b^2*ln(c)-b*n*ln(d*(1/d+f*x^m))*ln(x)^2*a+2
*b*ln(x)*ln(d*(1/d+f*x^m))*ln(x^n)*a+b*n*ln(x)^2*ln(d*f*x^m+1)*a+2*b*n/m^2*polylog(3,-d*f*x^m)*a-2*b/m*dilog(d
*f*x^m+1)*ln(x^n)*a-2*b*ln(x)*ln(x^n)*ln(d*f*x^m+1)*a+1/2*I*n*ln(x)^2*ln(d*f*x^m+1)*b^2*Pi*csgn(I*x^n)*csgn(I*
c*x^n)^2-2*ln(x)*ln(x^n)*ln(d*f*x^m+1)*b^2*ln(c)-n*ln(d*(1/d+f*x^m))*ln(x)^2*b^2*ln(c)+2*ln(x)*ln(d*(1/d+f*x^m
))*ln(x^n)*b^2*ln(c)-2/m*dilog(d*f*x^m+1)*ln(c)*a*b+2*n/m^2*polylog(3,-d*f*x^m)*b^2*ln(c)-2/m*dilog(d*f*x^m+1)
*ln(x^n)*b^2*ln(c)-I/m*dilog(d*f*x^m+1)*n*ln(x)*b^2*Pi*csgn(I*c*x^n)^3-b^2*n*ln(1/d+f*x^m)*ln(x)^2*ln(x^n)-b^2
/m*dilog(d*f*x^m+1)*ln(x)^2*n^2+2*b^2*n/m^2*polylog(3,-d*f*x^m)*ln(x^n)+b^2*ln(x)^2*ln(x^n)*ln(d*f*x^m+1)*n+1/
4/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*c*x^n)^6+b^2*n^2/m*ln(x)^2*polylog(2,-d*f*x^m)-I*ln(x)*ln(x^n)*ln(d*f*x^m
+1)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I/m*dilog(d*f*x^m+1)*ln(x^n)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I/m*dilog
(d*f*x^m+1)*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I/m*dilog(d*f*x^m+1)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I
/m*dilog(d*f*x^m+1)*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I/m*dilog(d*f*x^m+1)*Pi*a*b*csgn(I*c)*csgn(I*c*x^
n)^2-I/m*dilog(d*f*x^m+1)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*c*x^n)^2+I/m*dilog(d*f*x^m+1)*ln(x^n)*b^2*Pi*csgn(I*c*
x^n)^3+I*ln(x)*ln(x^n)*ln(d*f*x^m+1)*b^2*Pi*csgn(I*c*x^n)^3+1/4/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*c)^2*csgn(I
*x^n)^2*csgn(I*c*x^n)^2-1/2/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-2*b*n/m*ln(x)*
polylog(2,-d*f*x^m)*a+2*b/m*dilog(d*f*x^m+1)*n*ln(x)*a-2*n/m*ln(x)*polylog(2,-d*f*x^m)*b^2*ln(c)+2/m*dilog(d*f
*x^m+1)*n*ln(x)*b^2*ln(c)-2*b^2*n/m*ln(x)*polylog(2,-d*f*x^m)*ln(x^n)-I*ln(x)*ln(x^n)*ln(d*f*x^m+1)*b^2*Pi*csg
n(I*x^n)*csgn(I*c*x^n)^2-1/2*I*n*ln(x)^2*ln(d*f*x^m+1)*b^2*Pi*csgn(I*c*x^n)^3-1/2*I*n*ln(d*(1/d+f*x^m))*ln(x)^
2*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*n*ln(x)^2*ln(d*f*x^m+1)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*n*ln
(d*(1/d+f*x^m))*ln(x)^2*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*ln(x)*ln(d*(1/d+f*x^m))*ln(x^n)*b^2*Pi*csgn(I*x^n)*
csgn(I*c*x^n)^2+I*n/m*ln(x)*polylog(2,-d*f*x^m)*b^2*Pi*csgn(I*c*x^n)^3+I*n/m^2*polylog(3,-d*f*x^m)*b^2*Pi*csgn
(I*c)*csgn(I*c*x^n)^2+I*n/m^2*polylog(3,-d*f*x^m)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I*ln(x)*ln(d*(1/d+f*x^m))
*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*n*ln(x)^2*ln(d*f*x^m+1)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*
x^n)+1/2*I*n*ln(d*(1/d+f*x^m))*ln(x)^2*b^2*Pi*csgn(I*c*x^n)^3-I*ln(x)*ln(d*(1/d+f*x^m))*ln(x^n)*b^2*Pi*csgn(I*
c*x^n)^3-I*n/m^2*polylog(3,-d*f*x^m)*b^2*Pi*csgn(I*c*x^n)^3-1/m*dilog(d*f*x^m+1)*ln(c)^2*b^2-1/3*b^2*n^2*ln(x)
^3*ln(d*f*x^m+1)+1/3*b^2/n*ln(d*(1/d+f*x^m))*ln(x^n)^3+1/3*b^2*n^2*ln(1/d+f*x^m)*ln(x)^3-1/3*b^2/n*ln(1/d+f*x^
m)*ln(x^n)^3-b^2/m*dilog(d*f*x^m+1)*ln(x^n)^2-b^2*ln(x)*ln(x^n)^2*ln(d*f*x^m+1)+b^2*ln(1/d+f*x^m)*ln(x)*ln(x^n
)^2-1/2/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+I/m*dilog(d*f*x^m+1)*ln(c)*Pi*b^2*
csgn(I*c*x^n)^3+I/m*dilog(d*f*x^m+1)*Pi*a*b*csgn(I*c*x^n)^3+1/m*dilog(d*f*x^m+1)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n
)*csgn(I*c*x^n)^4+I/m*dilog(d*f*x^m+1)*n*ln(x)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I/m*dilog(d*f*x^m+1)*n*ln(x)*b
^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+I/m*dilog(d*f*x^m+1)*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*ln
(x)*ln(x^n)*ln(d*f*x^m+1)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I/m*dilog(d*f*x^m+1)*ln(c)*Pi*b^2*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)-I*n/m*ln(x)*polylog(2,-d*f*x^m)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I*ln(x)*ln(d*(1/
d+f*x^m))*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I/m*dilog(d*f*x^m+1)*Pi*a*b*csgn(I*c)*csgn(I*x^n)
*csgn(I*c*x^n)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \,{\left (b^{2} n^{2} \log \left (x\right )^{3} + 3 \, b^{2} \log \left (x\right ) \log \left (x^{n}\right )^{2} - 3 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )^{2} - 3 \,{\left (b^{2} n \log \left (x\right )^{2} - 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x\right )\right )} \log \left (x^{n}\right ) + 3 \,{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}\right )} \log \left (x\right )\right )} \log \left (d f x^{m} + 1\right ) - \int \frac{3 \, b^{2} d f m x^{m} \log \left (x\right ) \log \left (x^{n}\right )^{2} - 3 \,{\left (b^{2} d f m n \log \left (x\right )^{2} - 2 \,{\left (b^{2} d f m \log \left (c\right ) + a b d f m\right )} \log \left (x\right )\right )} x^{m} \log \left (x^{n}\right ) +{\left (b^{2} d f m n^{2} \log \left (x\right )^{3} - 3 \,{\left (b^{2} d f m n \log \left (c\right ) + a b d f m n\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{2} d f m \log \left (c\right )^{2} + 2 \, a b d f m \log \left (c\right ) + a^{2} d f m\right )} \log \left (x\right )\right )} x^{m}}{3 \,{\left (d f x x^{m} + x\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*(1/d+f*x^m))/x,x, algorithm="maxima")
[Out]
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^m + 1) - integrate
(1/3*(3*b^2*d*f*m*x^m*log(x)*log(x^n)^2 - 3*(b^2*d*f*m*n*log(x)^2 - 2*(b^2*d*f*m*log(c) + a*b*d*f*m)*log(x))*x
^m*log(x^n) + (b^2*d*f*m*n^2*log(x)^3 - 3*(b^2*d*f*m*n*log(c) + a*b*d*f*m*n)*log(x)^2 + 3*(b^2*d*f*m*log(c)^2
+ 2*a*b*d*f*m*log(c) + a^2*d*f*m)*log(x))*x^m)/(d*f*x*x^m + x), x)
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Fricas [C] time = 1.38364, size = 325, normalized size = 4.45 \begin{align*} -\frac{2 \, b^{2} n^{2}{\rm polylog}\left (4, -d f x^{m}\right ) +{\left (b^{2} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} m^{2} \log \left (c\right )^{2} + 2 \, a b m^{2} \log \left (c\right ) + a^{2} m^{2} + 2 \,{\left (b^{2} m^{2} n \log \left (c\right ) + a b m^{2} n\right )} \log \left (x\right )\right )}{\rm Li}_2\left (-d f x^{m}\right ) - 2 \,{\left (b^{2} m n^{2} \log \left (x\right ) + b^{2} m n \log \left (c\right ) + a b m n\right )}{\rm polylog}\left (3, -d f x^{m}\right )}{m^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))^2*log(d*(1/d+f*x^m))/x,x, algorithm="fricas")
[Out]
-(2*b^2*n^2*polylog(4, -d*f*x^m) + (b^2*m^2*n^2*log(x)^2 + b^2*m^2*log(c)^2 + 2*a*b*m^2*log(c) + a^2*m^2 + 2*(
b^2*m^2*n*log(c) + a*b*m^2*n)*log(x))*dilog(-d*f*x^m) - 2*(b^2*m*n^2*log(x) + b^2*m*n*log(c) + a*b*m*n)*polylo
g(3, -d*f*x^m))/m^3
________________________________________________________________________________________
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*(1/d+f*x**m))/x,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{m} + \frac{1}{d}\right )} d\right )}{x}\,{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*(1/d+f*x^m))/x,x, algorithm="giac")
[Out]
integrate((b*log(c*x^n) + a)^2*log((f*x^m + 1/d)*d)/x, x)