Integrand size = 22, antiderivative size = 287 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx=-\frac {7 b^2 e n^2}{4 x}-\frac {1}{4} b^2 e^2 n^2 \log (x)-\frac {3 b e n \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {1}{2} b e^2 n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 x}+\frac {1}{2} e^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} b^2 e^2 n^2 \log (1+e x)-\frac {b^2 n^2 \log (1+e x)}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}-\frac {1}{2} b^2 e^2 n^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-b e^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right ) \]
-7/4*b^2*e*n^2/x-1/4*b^2*e^2*n^2*ln(x)-3/2*b*e*n*(a+b*ln(c*x^n))/x+1/2*b*e ^2*n*ln(1+1/e/x)*(a+b*ln(c*x^n))-1/2*e*(a+b*ln(c*x^n))^2/x+1/2*e^2*ln(1+1/ e/x)*(a+b*ln(c*x^n))^2+1/4*b^2*e^2*n^2*ln(e*x+1)-1/4*b^2*n^2*ln(e*x+1)/x^2 -1/2*b*n*(a+b*ln(c*x^n))*ln(e*x+1)/x^2-1/2*(a+b*ln(c*x^n))^2*ln(e*x+1)/x^2 -1/2*b^2*e^2*n^2*polylog(2,-1/e/x)-b*e^2*n*(a+b*ln(c*x^n))*polylog(2,-1/e/ x)-b^2*e^2*n^2*polylog(3,-1/e/x)
Time = 0.13 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx=-\frac {6 a^2 e x+18 a b e n x+21 b^2 e n^2 x+6 a^2 e^2 x^2 \log (x)+6 a b e^2 n x^2 \log (x)+3 b^2 e^2 n^2 x^2 \log (x)-6 a b e^2 n x^2 \log ^2(x)-3 b^2 e^2 n^2 x^2 \log ^2(x)+2 b^2 e^2 n^2 x^2 \log ^3(x)+12 a b e x \log \left (c x^n\right )+18 b^2 e n x \log \left (c x^n\right )+12 a b e^2 x^2 \log (x) \log \left (c x^n\right )+6 b^2 e^2 n x^2 \log (x) \log \left (c x^n\right )-6 b^2 e^2 n x^2 \log ^2(x) \log \left (c x^n\right )+6 b^2 e x \log ^2\left (c x^n\right )+6 b^2 e^2 x^2 \log (x) \log ^2\left (c x^n\right )+6 a^2 \log (1+e x)+6 a b n \log (1+e x)+3 b^2 n^2 \log (1+e x)-6 a^2 e^2 x^2 \log (1+e x)-6 a b e^2 n x^2 \log (1+e x)-3 b^2 e^2 n^2 x^2 \log (1+e x)+12 a b \log \left (c x^n\right ) \log (1+e x)+6 b^2 n \log \left (c x^n\right ) \log (1+e x)-12 a b e^2 x^2 \log \left (c x^n\right ) \log (1+e x)-6 b^2 e^2 n x^2 \log \left (c x^n\right ) \log (1+e x)+6 b^2 \log ^2\left (c x^n\right ) \log (1+e x)-6 b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (1+e x)-6 b e^2 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)+12 b^2 e^2 n^2 x^2 \operatorname {PolyLog}(3,-e x)}{12 x^2} \]
-1/12*(6*a^2*e*x + 18*a*b*e*n*x + 21*b^2*e*n^2*x + 6*a^2*e^2*x^2*Log[x] + 6*a*b*e^2*n*x^2*Log[x] + 3*b^2*e^2*n^2*x^2*Log[x] - 6*a*b*e^2*n*x^2*Log[x] ^2 - 3*b^2*e^2*n^2*x^2*Log[x]^2 + 2*b^2*e^2*n^2*x^2*Log[x]^3 + 12*a*b*e*x* Log[c*x^n] + 18*b^2*e*n*x*Log[c*x^n] + 12*a*b*e^2*x^2*Log[x]*Log[c*x^n] + 6*b^2*e^2*n*x^2*Log[x]*Log[c*x^n] - 6*b^2*e^2*n*x^2*Log[x]^2*Log[c*x^n] + 6*b^2*e*x*Log[c*x^n]^2 + 6*b^2*e^2*x^2*Log[x]*Log[c*x^n]^2 + 6*a^2*Log[1 + e*x] + 6*a*b*n*Log[1 + e*x] + 3*b^2*n^2*Log[1 + e*x] - 6*a^2*e^2*x^2*Log[ 1 + e*x] - 6*a*b*e^2*n*x^2*Log[1 + e*x] - 3*b^2*e^2*n^2*x^2*Log[1 + e*x] + 12*a*b*Log[c*x^n]*Log[1 + e*x] + 6*b^2*n*Log[c*x^n]*Log[1 + e*x] - 12*a*b *e^2*x^2*Log[c*x^n]*Log[1 + e*x] - 6*b^2*e^2*n*x^2*Log[c*x^n]*Log[1 + e*x] + 6*b^2*Log[c*x^n]^2*Log[1 + e*x] - 6*b^2*e^2*x^2*Log[c*x^n]^2*Log[1 + e* x] - 6*b*e^2*n*x^2*(2*a + b*n + 2*b*Log[c*x^n])*PolyLog[2, -(e*x)] + 12*b^ 2*e^2*n^2*x^2*PolyLog[3, -(e*x)])/x^2
Time = 0.67 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -e \int \left (-\frac {b^2 n^2}{4 x^2 (e x+1)}-\frac {b \left (a+b \log \left (c x^n\right )\right ) n}{2 x^2 (e x+1)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2 (e x+1)}\right )dx-\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b^2 n^2 \log (e x+1)}{4 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -e \left (b e n \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b e n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x}+\frac {1}{2} b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )+\frac {1}{4} b^2 e n^2 \log (x)-\frac {1}{4} b^2 e n^2 \log (e x+1)+\frac {7 b^2 n^2}{4 x}\right )-\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b^2 n^2 \log (e x+1)}{4 x^2}\) |
-1/4*(b^2*n^2*Log[1 + e*x])/x^2 - (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/(2 *x^2) - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(2*x^2) - e*((7*b^2*n^2)/(4*x) + (b^2*e*n^2*Log[x])/4 + (3*b*n*(a + b*Log[c*x^n]))/(2*x) - (b*e*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n]))/2 + (a + b*Log[c*x^n])^2/(2*x) - (e*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2)/2 - (b^2*e*n^2*Log[1 + e*x])/4 + (b^2*e*n^ 2*PolyLog[2, -(1/(e*x))])/2 + b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x ))] + b^2*e*n^2*PolyLog[3, -(1/(e*x))])
3.1.16.3.1 Defintions of rubi rules used
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.16 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.46
| method | result | size |
| risch | \(-\frac {b^{2} n^{2} \ln \left (e x +1\right )}{4 x^{2}}-\frac {7 b^{2} e \,n^{2}}{4 x}-\frac {b^{2} e^{2} n^{2} \ln \left (x \right )}{4}+\frac {b^{2} e^{2} n^{2} \ln \left (e x +1\right )}{4}+\frac {b^{2} n^{2} e^{2} \ln \left (x \right )^{2}}{4}+\frac {b^{2} n^{2} e^{2} \operatorname {Li}_{2}\left (-e x \right )}{2}+\frac {b^{2} n^{2} e^{2} \ln \left (x \right )^{3}}{3}-b^{2} n^{2} e^{2} \operatorname {Li}_{3}\left (-e x \right )-\frac {b^{2} e^{2} \ln \left (e x \right ) \ln \left (x^{n}\right )^{2}}{2}-\frac {b^{2} e \ln \left (x^{n}\right )^{2}}{2 x}+\frac {b^{2} e^{2} \ln \left (e x +1\right ) \ln \left (x^{n}\right )^{2}}{2}-\frac {b^{2} e^{2} \ln \left (e x \right ) \ln \left (x \right )^{2} n^{2}}{2}-\frac {\ln \left (x^{n}\right )^{2} \ln \left (e x +1\right ) b^{2}}{2 x^{2}}-\frac {b^{2} n \ln \left (e x +1\right ) \ln \left (x^{n}\right )}{2 x^{2}}-\frac {3 b^{2} n e \ln \left (x^{n}\right )}{2 x}-\frac {b^{2} n \,e^{2} \ln \left (x \right ) \ln \left (x^{n}\right )}{2}+\frac {b^{2} n \,e^{2} \ln \left (e x +1\right ) \ln \left (x^{n}\right )}{2}-\frac {b^{2} n \,e^{2} \ln \left (x \right )^{2} \ln \left (x^{n}\right )}{2}+b^{2} n \,e^{2} \operatorname {Li}_{2}\left (-e x \right ) \ln \left (x^{n}\right )+b^{2} e^{2} \ln \left (e x \right ) \ln \left (x \right ) \ln \left (x^{n}\right ) n +\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) e^{2} \left (-\frac {\ln \left (e x \right )}{2}-\frac {1}{2 e x}+\frac {\ln \left (e x +1\right ) \left (e x +1\right ) \left (e x -1\right )}{2 x^{2} e^{2}}\right )+n \left (\frac {\left (-\frac {1}{4}-\frac {\ln \left (x \right )}{2}\right ) \ln \left (e x +1\right )}{x^{2}}-\frac {3 e}{4 x}-\frac {e^{2} \ln \left (x \right )}{4}+\frac {e^{2} \ln \left (e x +1\right )}{4}-\frac {e \ln \left (x \right )}{2 x}+\frac {e^{2} \ln \left (e x +1\right ) \ln \left (x \right )}{2}-\frac {e^{2} \ln \left (x \right )^{2}}{4}+\frac {e^{2} \operatorname {Li}_{2}\left (-e x \right )}{2}\right )\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} e^{2} \left (-\frac {\ln \left (e x \right )}{2}-\frac {1}{2 e x}+\frac {\ln \left (e x +1\right ) \left (e x +1\right ) \left (e x -1\right )}{2 x^{2} e^{2}}\right )}{4}\) | \(705\) |
-1/4*b^2*n^2*ln(e*x+1)/x^2-7/4*b^2*e*n^2/x-1/4*b^2*e^2*n^2*ln(x)+1/4*b^2*e ^2*n^2*ln(e*x+1)+1/4*b^2*n^2*e^2*ln(x)^2+1/2*b^2*n^2*e^2*polylog(2,-e*x)+1 /3*b^2*n^2*e^2*ln(x)^3-b^2*n^2*e^2*polylog(3,-e*x)-1/2*b^2*e^2*ln(e*x)*ln( x^n)^2-1/2*b^2*e/x*ln(x^n)^2+1/2*b^2*e^2*ln(e*x+1)*ln(x^n)^2-1/2*b^2*e^2*l n(e*x)*ln(x)^2*n^2-1/2*ln(x^n)^2/x^2*ln(e*x+1)*b^2-1/2*b^2*n/x^2*ln(e*x+1) *ln(x^n)-3/2*b^2*n*e/x*ln(x^n)-1/2*b^2*n*e^2*ln(x)*ln(x^n)+1/2*b^2*n*e^2*l n(e*x+1)*ln(x^n)-1/2*b^2*n*e^2*ln(x)^2*ln(x^n)+b^2*n*e^2*polylog(2,-e*x)*l n(x^n)+b^2*e^2*ln(e*x)*ln(x)*ln(x^n)*n+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn (I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n )^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*((ln(x^n)-n*ln(x))*e^2*(-1/2*l n(e*x)-1/2/e/x+1/2*ln(e*x+1)*(e*x+1)*(e*x-1)/x^2/e^2)+n*((-1/4-1/2*ln(x))/ x^2*ln(e*x+1)-3/4*e/x-1/4*e^2*ln(x)+1/4*e^2*ln(e*x+1)-1/2*e*ln(x)/x+1/2*e^ 2*ln(e*x+1)*ln(x)-1/4*e^2*ln(x)^2+1/2*e^2*polylog(2,-e*x)))+1/4*(-I*b*Pi*c sgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi *csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*e^2*( -1/2*ln(e*x)-1/2/e/x+1/2*ln(e*x+1)*(e*x+1)*(e*x-1)/x^2/e^2)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{x^{3}} \,d x } \]
integral((b^2*log(c*x^n)^2*log(e*x + 1) + 2*a*b*log(c*x^n)*log(e*x + 1) + a^2*log(e*x + 1))/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{x^{3}} \,d x } \]
-1/2*(b^2*e^2*x^2*log(x) + b^2*e*x - (b^2*e^2*x^2 - b^2)*log(e*x + 1))*log (x^n)^2/x^2 - integrate(-((b^2*log(c)^2 + 2*a*b*log(c) + a^2)*log(e*x + 1) + (b^2*e^2*n*x^2*log(x) + b^2*e*n*x - (b^2*e^2*n*x^2 - b^2*(n + 2*log(c)) - 2*a*b)*log(e*x + 1))*log(x^n))/x^3, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx=\int \frac {\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \]