Integrand size = 22, antiderivative size = 470 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=-\frac {45 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)-\frac {21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}+\frac {3}{4} b^2 e^2 n^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac {3}{4} b e^2 n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}+\frac {1}{2} e^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}-\frac {3}{4} b^3 e^2 n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-\frac {3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-\frac {3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-\frac {3}{2} b^3 e^2 n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )-3 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )-3 b^3 e^2 n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right ) \]
-45/8*b^3*e*n^3/x-3/8*b^3*e^2*n^3*ln(x)-21/4*b^2*e*n^2*(a+b*ln(c*x^n))/x+3 /4*b^2*e^2*n^2*ln(1+1/e/x)*(a+b*ln(c*x^n))-9/4*b*e*n*(a+b*ln(c*x^n))^2/x+3 /4*b*e^2*n*ln(1+1/e/x)*(a+b*ln(c*x^n))^2-1/2*e*(a+b*ln(c*x^n))^3/x+1/2*e^2 *ln(1+1/e/x)*(a+b*ln(c*x^n))^3+3/8*b^3*e^2*n^3*ln(e*x+1)-3/8*b^3*n^3*ln(e* x+1)/x^2-3/4*b^2*n^2*(a+b*ln(c*x^n))*ln(e*x+1)/x^2-3/4*b*n*(a+b*ln(c*x^n)) ^2*ln(e*x+1)/x^2-1/2*(a+b*ln(c*x^n))^3*ln(e*x+1)/x^2-3/4*b^3*e^2*n^3*polyl og(2,-1/e/x)-3/2*b^2*e^2*n^2*(a+b*ln(c*x^n))*polylog(2,-1/e/x)-3/2*b*e^2*n *(a+b*ln(c*x^n))^2*polylog(2,-1/e/x)-3/2*b^3*e^2*n^3*polylog(3,-1/e/x)-3*b ^2*e^2*n^2*(a+b*ln(c*x^n))*polylog(3,-1/e/x)-3*b^3*e^2*n^3*polylog(4,-1/e/ x)
Leaf count is larger than twice the leaf count of optimal. \(1047\) vs. \(2(470)=940\).
Time = 0.24 (sec) , antiderivative size = 1047, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=-\frac {4 a^3 e x+18 a^2 b e n x+42 a b^2 e n^2 x+45 b^3 e n^3 x+4 a^3 e^2 x^2 \log (x)+6 a^2 b e^2 n x^2 \log (x)+6 a b^2 e^2 n^2 x^2 \log (x)+3 b^3 e^2 n^3 x^2 \log (x)-6 a^2 b e^2 n x^2 \log ^2(x)-6 a b^2 e^2 n^2 x^2 \log ^2(x)-3 b^3 e^2 n^3 x^2 \log ^2(x)+4 a b^2 e^2 n^2 x^2 \log ^3(x)+2 b^3 e^2 n^3 x^2 \log ^3(x)-b^3 e^2 n^3 x^2 \log ^4(x)+12 a^2 b e x \log \left (c x^n\right )+36 a b^2 e n x \log \left (c x^n\right )+42 b^3 e n^2 x \log \left (c x^n\right )+12 a^2 b e^2 x^2 \log (x) \log \left (c x^n\right )+12 a b^2 e^2 n x^2 \log (x) \log \left (c x^n\right )+6 b^3 e^2 n^2 x^2 \log (x) \log \left (c x^n\right )-12 a b^2 e^2 n x^2 \log ^2(x) \log \left (c x^n\right )-6 b^3 e^2 n^2 x^2 \log ^2(x) \log \left (c x^n\right )+4 b^3 e^2 n^2 x^2 \log ^3(x) \log \left (c x^n\right )+12 a b^2 e x \log ^2\left (c x^n\right )+18 b^3 e n x \log ^2\left (c x^n\right )+12 a b^2 e^2 x^2 \log (x) \log ^2\left (c x^n\right )+6 b^3 e^2 n x^2 \log (x) \log ^2\left (c x^n\right )-6 b^3 e^2 n x^2 \log ^2(x) \log ^2\left (c x^n\right )+4 b^3 e x \log ^3\left (c x^n\right )+4 b^3 e^2 x^2 \log (x) \log ^3\left (c x^n\right )+4 a^3 \log (1+e x)+6 a^2 b n \log (1+e x)+6 a b^2 n^2 \log (1+e x)+3 b^3 n^3 \log (1+e x)-4 a^3 e^2 x^2 \log (1+e x)-6 a^2 b e^2 n x^2 \log (1+e x)-6 a b^2 e^2 n^2 x^2 \log (1+e x)-3 b^3 e^2 n^3 x^2 \log (1+e x)+12 a^2 b \log \left (c x^n\right ) \log (1+e x)+12 a b^2 n \log \left (c x^n\right ) \log (1+e x)+6 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)-12 a^2 b e^2 x^2 \log \left (c x^n\right ) \log (1+e x)-12 a b^2 e^2 n x^2 \log \left (c x^n\right ) \log (1+e x)-6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log (1+e x)+12 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)+6 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)-12 a b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (1+e x)-6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log (1+e x)+4 b^3 \log ^3\left (c x^n\right ) \log (1+e x)-4 b^3 e^2 x^2 \log ^3\left (c x^n\right ) \log (1+e x)-6 b e^2 n x^2 \left (2 a^2+2 a b n+b^2 n^2+2 b (2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)+12 b^2 e^2 n^2 x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,-e x)-24 b^3 e^2 n^3 x^2 \operatorname {PolyLog}(4,-e x)}{8 x^2} \]
-1/8*(4*a^3*e*x + 18*a^2*b*e*n*x + 42*a*b^2*e*n^2*x + 45*b^3*e*n^3*x + 4*a ^3*e^2*x^2*Log[x] + 6*a^2*b*e^2*n*x^2*Log[x] + 6*a*b^2*e^2*n^2*x^2*Log[x] + 3*b^3*e^2*n^3*x^2*Log[x] - 6*a^2*b*e^2*n*x^2*Log[x]^2 - 6*a*b^2*e^2*n^2* x^2*Log[x]^2 - 3*b^3*e^2*n^3*x^2*Log[x]^2 + 4*a*b^2*e^2*n^2*x^2*Log[x]^3 + 2*b^3*e^2*n^3*x^2*Log[x]^3 - b^3*e^2*n^3*x^2*Log[x]^4 + 12*a^2*b*e*x*Log[ c*x^n] + 36*a*b^2*e*n*x*Log[c*x^n] + 42*b^3*e*n^2*x*Log[c*x^n] + 12*a^2*b* e^2*x^2*Log[x]*Log[c*x^n] + 12*a*b^2*e^2*n*x^2*Log[x]*Log[c*x^n] + 6*b^3*e ^2*n^2*x^2*Log[x]*Log[c*x^n] - 12*a*b^2*e^2*n*x^2*Log[x]^2*Log[c*x^n] - 6* b^3*e^2*n^2*x^2*Log[x]^2*Log[c*x^n] + 4*b^3*e^2*n^2*x^2*Log[x]^3*Log[c*x^n ] + 12*a*b^2*e*x*Log[c*x^n]^2 + 18*b^3*e*n*x*Log[c*x^n]^2 + 12*a*b^2*e^2*x ^2*Log[x]*Log[c*x^n]^2 + 6*b^3*e^2*n*x^2*Log[x]*Log[c*x^n]^2 - 6*b^3*e^2*n *x^2*Log[x]^2*Log[c*x^n]^2 + 4*b^3*e*x*Log[c*x^n]^3 + 4*b^3*e^2*x^2*Log[x] *Log[c*x^n]^3 + 4*a^3*Log[1 + e*x] + 6*a^2*b*n*Log[1 + e*x] + 6*a*b^2*n^2* Log[1 + e*x] + 3*b^3*n^3*Log[1 + e*x] - 4*a^3*e^2*x^2*Log[1 + e*x] - 6*a^2 *b*e^2*n*x^2*Log[1 + e*x] - 6*a*b^2*e^2*n^2*x^2*Log[1 + e*x] - 3*b^3*e^2*n ^3*x^2*Log[1 + e*x] + 12*a^2*b*Log[c*x^n]*Log[1 + e*x] + 12*a*b^2*n*Log[c* x^n]*Log[1 + e*x] + 6*b^3*n^2*Log[c*x^n]*Log[1 + e*x] - 12*a^2*b*e^2*x^2*L og[c*x^n]*Log[1 + e*x] - 12*a*b^2*e^2*n*x^2*Log[c*x^n]*Log[1 + e*x] - 6*b^ 3*e^2*n^2*x^2*Log[c*x^n]*Log[1 + e*x] + 12*a*b^2*Log[c*x^n]^2*Log[1 + e*x] + 6*b^3*n*Log[c*x^n]^2*Log[1 + e*x] - 12*a*b^2*e^2*x^2*Log[c*x^n]^2*Lo...
Time = 1.02 (sec) , antiderivative size = 448, 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.
\(\displaystyle \int \frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx\) |
\(\Big \downarrow \) 2825 |
\(\displaystyle -e \int \left (-\frac {3 b^3 n^3}{8 x^2 (e x+1)}-\frac {3 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{4 x^2 (e x+1)}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 n}{4 x^2 (e x+1)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2 (e x+1)}\right )dx-\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log (e x+1)}{8 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-e \left (\frac {3}{2} b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+3 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b^2 e n^2 \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {21 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}+\frac {3}{2} b e n \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b e n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} e \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {9 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x}+\frac {3}{4} b^3 e n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+\frac {3}{2} b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )+3 b^3 e n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right )+\frac {3}{8} b^3 e n^3 \log (x)-\frac {3}{8} b^3 e n^3 \log (e x+1)+\frac {45 b^3 n^3}{8 x}\right )-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log (e x+1)}{8 x^2}\) |
(-3*b^3*n^3*Log[1 + e*x])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(2*x^2) - e*((45*b^3*n^3)/(8*x) + (3*b^3*e*n ^3*Log[x])/8 + (21*b^2*n^2*(a + b*Log[c*x^n]))/(4*x) - (3*b^2*e*n^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n]))/4 + (9*b*n*(a + b*Log[c*x^n])^2)/(4*x) - (3 *b*e*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2)/4 + (a + b*Log[c*x^n])^3/(2* x) - (e*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^3)/2 - (3*b^3*e*n^3*Log[1 + e* x])/8 + (3*b^3*e*n^3*PolyLog[2, -(1/(e*x))])/4 + (3*b^2*e*n^2*(a + b*Log[c *x^n])*PolyLog[2, -(1/(e*x))])/2 + (3*b*e*n*(a + b*Log[c*x^n])^2*PolyLog[2 , -(1/(e*x))])/2 + (3*b^3*e*n^3*PolyLog[3, -(1/(e*x))])/2 + 3*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] + 3*b^3*e*n^3*PolyLog[4, -(1/(e*x) )])
3.1.23.3.1 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 = 12.34 (sec) , antiderivative size = 1329, normalized size of antiderivative = 2.83
-3/4*b^3*n/x^2*ln(e*x+1)*ln(x^n)^2+3/4*b^3*n^2*e^2*ln(e*x+1)*ln(x^n)-3/4*b ^3*n^2/x^2*ln(e*x+1)*ln(x^n)+3/2*b^3*n^2*e^2*polylog(2,-e*x)*ln(x^n)-3*b^3 *n^2*e^2*polylog(3,-e*x)*ln(x^n)-3/4*b^3*n^2*e^2*ln(x)*ln(x^n)-3/4*b^3*n*e ^2*ln(x)*ln(x^n)^2+3/4*b^3*n*e^2*ln(e*x+1)*ln(x^n)^2-3/4*b^3*n*e^2*ln(x)^2 *ln(x^n)^2+1/2*b^3*e^2*ln(e*x)*ln(x)^3*n^3+3*b^3*n^3*polylog(4,-e*x)*e^2+3 /8*b^3*n^3*e^2*ln(x)^2+3/2*b^3*n*e^2*polylog(2,-e*x)*ln(x^n)^2+3/4*b^3*e^2 *ln(x)^2*ln(x^n)*n^2+b^3*e^2*ln(x)^3*ln(x^n)*n^2+3/2*(-I*b*Pi*csgn(I*c)*cs gn(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^2*((ln(x^n)-n*ln (x))^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)+n^ 2*((-1/4-1/2*ln(x)-1/2*ln(x)^2)/x^2*ln(e*x+1)-7/4*e/x-1/4*e^2*ln(x)+1/4*e^ 2*ln(e*x+1)-3/2*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)-1/2*e*ln(x)^2/x-1/6*e^2*ln(x)^3+1/2*e^2*ln(e*x+1)*ln(x)^2+ e^2*ln(x)*polylog(2,-e*x)-e^2*polylog(3,-e*x))+2*n*(ln(x^n)-n*ln(x))*((-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*l n(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 /8*(-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)^3*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)-1 /2*ln(x^n)^3/x^2*ln(e*x+1)*b^3-1/4*b^3*n^3*e^2*ln(x)^3-3/8*b^3*n^3*e^2*...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{3}} \,d x } \]
integral((b^3*log(c*x^n)^3*log(e*x + 1) + 3*a*b^2*log(c*x^n)^2*log(e*x + 1 ) + 3*a^2*b*log(c*x^n)*log(e*x + 1) + a^3*log(e*x + 1))/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{3}} \,d x } \]
-1/2*(b^3*e^2*x^2*log(x) + b^3*e*x - (b^3*e^2*x^2 - b^3)*log(e*x + 1))*log (x^n)^3/x^2 - 1/2*integrate(-(6*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*lo g(e*x + 1)*log(x^n) + 3*(b^3*e^2*n*x^2*log(x) + b^3*e*n*x - (b^3*e^2*n*x^2 - b^3*(n + 2*log(c)) - 2*a*b^2)*log(e*x + 1))*log(x^n)^2 + 2*(b^3*log(c)^ 3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + a^3)*log(e*x + 1))/x^3, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=\int \frac {\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]