Integrand size = 22, antiderivative size = 342 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=6 b^3 e n^3 \log (x)-6 b^2 e n^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-3 b e n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-e \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 e n^3 \log (1+e x)-\frac {6 b^3 n^3 \log (1+e x)}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+6 b^3 e n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )+6 b^3 e n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right ) \]
6*b^3*e*n^3*ln(x)-6*b^2*e*n^2*ln(1+1/e/x)*(a+b*ln(c*x^n))-3*b*e*n*ln(1+1/e /x)*(a+b*ln(c*x^n))^2-e*ln(1+1/e/x)*(a+b*ln(c*x^n))^3-6*b^3*e*n^3*ln(e*x+1 )-6*b^3*n^3*ln(e*x+1)/x-6*b^2*n^2*(a+b*ln(c*x^n))*ln(e*x+1)/x-3*b*n*(a+b*l n(c*x^n))^2*ln(e*x+1)/x-(a+b*ln(c*x^n))^3*ln(e*x+1)/x+6*b^3*e*n^3*polylog( 2,-1/e/x)+6*b^2*e*n^2*(a+b*ln(c*x^n))*polylog(2,-1/e/x)+3*b*e*n*(a+b*ln(c* x^n))^2*polylog(2,-1/e/x)+6*b^3*e*n^3*polylog(3,-1/e/x)+6*b^2*e*n^2*(a+b*l n(c*x^n))*polylog(3,-1/e/x)+6*b^3*e*n^3*polylog(4,-1/e/x)
Leaf count is larger than twice the leaf count of optimal. \(770\) vs. \(2(342)=684\).
Time = 0.19 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=a^3 e \log (x)+3 a^2 b e n \log (x)+6 a b^2 e n^2 \log (x)+6 b^3 e n^3 \log (x)-\frac {3}{2} a^2 b e n \log ^2(x)-3 a b^2 e n^2 \log ^2(x)-3 b^3 e n^3 \log ^2(x)+a b^2 e n^2 \log ^3(x)+b^3 e n^3 \log ^3(x)-\frac {1}{4} b^3 e n^3 \log ^4(x)+3 a^2 b e \log (x) \log \left (c x^n\right )+6 a b^2 e n \log (x) \log \left (c x^n\right )+6 b^3 e n^2 \log (x) \log \left (c x^n\right )-3 a b^2 e n \log ^2(x) \log \left (c x^n\right )-3 b^3 e n^2 \log ^2(x) \log \left (c x^n\right )+b^3 e n^2 \log ^3(x) \log \left (c x^n\right )+3 a b^2 e \log (x) \log ^2\left (c x^n\right )+3 b^3 e n \log (x) \log ^2\left (c x^n\right )-\frac {3}{2} b^3 e n \log ^2(x) \log ^2\left (c x^n\right )+b^3 e \log (x) \log ^3\left (c x^n\right )-a^3 e \log (1+e x)-3 a^2 b e n \log (1+e x)-6 a b^2 e n^2 \log (1+e x)-6 b^3 e n^3 \log (1+e x)-\frac {a^3 \log (1+e x)}{x}-\frac {3 a^2 b n \log (1+e x)}{x}-\frac {6 a b^2 n^2 \log (1+e x)}{x}-\frac {6 b^3 n^3 \log (1+e x)}{x}-3 a^2 b e \log \left (c x^n\right ) \log (1+e x)-6 a b^2 e n \log \left (c x^n\right ) \log (1+e x)-6 b^3 e n^2 \log \left (c x^n\right ) \log (1+e x)-\frac {3 a^2 b \log \left (c x^n\right ) \log (1+e x)}{x}-\frac {6 a b^2 n \log \left (c x^n\right ) \log (1+e x)}{x}-\frac {6 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)}{x}-3 a b^2 e \log ^2\left (c x^n\right ) \log (1+e x)-3 b^3 e n \log ^2\left (c x^n\right ) \log (1+e x)-\frac {3 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)}{x}-\frac {3 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)}{x}-b^3 e \log ^3\left (c x^n\right ) \log (1+e x)-\frac {b^3 \log ^3\left (c x^n\right ) \log (1+e x)}{x}-3 b e n \left (a^2+2 a b n+2 b^2 n^2+2 b (a+b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)+6 b^2 e n^2 \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,-e x)-6 b^3 e n^3 \operatorname {PolyLog}(4,-e x) \]
a^3*e*Log[x] + 3*a^2*b*e*n*Log[x] + 6*a*b^2*e*n^2*Log[x] + 6*b^3*e*n^3*Log [x] - (3*a^2*b*e*n*Log[x]^2)/2 - 3*a*b^2*e*n^2*Log[x]^2 - 3*b^3*e*n^3*Log[ x]^2 + a*b^2*e*n^2*Log[x]^3 + b^3*e*n^3*Log[x]^3 - (b^3*e*n^3*Log[x]^4)/4 + 3*a^2*b*e*Log[x]*Log[c*x^n] + 6*a*b^2*e*n*Log[x]*Log[c*x^n] + 6*b^3*e*n^ 2*Log[x]*Log[c*x^n] - 3*a*b^2*e*n*Log[x]^2*Log[c*x^n] - 3*b^3*e*n^2*Log[x] ^2*Log[c*x^n] + b^3*e*n^2*Log[x]^3*Log[c*x^n] + 3*a*b^2*e*Log[x]*Log[c*x^n ]^2 + 3*b^3*e*n*Log[x]*Log[c*x^n]^2 - (3*b^3*e*n*Log[x]^2*Log[c*x^n]^2)/2 + b^3*e*Log[x]*Log[c*x^n]^3 - a^3*e*Log[1 + e*x] - 3*a^2*b*e*n*Log[1 + e*x ] - 6*a*b^2*e*n^2*Log[1 + e*x] - 6*b^3*e*n^3*Log[1 + e*x] - (a^3*Log[1 + e *x])/x - (3*a^2*b*n*Log[1 + e*x])/x - (6*a*b^2*n^2*Log[1 + e*x])/x - (6*b^ 3*n^3*Log[1 + e*x])/x - 3*a^2*b*e*Log[c*x^n]*Log[1 + e*x] - 6*a*b^2*e*n*Lo g[c*x^n]*Log[1 + e*x] - 6*b^3*e*n^2*Log[c*x^n]*Log[1 + e*x] - (3*a^2*b*Log [c*x^n]*Log[1 + e*x])/x - (6*a*b^2*n*Log[c*x^n]*Log[1 + e*x])/x - (6*b^3*n ^2*Log[c*x^n]*Log[1 + e*x])/x - 3*a*b^2*e*Log[c*x^n]^2*Log[1 + e*x] - 3*b^ 3*e*n*Log[c*x^n]^2*Log[1 + e*x] - (3*a*b^2*Log[c*x^n]^2*Log[1 + e*x])/x - (3*b^3*n*Log[c*x^n]^2*Log[1 + e*x])/x - b^3*e*Log[c*x^n]^3*Log[1 + e*x] - (b^3*Log[c*x^n]^3*Log[1 + e*x])/x - 3*b*e*n*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2 *b*(a + b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog[2, -(e*x)] + 6*b^2*e*n ^2*(a + b*n + b*Log[c*x^n])*PolyLog[3, -(e*x)] - 6*b^3*e*n^3*PolyLog[4, -( e*x)]
Time = 0.73 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -e \int \left (-\frac {6 b^3 n^3}{x (e x+1)}-\frac {6 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{x (e x+1)}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 n}{x (e x+1)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (e x+1)}\right )dx-\frac {6 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {6 b^3 n^3 \log (e x+1)}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-e \left (-6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-3 b n \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+3 b n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )-6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right )+6 b^3 n^3 \log (e x+1)-6 b^3 n^3 \log (x)\right )-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {6 b^3 n^3 \log (e x+1)}{x}\) |
(-6*b^3*n^3*Log[1 + e*x])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/ x - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/x - ((a + b*Log[c*x^n])^3*Lo g[1 + e*x])/x - e*(-6*b^3*n^3*Log[x] + 6*b^2*n^2*Log[1 + 1/(e*x)]*(a + b*L og[c*x^n]) + 3*b*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2 + Log[1 + 1/(e*x) ]*(a + b*Log[c*x^n])^3 + 6*b^3*n^3*Log[1 + e*x] - 6*b^3*n^3*PolyLog[2, -(1 /(e*x))] - 6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))] - 3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(e*x))] - 6*b^3*n^3*PolyLog[3, -(1/(e*x)) ] - 6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] - 6*b^3*n^3*PolyLo g[4, -(1/(e*x))])
3.1.22.3.1 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 = 12.70 (sec) , antiderivative size = 1080, normalized size of antiderivative = 3.16
3/4*(-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)^2*b*((ln(x^n)-n*ln(x))*e*(ln(e*x)-ln(e*x+1)/x/e*(e*x+1))+n*((-1-ln( x))/x*ln(e*x+1)+e*ln(x)-ln(e*x+1)*e+1/2*e*ln(x)^2-e*ln(e*x+1)*ln(x)-e*poly log(2,-e*x)))-ln(x^n)^3/x*ln(e*x+1)*b^3+3/2*(-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^2*((ln(x^n)-n*ln(x))^2*e* (ln(e*x)-ln(e*x+1)/x/e*(e*x+1))+n^2*((-ln(x)^2-2*ln(x)-2)/x*ln(e*x+1)+2*e* ln(x)-2*ln(e*x+1)*e+e*ln(x)^2-2*e*ln(e*x+1)*ln(x)-2*e*polylog(2,-e*x)+1/3* e*ln(x)^3-e*ln(e*x+1)*ln(x)^2-2*e*ln(x)*polylog(2,-e*x)+2*e*polylog(3,-e*x ))+2*n*(ln(x^n)-n*ln(x))*((-1-ln(x))/x*ln(e*x+1)+e*ln(x)-ln(e*x+1)*e+1/2*e *ln(x)^2-e*ln(e*x+1)*ln(x)-e*polylog(2,-e*x)))-3*b^3*n^3*e*ln(x)^2-6*b^3*n ^3*e*polylog(2,-e*x)+b^3*e*ln(e*x)*ln(x^n)^3-b^3*e*ln(e*x+1)*ln(x^n)^3+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*(ln(e*x)-ln(e*x+1)/x/e*(e*x+1))+b^3*n^3*e*ln(x)^3+6*b^3*n^3*e*poly log(3,-e*x)+3/4*b^3*n^3*e*ln(x)^4-6*b^3*n^3*e*polylog(4,-e*x)-b^3*e*ln(e*x )*ln(x)^3*n^3-3*b^3*n/x*ln(e*x+1)*ln(x^n)^2-3*b^3*e*ln(x)^2*ln(x^n)*n^2+3* b^3*n*e*ln(x)*ln(x^n)^2-3*b^3*n*ln(e*x+1)*e*ln(x^n)^2-2*b^3*e*ln(x)^3*ln(x ^n)*n^2+3/2*b^3*n*e*ln(x)^2*ln(x^n)^2-3*b^3*n*e*polylog(2,-e*x)*ln(x^n)...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{2}} \,d x } \]
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^2, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3} \log {\left (e x + 1 \right )}}{x^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{2}} \,d x } \]
(b^3*e*x*log(x) - (b^3*e*x + b^3)*log(e*x + 1))*log(x^n)^3/x + integrate(( 3*(b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(e*x + 1)*log(x^n) - 3*(b^3*e *n*x*log(x) - (b^3*e*n*x + b^3*(n + log(c)) + a*b^2)*log(e*x + 1))*log(x^n )^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^2, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int \frac {\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \]