Integrand size = 23, antiderivative size = 151 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^2} \]
-e*x*(a+b*ln(c*x^n))^2/d^2/(e*x+d)-ln(1+d/e/x)*(a+b*ln(c*x^n))^2/d^2+2*b*n *(a+b*ln(c*x^n))*ln(1+e*x/d)/d^2+2*b*n*(a+b*ln(c*x^n))*polylog(2,-d/e/x)/d ^2+2*b^2*n^2*polylog(2,-e*x/d)/d^2+2*b^2*n^2*polylog(3,-d/e/x)/d^2
Time = 0.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\frac {-3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{b n}+6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{3 d^2} \]
(-3*(a + b*Log[c*x^n])^2 + (3*d*(a + b*Log[c*x^n])^2)/(d + e*x) + (a + b*L og[c*x^n])^3/(b*n) + 6*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 3*(a + b* Log[c*x^n])^2*Log[1 + (e*x)/d] + 6*b^2*n^2*PolyLog[2, -((e*x)/d)] - 6*b*n* (a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 6*b^2*n^2*PolyLog[3, -((e*x)/d )])/(3*d^2)
Time = 0.85 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2789, 2755, 2754, 2779, 2821, 2838, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}dx}{d}\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{d+e x}dx}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {2 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{d}\right )}{d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{d}\right )}{d}\) |
-((e*((x*(a + b*Log[c*x^n])^2)/(d*(d + e*x)) - (2*b*n*(((a + b*Log[c*x^n]) *Log[1 + (e*x)/d])/e + (b*n*PolyLog[2, -((e*x)/d)])/e))/d))/d) + (-((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d) + (2*b*n*((a + b*Log[c*x^n])*PolyLog[ 2, -(d/(e*x))] + b*n*PolyLog[3, -(d/(e*x))]))/d)/d
3.2.4.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.43 (sec) , antiderivative size = 683, normalized size of antiderivative = 4.52
| method | result | size |
| risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{d^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2}}{d \left (e x +d \right )}+\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (x \right )}{d^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}+\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{d^{2}}-\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d^{2}}+\frac {b^{2} \ln \left (x \right )^{3} n^{2}}{3 d^{2}}-\frac {2 b^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d^{2}}-\frac {2 b^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d^{2}}-\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{2}}-\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d^{2}}+\frac {2 b^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d^{2}}+\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 (-\frac {\ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{2}}+\frac {\ln \left (x^{n}\right )}{d \left (e x +d \right )}+\frac {\ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}-n \left (\frac {\ln \left (x \right )^{2}}{2 d^{2}}-\frac {\ln \left (e x +d \right )}{d^{2}}+\frac {\ln \left (x \right )}{d^{2}}-\frac {\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{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} \left (-\frac {\ln \left (e x +d \right )}{d^{2}}+\frac {1}{d \left (e x +d \right )}+\frac {\ln \left (x \right )}{d^{2}}\right )}{4}\) | \(683\) |
-b^2*ln(x^n)^2/d^2*ln(e*x+d)+b^2*ln(x^n)^2/d/(e*x+d)+b^2*ln(x^n)^2/d^2*ln( x)+2*b^2*n*ln(x^n)/d^2*ln(e*x+d)-2*b^2*n*ln(x^n)/d^2*ln(x)+b^2/d^2*n^2*ln( x)^2-2*b^2/d^2*n^2*ln(e*x+d)*ln(-e*x/d)-2*b^2/d^2*n^2*dilog(-e*x/d)-b^2*n/ d^2*ln(x^n)*ln(x)^2+1/3*b^2/d^2*ln(x)^3*n^2-2*b^2/d^2*ln(x)*ln(e*x+d)*ln(- e*x/d)*n^2-2*b^2/d^2*ln(x)*dilog(-e*x/d)*n^2+2*b^2*n/d^2*ln(x^n)*ln(e*x+d) *ln(-e*x/d)+2*b^2*n/d^2*ln(x^n)*dilog(-e*x/d)+b^2/d^2*n^2*ln(e*x+d)*ln(x)^ 2-b^2/d^2*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2/d^2*n^2*ln(x)*polylog(2,-e*x/d)+2* b^2/d^2*n^2*polylog(3,-e*x/d)+(-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*P i*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(-ln(x^n)/d^2*ln(e*x+d)+ln(x^n)/d/(e*x+ d)+ln(x^n)/d^2*ln(x)-n*(1/2/d^2*ln(x)^2-1/d^2*ln(e*x+d)+1/d^2*ln(x)-1/d^2* ln(e*x+d)*ln(-e*x/d)-1/d^2*dilog(-e*x/d)))+1/4*(-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*(-1/d^2*ln(e*x+d)+1/d/ (e*x+d)+1/d^2*ln(x))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x \left (d + e x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x} \,d x } \]
a^2*(1/(d*e*x + d^2) - log(e*x + d)/d^2 + log(x)/d^2) + integrate((b^2*log (c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e^ 2*x^3 + 2*d*e*x^2 + d^2*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x\,{\left (d+e\,x\right )}^2} \,d x \]