Integrand size = 18, antiderivative size = 86 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx=\frac {2 e p \log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 d x^3}+\frac {2 e p^2 \operatorname {PolyLog}\left (2,1+\frac {e x^3}{d}\right )}{3 d} \]
2/3*e*p*ln(-e*x^3/d)*ln(c*(e*x^3+d)^p)/d-1/3*(e*x^3+d)*ln(c*(e*x^3+d)^p)^2 /d/x^3+2/3*e*p^2*polylog(2,1+e*x^3/d)/d
Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx=\frac {2 e p \log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 d}-\frac {e \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 d}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{3 x^3}+\frac {2 e p^2 \operatorname {PolyLog}\left (2,\frac {d+e x^3}{d}\right )}{3 d} \]
(2*e*p*Log[-((e*x^3)/d)]*Log[c*(d + e*x^3)^p])/(3*d) - (e*Log[c*(d + e*x^3 )^p]^2)/(3*d) - Log[c*(d + e*x^3)^p]^2/(3*x^3) + (2*e*p^2*PolyLog[2, (d + e*x^3)/d])/(3*d)
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2904, 2844, 2841, 2752}
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.
\(\displaystyle \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {1}{3} \int \frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{x^6}dx^3\) |
\(\Big \downarrow \) 2844 |
\(\displaystyle \frac {1}{3} \left (\frac {2 e p \int \frac {\log \left (c \left (e x^3+d\right )^p\right )}{x^3}dx^3}{d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{d x^3}\right )\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle \frac {1}{3} \left (\frac {2 e p \left (\log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )-e p \int \frac {\log \left (-\frac {e x^3}{d}\right )}{e x^3+d}dx^3\right )}{d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{d x^3}\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {1}{3} \left (\frac {2 e p \left (\log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {e x^3}{d}+1\right )\right )}{d}-\frac {\left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{d x^3}\right )\) |
(-(((d + e*x^3)*Log[c*(d + e*x^3)^p]^2)/(d*x^3)) + (2*e*p*(Log[-((e*x^3)/d )]*Log[c*(d + e*x^3)^p] + p*PolyLog[2, 1 + (e*x^3)/d]))/d)/3
3.2.32.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ )), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x )^n])/g), x] - Simp[b*e*(n/g) Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_))^2, x_Symbol] :> Simp[(d + e*x)*((a + b*Log[c*(d + e*x)^n])^p/((e*f - d*g)*(f + g*x))), x] - Simp[b*e*n*(p/(e*f - d*g)) Int[(a + b*Log[c*(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] & & NeQ[e*f - d*g, 0] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.82 (sec) , antiderivative size = 411, normalized size of antiderivative = 4.78
method | result | size |
risch | \(-\frac {{\ln \left (\left (e \,x^{3}+d \right )^{p}\right )}^{2}}{3 x^{3}}+\frac {2 p e \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \ln \left (x \right )}{d}-\frac {2 p e \ln \left (\left (e \,x^{3}+d \right )^{p}\right ) \ln \left (e \,x^{3}+d \right )}{3 d}-\frac {2 p^{2} e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e +d \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{d}+\frac {p^{2} e \ln \left (e \,x^{3}+d \right )^{2}}{3 d}+\left (i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (\left (e \,x^{3}+d \right )^{p}\right )}{3 x^{3}}+p e \left (\frac {\ln \left (x \right )}{d}-\frac {\ln \left (e \,x^{3}+d \right )}{3 d}\right )\right )-\frac {{\left (i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2}}{12 x^{3}}\) | \(411\) |
-1/3*ln((e*x^3+d)^p)^2/x^3+2*p*e*ln((e*x^3+d)^p)/d*ln(x)-2/3*p*e*ln((e*x^3 +d)^p)/d*ln(e*x^3+d)-2*p^2*e/d*sum(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1 ),_R1=RootOf(_Z^3*e+d))+1/3*p^2*e/d*ln(e*x^3+d)^2+(I*Pi*csgn(I*(e*x^3+d)^p )*csgn(I*c*(e*x^3+d)^p)^2-I*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+d)^p)*c sgn(I*c)-I*Pi*csgn(I*c*(e*x^3+d)^p)^3+I*Pi*csgn(I*c*(e*x^3+d)^p)^2*csgn(I* c)+2*ln(c))*(-1/3*ln((e*x^3+d)^p)/x^3+p*e*(1/d*ln(x)-1/3/d*ln(e*x^3+d)))-1 /12*(I*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+d)^p)^2-I*Pi*csgn(I*(e*x^3+d )^p)*csgn(I*c*(e*x^3+d)^p)*csgn(I*c)-I*Pi*csgn(I*c*(e*x^3+d)^p)^3+I*Pi*csg n(I*c*(e*x^3+d)^p)^2*csgn(I*c)+2*ln(c))^2/x^3
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx=\int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{4}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.37 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx=\frac {1}{3} \, e^{2} p^{2} {\left (\frac {\log \left (e x^{3} + d\right )^{2}}{d e} - \frac {2 \, {\left (3 \, \log \left (\frac {e x^{3}}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x^{3}}{d}\right )\right )}}{d e}\right )} - \frac {2}{3} \, e p {\left (\frac {\log \left (e x^{3} + d\right )}{d} - \frac {\log \left (x^{3}\right )}{d}\right )} \log \left ({\left (e x^{3} + d\right )}^{p} c\right ) - \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{3 \, x^{3}} \]
1/3*e^2*p^2*(log(e*x^3 + d)^2/(d*e) - 2*(3*log(e*x^3/d + 1)*log(x) + dilog (-e*x^3/d))/(d*e)) - 2/3*e*p*(log(e*x^3 + d)/d - log(x^3)/d)*log((e*x^3 + d)^p*c) - 1/3*log((e*x^3 + d)^p*c)^2/x^3
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^4} \, dx=\int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^4} \,d x \]