Integrand size = 16, antiderivative size = 100 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3 x^2}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )} \]
2*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-2*(a+b*ln(c*x^n))/b/n)/b^3/n^3/x^2-1/2/b/ n/x^2/(a+b*ln(c*x^n))^2+1/b^2/n^2/x^2/(a+b*ln(c*x^n))
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {4 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {b n \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}}{2 b^3 n^3 x^2} \]
(4*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[(-2*(a + b*Log[c*x^n]))/(b* n)] + (b*n*(2*a - b*n + 2*b*Log[c*x^n]))/(a + b*Log[c*x^n])^2)/(2*b^3*n^3* x^2)
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2743, 2743, 2747, 2609}
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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle -\frac {\int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2}dx}{b n}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )}dx}{b n}-\frac {1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle -\frac {-\frac {2 \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-2/n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{b n^2 x^2}-\frac {1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle -\frac {-\frac {2 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac {1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {1}{2 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}\) |
-1/2*1/(b*n*x^2*(a + b*Log[c*x^n])^2) - ((-2*E^((2*a)/(b*n))*(c*x^n)^(2/n) *ExpIntegralEi[(-2*(a + b*Log[c*x^n]))/(b*n)])/(b^2*n^2*x^2) - 1/(b*n*x^2* (a + b*Log[c*x^n])))/(b*n)
3.1.87.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^(p + 1)/(b*d*n*(p + 1))), x] - Simp[(m + 1)/(b*n*(p + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)) Subst[Int[E^(((m + 1)/n )*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.62 (sec) , antiderivative size = 454, normalized size of antiderivative = 4.54
| method | result | size |
| risch | \(\frac {-2 b n +4 a +4 b \ln \left (c \right )+4 \ln \left (x^{n}\right ) b -2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{{\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 \ln \left (x^{n}\right ) b +2 a \right )}^{2} b^{2} n^{2} x^{2}}-\frac {2 c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {-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 a}{b n}} \operatorname {Ei}_{1}\left (2 \ln \left (x \right )+\frac {-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 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{b n}\right )}{b^{3} n^{3} x^{2}}\) | \(454\) |
2*(-b*n+2*a+2*b*ln(c)+2*ln(x^n)*b-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)/(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*c sgn(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*ln(x^n)*b+2*a)^2/b^2/n^2/x^2-2/b^3/n^3/x^2*c^(2/n)*(x ^n)^(2/n)*exp((-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*a)/b/n)*Ei(1,2*ln(x)+(-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*csg n(I*c*x^n)^3+2*b*ln(c)+2*b*(ln(x^n)-n*ln(x))+2*a)/b/n)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (97) = 194\).
Time = 0.33 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 \, b^{2} n^{2} \log \left (x\right ) - b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n + 4 \, {\left (b^{2} n^{2} x^{2} \log \left (x\right )^{2} + b^{2} x^{2} \log \left (c\right )^{2} + 2 \, a b x^{2} \log \left (c\right ) + a^{2} x^{2} + 2 \, {\left (b^{2} n x^{2} \log \left (c\right ) + a b n x^{2}\right )} \log \left (x\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{2}}\right )}{2 \, {\left (b^{5} n^{5} x^{2} \log \left (x\right )^{2} + b^{5} n^{3} x^{2} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} x^{2} \log \left (c\right ) + a^{2} b^{3} n^{3} x^{2} + 2 \, {\left (b^{5} n^{4} x^{2} \log \left (c\right ) + a b^{4} n^{4} x^{2}\right )} \log \left (x\right )\right )}} \]
1/2*(2*b^2*n^2*log(x) - b^2*n^2 + 2*b^2*n*log(c) + 2*a*b*n + 4*(b^2*n^2*x^ 2*log(x)^2 + b^2*x^2*log(c)^2 + 2*a*b*x^2*log(c) + a^2*x^2 + 2*(b^2*n*x^2* log(c) + a*b*n*x^2)*log(x))*e^(2*(b*log(c) + a)/(b*n))*log_integral(e^(-2* (b*log(c) + a)/(b*n))/x^2))/(b^5*n^5*x^2*log(x)^2 + b^5*n^3*x^2*log(c)^2 + 2*a*b^4*n^3*x^2*log(c) + a^2*b^3*n^3*x^2 + 2*(b^5*n^4*x^2*log(c) + a*b^4* n^4*x^2)*log(x))
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3}} \,d x } \]
-1/2*(b*(n - 2*log(c)) - 2*b*log(x^n) - 2*a)/(b^4*n^2*x^2*log(x^n)^2 + 2*( b^4*n^2*log(c) + a*b^3*n^2)*x^2*log(x^n) + (b^4*n^2*log(c)^2 + 2*a*b^3*n^2 *log(c) + a^2*b^2*n^2)*x^2) + 2*integrate(1/(b^3*n^2*x^3*log(x^n) + (b^3*n ^2*log(c) + a*b^2*n^2)*x^3), x)
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]