Integrand size = 16, antiderivative size = 105 \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {9 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 n^3 x^3}-\frac {1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}+\frac {3}{2 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )} \]
9/2*exp(3*a/b/n)*(c*x^n)^(3/n)*Ei(-3*(a+b*ln(c*x^n))/b/n)/b^3/n^3/x^3-1/2/ b/n/x^3/(a+b*ln(c*x^n))^2+3/2/b^2/n^2/x^3/(a+b*ln(c*x^n))
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {9 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {b n \left (3 a-b n+3 b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}}{2 b^3 n^3 x^3} \]
(9*E^((3*a)/(b*n))*(c*x^n)^(3/n)*ExpIntegralEi[(-3*(a + b*Log[c*x^n]))/(b* n)] + (b*n*(3*a - b*n + 3*b*Log[c*x^n]))/(a + b*Log[c*x^n])^2)/(2*b^3*n^3* x^3)
Time = 0.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.07, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle -\frac {3 \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2}dx}{2 b n}-\frac {1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle -\frac {3 \left (-\frac {3 \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )}dx}{b n}-\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}-\frac {1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle -\frac {3 \left (-\frac {3 \left (c x^n\right )^{3/n} \int \frac {\left (c x^n\right )^{-3/n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{b n^2 x^3}-\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}-\frac {1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle -\frac {3 \left (-\frac {3 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^3}-\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}-\frac {1}{2 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}\) |
-1/2*1/(b*n*x^3*(a + b*Log[c*x^n])^2) - (3*((-3*E^((3*a)/(b*n))*(c*x^n)^(3 /n)*ExpIntegralEi[(-3*(a + b*Log[c*x^n]))/(b*n)])/(b^2*n^2*x^3) - 1/(b*n*x ^3*(a + b*Log[c*x^n]))))/(2*b*n)
3.1.88.3.1 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 = 0.85 (sec) , antiderivative size = 455, normalized size of antiderivative = 4.33
| method | result | size |
| risch | \(\frac {6 a +6 b \ln \left (c \right )+6 \ln \left (x^{n}\right ) b -3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 b n}{b^{2} n^{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 \ln \left (x^{n}\right ) b +2 a \right )}^{2} x^{3}}-\frac {9 c^{\frac {3}{n}} \left (x^{n}\right )^{\frac {3}{n}} {\mathrm e}^{\frac {-\frac {3 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}+\frac {3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+3 a}{b n}} \operatorname {Ei}_{1}\left (3 \ln \left (x \right )+\frac {-\frac {3 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}+\frac {3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+3 b \ln \left (c \right )+3 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+3 a}{b n}\right )}{2 b^{3} n^{3} x^{3}}\) | \(455\) |
(6*a+6*b*ln(c)+6*ln(x^n)*b-3*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3* I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I* b*Pi*csgn(I*c*x^n)^3-2*b*n)/b^2/n^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*ln(x^n)*b+2*a)^2/x^3-9/2/b^3/n^3/x^3*c ^(3/n)*(x^n)^(3/n)*exp(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*c sgn(I*c*x^n)^3+2*a)/b/n)*Ei(1,3*ln(x)+3/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*c sgn(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*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. \(2 (98) = 196\).
Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {3 \, b^{2} n^{2} \log \left (x\right ) - b^{2} n^{2} + 3 \, b^{2} n \log \left (c\right ) + 3 \, a b n + 9 \, {\left (b^{2} n^{2} x^{3} \log \left (x\right )^{2} + b^{2} x^{3} \log \left (c\right )^{2} + 2 \, a b x^{3} \log \left (c\right ) + a^{2} x^{3} + 2 \, {\left (b^{2} n x^{3} \log \left (c\right ) + a b n x^{3}\right )} \log \left (x\right )\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{3}}\right )}{2 \, {\left (b^{5} n^{5} x^{3} \log \left (x\right )^{2} + b^{5} n^{3} x^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} x^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} x^{3} + 2 \, {\left (b^{5} n^{4} x^{3} \log \left (c\right ) + a b^{4} n^{4} x^{3}\right )} \log \left (x\right )\right )}} \]
1/2*(3*b^2*n^2*log(x) - b^2*n^2 + 3*b^2*n*log(c) + 3*a*b*n + 9*(b^2*n^2*x^ 3*log(x)^2 + b^2*x^3*log(c)^2 + 2*a*b*x^3*log(c) + a^2*x^3 + 2*(b^2*n*x^3* log(c) + a*b*n*x^3)*log(x))*e^(3*(b*log(c) + a)/(b*n))*log_integral(e^(-3* (b*log(c) + a)/(b*n))/x^3))/(b^5*n^5*x^3*log(x)^2 + b^5*n^3*x^3*log(c)^2 + 2*a*b^4*n^3*x^3*log(c) + a^2*b^3*n^3*x^3 + 2*(b^5*n^4*x^3*log(c) + a*b^4* n^4*x^3)*log(x))
\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{4}} \,d x } \]
-1/2*(b*(n - 3*log(c)) - 3*b*log(x^n) - 3*a)/(b^4*n^2*x^3*log(x^n)^2 + 2*( b^4*n^2*log(c) + a*b^3*n^2)*x^3*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^3) + 9*integrate(1/2/(b^3*n^2*x^4*log(x^n) + (b^3 *n^2*log(c) + a*b^2*n^2)*x^4), x)
\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^4\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]