Integrand size = 16, antiderivative size = 102 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3 x}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \]
1/2*exp(a/b/n)*(c*x^n)^(1/n)*Ei((-a-b*ln(c*x^n))/b/n)/b^3/n^3/x-1/2/b/n/x/ (a+b*ln(c*x^n))^2+1/2/b^2/n^2/x/(a+b*ln(c*x^n))
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^2+b n \left (a-b n+b \log \left (c x^n\right )\right )}{2 b^3 n^3 x \left (a+b \log \left (c x^n\right )\right )^2} \]
(E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))]*(a + b*Log[c*x^n])^2 + b*n*(a - b*n + b*Log[c*x^n]))/(2*b^3*n^3*x*(a + b*Log[ c*x^n])^2)
Time = 0.36 (sec) , antiderivative size = 109, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle -\frac {\int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2}dx}{2 b n}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle -\frac {-\frac {\int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )}dx}{b n}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )}}{2 b n}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle -\frac {-\frac {\left (c x^n\right )^{\frac {1}{n}} \int \frac {\left (c x^n\right )^{-1/n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{b n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )}}{2 b n}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle -\frac {-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )}}{2 b n}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}\) |
-1/2*1/(b*n*x*(a + b*Log[c*x^n])^2) - (-((E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIn tegralEi[-((a + b*Log[c*x^n])/(b*n))])/(b^2*n^2*x)) - 1/(b*n*x*(a + b*Log[ c*x^n])))/(2*b*n)
3.1.86.3.1 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 = 0.68 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.40
| method | result | size |
| risch | \(\frac {-2 b n +2 a +2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b -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}}{{\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}-\frac {c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{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}{2 b n}} \operatorname {Ei}_{1}\left (\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}{2 b n}\right )}{2 b^{3} n^{3} x}\) | \(449\) |
(-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-1/2/b^3/n^3/x*c^(1/n)*(x^n )^(1/n)*exp(1/2*(-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*a)/b/n)*Ei(1,ln(x)+1/2*(-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*b*(ln(x^n)-n*ln(x))+2*a)/b/n)
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (95) = 190\).
Time = 0.31 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.88 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {b^{2} n^{2} \log \left (x\right ) - b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n + {\left (b^{2} n^{2} x \log \left (x\right )^{2} + b^{2} x \log \left (c\right )^{2} + 2 \, a b x \log \left (c\right ) + a^{2} x + 2 \, {\left (b^{2} n x \log \left (c\right ) + a b n x\right )} \log \left (x\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{x}\right )}{2 \, {\left (b^{5} n^{5} x \log \left (x\right )^{2} + b^{5} n^{3} x \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} x \log \left (c\right ) + a^{2} b^{3} n^{3} x + 2 \, {\left (b^{5} n^{4} x \log \left (c\right ) + a b^{4} n^{4} x\right )} \log \left (x\right )\right )}} \]
1/2*(b^2*n^2*log(x) - b^2*n^2 + b^2*n*log(c) + a*b*n + (b^2*n^2*x*log(x)^2 + b^2*x*log(c)^2 + 2*a*b*x*log(c) + a^2*x + 2*(b^2*n*x*log(c) + a*b*n*x)* log(x))*e^((b*log(c) + a)/(b*n))*log_integral(e^(-(b*log(c) + a)/(b*n))/x) )/(b^5*n^5*x*log(x)^2 + b^5*n^3*x*log(c)^2 + 2*a*b^4*n^3*x*log(c) + a^2*b^ 3*n^3*x + 2*(b^5*n^4*x*log(c) + a*b^4*n^4*x)*log(x))
\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2}} \,d x } \]
-1/2*(b*(n - log(c)) - b*log(x^n) - a)/(b^4*n^2*x*log(x^n)^2 + 2*(b^4*n^2* log(c) + a*b^3*n^2)*x*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) + integrate(1/2/(b^3*n^2*x^2*log(x^n) + (b^3*n^2*log(c) + a*b^2*n^2)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]