Integrand size = 16, antiderivative size = 101 \[ \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {8 e^{-\frac {4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac {x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {2 x^4}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )} \]
8*x^4*Ei(4*(a+b*ln(c*x^n))/b/n)/b^3/exp(4*a/b/n)/n^3/((c*x^n)^(4/n))-1/2*x ^4/b/n/(a+b*ln(c*x^n))^2-2*x^4/b^2/n^2/(a+b*ln(c*x^n))
Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {x^4 \left (16 e^{-\frac {4 a}{b n}} \left (c x^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]
(x^4*((16*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(E^((4*a)/(b*n))*(c *x^n)^(4/n)) - (b*n*(4*a + b*n + 4*b*Log[c*x^n]))/(a + b*Log[c*x^n])^2))/( 2*b^3*n^3)
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09, 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.
| \(\displaystyle \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle \frac {2 \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^2}dx}{b n}-\frac {x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle \frac {2 \left (\frac {4 \int \frac {x^3}{a+b \log \left (c x^n\right )}dx}{b n}-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle \frac {2 \left (\frac {4 x^4 \left (c x^n\right )^{-4/n} \int \frac {\left (c x^n\right )^{4/n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{b n^2}-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {2 \left (\frac {4 x^4 e^{-\frac {4 a}{b n}} \left (c x^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {x^4}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
-1/2*x^4/(b*n*(a + b*Log[c*x^n])^2) + (2*((4*x^4*ExpIntegralEi[(4*(a + b*L og[c*x^n]))/(b*n)])/(b^2*E^((4*a)/(b*n))*n^2*(c*x^n)^(4/n)) - x^4/(b*n*(a + b*Log[c*x^n]))))/(b*n)
3.1.81.3.1 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 = 0.44 (sec) , antiderivative size = 473, normalized size of antiderivative = 4.68
| method | result | size |
| risch | \(-\frac {2 i \left (i x^{4} b n +2 \pi b \,x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-2 \pi b \,x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 \pi b \,x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 \pi b \,x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 i \ln \left (c \right ) b \,x^{4}+4 i b \,x^{4} \ln \left (x^{n}\right )+4 i a \,x^{4}\right )}{{\left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 i b \ln \left (c \right )+2 i b \ln \left (x^{n}\right )+2 i a \right )}^{2} b^{2} n^{2}}-\frac {8 x^{4} c^{-\frac {4}{n}} \left (x^{n}\right )^{-\frac {4}{n}} {\mathrm e}^{-\frac {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 a \right )}{b n}} \operatorname {Ei}_{1}\left (-4 \ln \left (x \right )+\frac {2 i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 i b \ln \left (c \right )+2 i b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 i a \right )}{b n}\right )}{b^{3} n^{3}}\) | \(473\) |
-2*I*(I*x^4*b*n+2*Pi*b*x^4*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-2*Pi*b*x^4* csgn(I*c)*csgn(I*c*x^n)^2-2*Pi*b*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2+2*Pi*b*x^ 4*csgn(I*c*x^n)^3+4*I*ln(c)*b*x^4+4*I*b*x^4*ln(x^n)+4*I*a*x^4)/(b*Pi*csgn( I*c)*csgn(I*x^n)*csgn(I*c*x^n)-b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-b*Pi*csgn(I* x^n)*csgn(I*c*x^n)^2+b*Pi*csgn(I*c*x^n)^3+2*I*b*ln(c)+2*I*b*ln(x^n)+2*I*a) ^2/b^2/n^2-8/b^3/n^3*x^4*c^(-4/n)*(x^n)^(-4/n)*exp(-2*(-I*b*Pi*csgn(I*c)*c sgn(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,-4*ln(x)+2*I*(b*P i*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-b*Pi* csgn(I*x^n)*csgn(I*c*x^n)^2+b*Pi*csgn(I*c*x^n)^3+2*I*b*ln(c)+2*I*b*(ln(x^n )-n*ln(x))+2*I*a)/b/n)
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (100) = 200\).
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.09 \[ \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {{\left ({\left (4 \, b^{2} n^{2} x^{4} \log \left (x\right ) + 4 \, b^{2} n x^{4} \log \left (c\right ) + {\left (b^{2} n^{2} + 4 \, a b n\right )} x^{4}\right )} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 16 \, {\left (b^{2} n^{2} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \operatorname {log\_integral}\left (x^{4} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \, {\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \]
-1/2*((4*b^2*n^2*x^4*log(x) + 4*b^2*n*x^4*log(c) + (b^2*n^2 + 4*a*b*n)*x^4 )*e^(4*(b*log(c) + a)/(b*n)) - 16*(b^2*n^2*log(x)^2 + b^2*log(c)^2 + 2*a*b *log(c) + a^2 + 2*(b^2*n*log(c) + a*b*n)*log(x))*log_integral(x^4*e^(4*(b* log(c) + a)/(b*n))))*e^(-4*(b*log(c) + a)/(b*n))/(b^5*n^5*log(x)^2 + b^5*n ^3*log(c)^2 + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3 + 2*(b^5*n^4*log(c) + a*b^4 *n^4)*log(x))
\[ \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x^{3}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
\[ \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {x^{3}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}} \,d x } \]
-1/2*(4*b*x^4*log(x^n) + (b*(n + 4*log(c)) + 4*a)*x^4)/(b^4*n^2*log(c)^2 + b^4*n^2*log(x^n)^2 + 2*a*b^3*n^2*log(c) + a^2*b^2*n^2 + 2*(b^4*n^2*log(c) + a*b^3*n^2)*log(x^n)) + 8*integrate(x^3/(b^3*n^2*log(c) + b^3*n^2*log(x^ n) + a*b^2*n^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 1029 vs. \(2 (100) = 200\).
Time = 0.36 (sec) , antiderivative size = 1029, normalized size of antiderivative = 10.19 \[ \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\text {Too large to display} \]
-2*b^2*n^2*x^4*log(x)/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^ 3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) - 1/2* b^2*n^2*x^4/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) - 2*b^2*n*x^4*lo g(c)/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a* b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) - 2*a*b*n*x^4/(b^5*n^5* log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) + 8*b^2*n^2*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))*log(x)^2/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c) *log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2 *b^3*n^3)*c^(4/n)) + 16*b^2*n*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4* a/(b*n))*log(c)*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5* n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(4 /n)) + 8*b^2*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))*log(c)^2 /((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4 *n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(4/n)) + 16*a*b*n*Ei(4*l og(c)/n + 4*a/(b*n) + 4*log(x))*e^(-4*a/(b*n))*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4 *n^3*log(c) + a^2*b^3*n^3)*c^(4/n)) + 16*a*b*Ei(4*log(c)/n + 4*a/(b*n) + 4 *log(x))*e^(-4*a/(b*n))*log(c)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*lo...
Timed out. \[ \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x^3}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]