Integrand size = 14, antiderivative size = 101 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {2 e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^3 n^3}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x^2}{b^2 n^2 \left (a+b \log \left (c x^n\right )\right )} \]
2*x^2*Ei(2*(a+b*ln(c*x^n))/b/n)/b^3/exp(2*a/b/n)/n^3/((c*x^n)^(2/n))-1/2*x ^2/b/n/(a+b*ln(c*x^n))^2-x^2/b^2/n^2/(a+b*ln(c*x^n))
Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {x^2 \left (4 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]
(x^2*((4*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(E^((2*a)/(b*n))*(c* x^n)^(2/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)
Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, 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.
| \(\displaystyle \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle \frac {\int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^2}dx}{b n}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle \frac {\frac {2 \int \frac {x}{a+b \log \left (c x^n\right )}dx}{b n}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle \frac {\frac {2 x^2 \left (c x^n\right )^{-2/n} \int \frac {\left (c x^n\right )^{2/n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{b n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {\frac {2 x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^2}{b n \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {x^2}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\) |
-1/2*x^2/(b*n*(a + b*Log[c*x^n])^2) + ((2*x^2*ExpIntegralEi[(2*(a + b*Log[ c*x^n]))/(b*n)])/(b^2*E^((2*a)/(b*n))*n^2*(c*x^n)^(2/n)) - x^2/(b*n*(a + b *Log[c*x^n])))/(b*n)
3.1.83.3.1 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 = 0.56 (sec) , antiderivative size = 476, normalized size of antiderivative = 4.71
| method | result | size |
| risch | \(-\frac {2 \left (b n \,x^{2}-i \pi b \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi b \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \left (c \right ) b \,x^{2}+2 b \,x^{2} \ln \left (x^{n}\right )+2 x^{2} a \right )}{{\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}}-\frac {2 x^{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}}\) | \(476\) |
-2*(b*n*x^2-I*Pi*b*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*Pi*b*x^2*csgn (I*c)*csgn(I*c*x^n)^2+I*Pi*b*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*x^2*cs gn(I*c*x^n)^3+2*ln(c)*b*x^2+2*b*x^2*ln(x^n)+2*x^2*a)/(-I*b*Pi*csgn(I*c)*cs gn(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/b^2/ n^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*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. 211 vs. \(2 (100) = 200\).
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.09 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {{\left ({\left (2 \, b^{2} n^{2} x^{2} \log \left (x\right ) + 2 \, b^{2} n x^{2} \log \left (c\right ) + {\left (b^{2} n^{2} + 2 \, a b n\right )} x^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 4 \, {\left (b^{2} n^{2} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \operatorname {log\_integral}\left (x^{2} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \, {\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \]
-1/2*((2*b^2*n^2*x^2*log(x) + 2*b^2*n*x^2*log(c) + (b^2*n^2 + 2*a*b*n)*x^2 )*e^(2*(b*log(c) + a)/(b*n)) - 4*(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^2*e^(2*(b*l og(c) + a)/(b*n))))*e^(-2*(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))
\[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x}{\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
\[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int { \frac {x}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}} \,d x } \]
-1/2*(2*b*x^2*log(x^n) + (b*(n + 2*log(c)) + 2*a)*x^2)/(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)) + 2*integrate(x/(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 = 0.46 (sec) , antiderivative size = 1029, normalized size of antiderivative = 10.19 \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\text {Too large to display} \]
-b^2*n^2*x^2*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) + 2*b^2* n^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*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^(2/n)) - 1/2*b^2*n^2*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) - b^2*n*x^2*log(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) + 4*b^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x ))*e^(-2*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^(2/n)) - a*b*n*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) + 2*b^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*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*lo g(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(2/n)) + 4*a*b*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*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^(2/n)) + 4*a*b*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x))* e^(-2*a/(b*n))*log(c)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^...
Timed out. \[ \int \frac {x}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {x}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]