Integrand size = 16, antiderivative size = 135 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
1/2*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e/exp(a/b/n)/n^3/((c*(e*x+d) ^n)^(1/n))+1/2*(-e*x-d)/b/e/n/(a+b*ln(c*(e*x+d)^n))^2+1/2*(-e*x-d)/b^2/e/n ^2/(a+b*ln(c*(e*x+d)^n))
Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=-\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (-\operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b e^{\frac {a}{b n}} n \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (a+b n+b \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
-1/2*((d + e*x)*(-(ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*(a + b* Log[c*(d + e*x)^n])^2) + b*E^(a/(b*n))*n*(c*(d + e*x)^n)^n^(-1)*(a + b*n + b*Log[c*(d + e*x)^n])))/(b^3*e*E^(a/(b*n))*n^3*(c*(d + e*x)^n)^n^(-1)*(a + b*Log[c*(d + e*x)^n])^2)
Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2836, 2734, 2734, 2737, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx\) |
| \(\Big \downarrow \) 2836 |
| \(\displaystyle \frac {\int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}d(d+e x)}{e}\) |
| \(\Big \downarrow \) 2734 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}d(d+e x)}{2 b n}-\frac {d+e x}{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{e}\) |
| \(\Big \downarrow \) 2734 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{a+b \log \left (c (d+e x)^n\right )}d(d+e x)}{b n}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{2 b n}-\frac {d+e x}{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{e}\) |
| \(\Big \downarrow \) 2737 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{a+b \log \left (c (d+e x)^n\right )}d\log \left (c (d+e x)^n\right )}{b n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{2 b n}-\frac {d+e x}{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{e}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {\frac {\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{2 b n}-\frac {d+e x}{2 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{e}\) |
(-1/2*(d + e*x)/(b*n*(a + b*Log[c*(d + e*x)^n])^2) + (((d + e*x)*ExpIntegr alEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b^2*E^(a/(b*n))*n^2*(c*(d + e*x)^ n)^n^(-1)) - (d + e*x)/(b*n*(a + b*Log[c*(d + e*x)^n])))/(2*b*n))/e
3.1.23.3.1 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_), x_Symbol] :> Simp[x*((a + b *Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1)) Int[(a + b *Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && Int egerQ[2*p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x ^n)^(1/n)) Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ [{a, b, c, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.14 (sec) , antiderivative size = 734, normalized size of antiderivative = 5.44
| method | result | size |
| risch | \(-\frac {2 b e n x +2 b d n +i \pi b d \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b d \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b e x \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b e x \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+2 \ln \left (c \right ) b e x +2 b e x \ln \left (\left (e x +d \right )^{n}\right )+2 d b \ln \left (c \right )+2 a e x +2 b d \ln \left (\left (e x +d \right )^{n}\right )+2 a d}{{\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (\left (e x +d \right )^{n}\right )+2 b \ln \left (c \right )+2 a \right )}^{2} b^{2} n^{2} e}-\frac {\left (e x +d \right ) c^{-\frac {1}{n}} \left (\left (e x +d \right )^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 a}{2 b n}} \operatorname {Ei}_{1}\left (-\ln \left (e x +d \right )-\frac {-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 b \left (\ln \left (\left (e x +d \right )^{n}\right )-n \ln \left (e x +d \right )\right )+2 a}{2 b n}\right )}{2 b^{3} n^{3} e}\) | \(734\) |
-(2*b*e*n*x+2*b*d*n+I*Pi*b*d*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*Pi* b*d*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*d*csgn(I*c)*csgn(I*(e*x+d)^n)*c sgn(I*c*(e*x+d)^n)-I*Pi*b*d*csgn(I*c*(e*x+d)^n)^3-I*Pi*b*e*x*csgn(I*c*(e*x +d)^n)^3+I*Pi*b*e*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*e*x*csg n(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*Pi*b*e*x*csgn(I*c)*csgn(I*c *(e*x+d)^n)^2+2*ln(c)*b*e*x+2*b*e*x*ln((e*x+d)^n)+2*d*b*ln(c)+2*a*e*x+2*b* d*ln((e*x+d)^n)+2*a*d)/(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+ d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I *c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln((e*x+d)^n)+2*b*ln(c) +2*a)^2/b^2/n^2/e-1/2/b^3/n^3/e*(e*x+d)*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1 /2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c) *csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I* Pi*csgn(I*c*(e*x+d)^n)^3*b+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c *(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n) ^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^ n)^3*b+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (128) = 256\).
Time = 0.30 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.95 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=-\frac {{\left ({\left (b^{2} d n^{2} + a b d n + {\left (b^{2} e n^{2} + a b e n\right )} x + {\left (b^{2} e n^{2} x + b^{2} d n^{2}\right )} \log \left (e x + d\right ) + {\left (b^{2} e n x + b^{2} d n\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b^{2} n^{2} \log \left (e x + d\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (e x + d\right )\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{2 \, {\left (b^{5} e n^{5} \log \left (e x + d\right )^{2} + b^{5} e n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} e n^{3} \log \left (c\right ) + a^{2} b^{3} e n^{3} + 2 \, {\left (b^{5} e n^{4} \log \left (c\right ) + a b^{4} e n^{4}\right )} \log \left (e x + d\right )\right )}} \]
-1/2*((b^2*d*n^2 + a*b*d*n + (b^2*e*n^2 + a*b*e*n)*x + (b^2*e*n^2*x + b^2* d*n^2)*log(e*x + d) + (b^2*e*n*x + b^2*d*n)*log(c))*e^((b*log(c) + a)/(b*n )) - (b^2*n^2*log(e*x + d)^2 + b^2*log(c)^2 + 2*a*b*log(c) + a^2 + 2*(b^2* n*log(c) + a*b*n)*log(e*x + d))*log_integral((e*x + d)*e^((b*log(c) + a)/( b*n))))*e^(-(b*log(c) + a)/(b*n))/(b^5*e*n^5*log(e*x + d)^2 + b^5*e*n^3*lo g(c)^2 + 2*a*b^4*e*n^3*log(c) + a^2*b^3*e*n^3 + 2*(b^5*e*n^4*log(c) + a*b^ 4*e*n^4)*log(e*x + d))
\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \]
-1/2*((d*n + d*log(c))*b + a*d + ((e*n + e*log(c))*b + a*e)*x + (b*e*x + b *d)*log((e*x + d)^n))/(b^4*e*n^2*log((e*x + d)^n)^2 + b^4*e*n^2*log(c)^2 + 2*a*b^3*e*n^2*log(c) + a^2*b^2*e*n^2 + 2*(b^4*e*n^2*log(c) + a*b^3*e*n^2) *log((e*x + d)^n)) + integrate(1/2/(b^3*n^2*log((e*x + d)^n) + b^3*n^2*log (c) + a*b^2*n^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 1218 vs. \(2 (128) = 256\).
Time = 0.41 (sec) , antiderivative size = 1218, normalized size of antiderivative = 9.02 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \]
1/2*b^2*n^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d )^2/((b^5*e*n^5*log(e*x + d)^2 + 2*b^5*e*n^4*log(e*x + d)*log(c) + 2*a*b^4 *e*n^4*log(e*x + d) + b^5*e*n^3*log(c)^2 + 2*a*b^4*e*n^3*log(c) + a^2*b^3* e*n^3)*c^(1/n)) - 1/2*(e*x + d)*b^2*n^2*log(e*x + d)/(b^5*e*n^5*log(e*x + d)^2 + 2*b^5*e*n^4*log(e*x + d)*log(c) + 2*a*b^4*e*n^4*log(e*x + d) + b^5* e*n^3*log(c)^2 + 2*a*b^4*e*n^3*log(c) + a^2*b^3*e*n^3) + b^2*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)*log(c)/((b^5*e*n^5*lo g(e*x + d)^2 + 2*b^5*e*n^4*log(e*x + d)*log(c) + 2*a*b^4*e*n^4*log(e*x + d ) + b^5*e*n^3*log(c)^2 + 2*a*b^4*e*n^3*log(c) + a^2*b^3*e*n^3)*c^(1/n)) - 1/2*(e*x + d)*b^2*n^2/(b^5*e*n^5*log(e*x + d)^2 + 2*b^5*e*n^4*log(e*x + d) *log(c) + 2*a*b^4*e*n^4*log(e*x + d) + b^5*e*n^3*log(c)^2 + 2*a*b^4*e*n^3* log(c) + a^2*b^3*e*n^3) + a*b*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(- a/(b*n))*log(e*x + d)/((b^5*e*n^5*log(e*x + d)^2 + 2*b^5*e*n^4*log(e*x + d )*log(c) + 2*a*b^4*e*n^4*log(e*x + d) + b^5*e*n^3*log(c)^2 + 2*a*b^4*e*n^3 *log(c) + a^2*b^3*e*n^3)*c^(1/n)) - 1/2*(e*x + d)*b^2*n*log(c)/(b^5*e*n^5* log(e*x + d)^2 + 2*b^5*e*n^4*log(e*x + d)*log(c) + 2*a*b^4*e*n^4*log(e*x + d) + b^5*e*n^3*log(c)^2 + 2*a*b^4*e*n^3*log(c) + a^2*b^3*e*n^3) + 1/2*b^2 *Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(c)^2/((b^5*e*n^5*l og(e*x + d)^2 + 2*b^5*e*n^4*log(e*x + d)*log(c) + 2*a*b^4*e*n^4*log(e*x + d) + b^5*e*n^3*log(c)^2 + 2*a*b^4*e*n^3*log(c) + a^2*b^3*e*n^3)*c^(1/n)...
Timed out. \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \]