Integrand size = 23, antiderivative size = 89 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {d}{e n}} (-b d+a e+b e n) x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^3 n^2}+\frac {(b d-a e) x}{e^2 n \left (d+e \log \left (c x^n\right )\right )} \]
(b*e*n+a*e-b*d)*x*Ei((d+e*ln(c*x^n))/e/n)/e^3/exp(d/e/n)/n^2/((c*x^n)^(1/n ))+(-a*e+b*d)*x/e^2/n/(d+e*ln(c*x^n))
Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {d}{e n}} (-b d+a e+b e n) x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )-\frac {e (-b d+a e) n x}{d+e \log \left (c x^n\right )}}{e^3 n^2} \]
(((-(b*d) + a*e + b*e*n)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(E^(d/ (e*n))*(c*x^n)^n^(-1)) - (e*(-(b*d) + a*e)*n*x)/(d + e*Log[c*x^n]))/(e^3*n ^2)
Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2807, 2009}
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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{\left (e \log \left (c x^n\right )+d\right )^2} \, dx\) |
| \(\Big \downarrow \) 2807 |
| \(\displaystyle \int \left (\frac {a e-b d}{e \left (e \log \left (c x^n\right )+d\right )^2}+\frac {b}{e \left (e \log \left (c x^n\right )+d\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} (b d-a e) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^3 n^2}+\frac {x (b d-a e)}{e^2 n \left (e \log \left (c x^n\right )+d\right )}+\frac {b x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^2 n}\) |
-(((b*d - a*e)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(e^3*E^(d/(e*n)) *n^2*(c*x^n)^n^(-1))) + (b*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(e^2 *E^(d/(e*n))*n*(c*x^n)^n^(-1)) + ((b*d - a*e)*x)/(e^2*n*(d + e*Log[c*x^n]) )
3.2.76.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]*(e_. ) + (d_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p] && IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.65 (sec) , antiderivative size = 370, normalized size of antiderivative = 4.16
| method | result | size |
| risch | \(-\frac {2 x \left (a e -b d \right )}{e^{2} n \left (2 d +2 e \ln \left (c \right )+2 e \ln \left (x^{n}\right )-i e \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i e \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}\right )}-\frac {\left (b e n +a e -b d \right ) x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {-i e \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i e \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 d}{2 e n}} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-\frac {-i e \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i e \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 e \ln \left (c \right )+2 e \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 d}{2 e n}\right )}{e^{3} n^{2}}\) | \(370\) |
-2/e^2/n*x*(a*e-b*d)/(2*d+2*e*ln(c)+2*e*ln(x^n)-I*e*Pi*csgn(I*c)*csgn(I*x^ n)*csgn(I*c*x^n)+I*e*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*e*Pi*csgn(I*x^n)*csgn( I*c*x^n)^2-I*e*Pi*csgn(I*c*x^n)^3)-(b*e*n+a*e-b*d)/e^3/n^2*x*c^(-1/n)*(x^n )^(-1/n)*exp(-1/2*(-I*e*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*e*Pi*csgn (I*c)*csgn(I*c*x^n)^2+I*e*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e*Pi*csgn(I*c*x ^n)^3+2*d)/e/n)*Ei(1,-ln(x)-1/2*(-I*e*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^ n)+I*e*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*e*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e *Pi*csgn(I*c*x^n)^3+2*e*ln(c)+2*e*(ln(x^n)-n*ln(x))+2*d)/e/n)
Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.73 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\frac {{\left ({\left (b d e - a e^{2}\right )} n x e^{\left (\frac {e \log \left (c\right ) + d}{e n}\right )} + {\left (b d e n - b d^{2} + a d e + {\left (b e^{2} n - b d e + a e^{2}\right )} \log \left (c\right ) + {\left (b e^{2} n^{2} - {\left (b d e - a e^{2}\right )} n\right )} \log \left (x\right )\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {e \log \left (c\right ) + d}{e n}\right )}\right )\right )} e^{\left (-\frac {e \log \left (c\right ) + d}{e n}\right )}}{e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}} \]
((b*d*e - a*e^2)*n*x*e^((e*log(c) + d)/(e*n)) + (b*d*e*n - b*d^2 + a*d*e + (b*e^2*n - b*d*e + a*e^2)*log(c) + (b*e^2*n^2 - (b*d*e - a*e^2)*n)*log(x) )*log_integral(x*e^((e*log(c) + d)/(e*n))))*e^(-(e*log(c) + d)/(e*n))/(e^4 *n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (c x^{n}\right ) + d\right )}^{2}} \,d x } \]
((e*n - d)*b + a*e)*integrate(1/(e^3*n*log(c) + e^3*n*log(x^n) + d*e^2*n), x) + (b*d - a*e)*x/(e^3*n*log(c) + e^3*n*log(x^n) + d*e^2*n)
Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (88) = 176\).
Time = 0.33 (sec) , antiderivative size = 712, normalized size of antiderivative = 8.00 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\frac {b e^{2} n^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )} \log \left (x\right )}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {b e^{2} n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )} \log \left (c\right )}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b d e n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )} \log \left (x\right )}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a e^{2} n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )} \log \left (x\right )}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {b d e n x}{e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}} - \frac {a e^{2} n x}{e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}} + \frac {b d e n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )}}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b d e {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )} \log \left (c\right )}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a e^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )} \log \left (c\right )}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b d^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )}}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a d e {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {d}{e n} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e n}\right )}}{{\left (e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} \]
b*e^2*n^2*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(x)/((e^4*n^3*lo g(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) + b*e^2*n*Ei(log(c)/n + d/(e*n ) + log(x))*e^(-d/(e*n))*log(c)/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3* n^2)*c^(1/n)) - b*d*e*n*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(x )/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) + a*e^2*n*Ei(log (c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(x)/((e^4*n^3*log(x) + e^4*n^2*l og(c) + d*e^3*n^2)*c^(1/n)) + b*d*e*n*x/(e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2) - a*e^2*n*x/(e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2) + b* d*e*n*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))/((e^4*n^3*log(x) + e^4* n^2*log(c) + d*e^3*n^2)*c^(1/n)) - b*d*e*Ei(log(c)/n + d/(e*n) + log(x))*e ^(-d/(e*n))*log(c)/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) + a*e^2*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(c)/((e^4*n^3*log (x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) - b*d^2*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/ n)) + a*d*e*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n))
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (d+e\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]