Integrand size = 23, antiderivative size = 130 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {b n x}{e m}-\frac {b e^{-\frac {d}{e m}} n x \left (f x^m\right )^{-1/m} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {e^{-\frac {d}{e m}} x \left (f x^m\right )^{-1/m} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m} \]
b*n*x/e/m-b*n*x*Ei((d+e*ln(f*x^m))/e/m)*(d+e*ln(f*x^m))/e^2/exp(d/e/m)/m^2 /((f*x^m)^(1/m))+x*Ei((d+e*ln(f*x^m))/e/m)*(a+b*ln(c*x^n))/e/exp(d/e/m)/m/ ((f*x^m)^(1/m))
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {x \left (b e m n+e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a e m-b d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right )\right )}{e^2 m^2} \]
(x*(b*e*m*n + (ExpIntegralEi[(d + e*Log[f*x^m])/(e*m)]*(a*e*m - b*d*n - b* e*n*Log[f*x^m] + b*e*m*Log[c*x^n]))/(E^(d/(e*m))*(f*x^m)^m^(-1))))/(e^2*m^ 2)
Time = 0.55 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2808, 27, 34, 3039, 7281, 7036}
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.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx\) |
\(\Big \downarrow \) 2808 |
\(\displaystyle \frac {x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m}-b n \int \frac {e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m}-\frac {b n e^{-\frac {d}{e m}} \int \left (f x^m\right )^{-1/m} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )dx}{e m}\) |
\(\Big \downarrow \) 34 |
\(\displaystyle \frac {x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m}-\frac {b n x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \int \frac {\operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{x}dx}{e m}\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m}-\frac {b n x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \int \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )d\log \left (f x^m\right )}{e m^2}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m}-\frac {b n x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \int \operatorname {ExpIntegralEi}\left (\frac {d}{e m}+\frac {\log \left (f x^m\right )}{m}\right )d\left (\frac {d}{e m}+\frac {\log \left (f x^m\right )}{m}\right )}{e m}\) |
\(\Big \downarrow \) 7036 |
\(\displaystyle \frac {x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m}-\frac {b n x e^{-\frac {d}{e m}} \left (f x^m\right )^{-1/m} \left (\left (\frac {d}{e m}+\frac {\log \left (f x^m\right )}{m}\right ) \operatorname {ExpIntegralEi}\left (\frac {d}{e m}+\frac {\log \left (f x^m\right )}{m}\right )-f x^m\right )}{e m}\) |
-((b*n*x*(-(f*x^m) + ExpIntegralEi[d/(e*m) + Log[f*x^m]/m]*(d/(e*m) + Log[ f*x^m]/m)))/(e*E^(d/(e*m))*m*(f*x^m)^m^(-1))) + (x*ExpIntegralEi[(d + e*Lo g[f*x^m])/(e*m)]*(a + b*Log[c*x^n]))/(e*E^(d/(e*m))*m*(f*x^m)^m^(-1))
3.2.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F racPart[p]/x^(m*FracPart[p])) Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x ] && !IntegerQ[p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.)), x_Symbol] :> With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Simp[ (d + e*Log[f*x^r]) u, x] - Simp[e*r Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(ExpInte gralEi[a + b*x]/b), x] - Simp[E^(a + b*x)/b, x] /; FreeQ[{a, b}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.96 (sec) , antiderivative size = 2329, normalized size of antiderivative = 17.92
-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*a)/e/m*x*f^(-1/m)*(x^m)^(-1/m)*exp(-1/2*(-I*Pi*csgn(I*f)*csgn(I*x^m)* csgn(I*f*x^m)*e+I*Pi*csgn(I*f)*csgn(I*f*x^m)^2*e+I*Pi*csgn(I*x^m)*csgn(I*f *x^m)^2*e-I*Pi*csgn(I*f*x^m)^3*e+2*d)/e/m)*Ei(1,-ln(x)+1/2*I*(e*Pi*csgn(I* f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f*x^m)^2-e*Pi*csgn(I*x^ m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*e*ln(f)+2*I*e*(ln(x^m)-m*ln(x) )+2*I*d)/e/m)-b/e/m*x*f^(-1/m)*(x^m)^(-1/m)*exp(-1/2*(-I*Pi*csgn(I*f)*csgn (I*x^m)*csgn(I*f*x^m)*e+I*Pi*csgn(I*f)*csgn(I*f*x^m)^2*e+I*Pi*csgn(I*x^m)* csgn(I*f*x^m)^2*e-I*Pi*csgn(I*f*x^m)^3*e+2*d)/e/m)*Ei(1,-ln(x)+1/2*I*(e*Pi *csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f*x^m)^2-e*Pi*c sgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*e*ln(f)+2*I*e*(ln(x^m) -m*ln(x))+2*I*d)/e/m)*ln(x^n)+b*n*x/e/m-1/2*I*b*n/e/m^2*x*f^(-1/m)*(x^m)^( -1/m)*exp(-1/2*(-I*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)*e+I*Pi*csgn(I*f) *csgn(I*f*x^m)^2*e+I*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2*e-I*Pi*csgn(I*f*x^m)^3 *e+2*d)/e/m)*Ei(1,-ln(x)+1/2*I*(e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e *Pi*csgn(I*f)*csgn(I*f*x^m)^2-e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I *f*x^m)^3+2*I*e*ln(f)+2*I*e*(ln(x^m)-m*ln(x))+2*I*d)/e/m)*Pi*csgn(I*f)*csg n(I*x^m)*csgn(I*f*x^m)+1/2*I*b*n/e/m^2*x*f^(-1/m)*(x^m)^(-1/m)*exp(-1/2*(- I*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)*e+I*Pi*csgn(I*f)*csgn(I*f*x^m)...
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {{\left (b e m n x e^{\left (\frac {e \log \left (f\right ) + d}{e m}\right )} + {\left (b e m \log \left (c\right ) - b e n \log \left (f\right ) + a e m - b d n\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {e \log \left (f\right ) + d}{e m}\right )}\right )\right )} e^{\left (-\frac {e \log \left (f\right ) + d}{e m}\right )}}{e^{2} m^{2}} \]
(b*e*m*n*x*e^((e*log(f) + d)/(e*m)) + (b*e*m*log(c) - b*e*n*log(f) + a*e*m - b*d*n)*log_integral(x*e^((e*log(f) + d)/(e*m))))*e^(-(e*log(f) + d)/(e* m))/(e^2*m^2)
\[ \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{d + e \log {\left (f x^{m} \right )}}\, dx \]
\[ \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{e \log \left (f x^{m}\right ) + d} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.48 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx=\frac {b n x}{e m} + \frac {b {\rm Ei}\left (\frac {\log \left (f\right )}{m} + \frac {d}{e m} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e m}\right )} \log \left (c\right )}{e f^{\left (\frac {1}{m}\right )} m} - \frac {b n {\rm Ei}\left (\frac {\log \left (f\right )}{m} + \frac {d}{e m} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e m}\right )} \log \left (f\right )}{e f^{\left (\frac {1}{m}\right )} m^{2}} + \frac {a {\rm Ei}\left (\frac {\log \left (f\right )}{m} + \frac {d}{e m} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e m}\right )}}{e f^{\left (\frac {1}{m}\right )} m} - \frac {b d n {\rm Ei}\left (\frac {\log \left (f\right )}{m} + \frac {d}{e m} + \log \left (x\right )\right ) e^{\left (-\frac {d}{e m}\right )}}{e^{2} f^{\left (\frac {1}{m}\right )} m^{2}} \]
b*n*x/(e*m) + b*Ei(log(f)/m + d/(e*m) + log(x))*e^(-d/(e*m))*log(c)/(e*f^( 1/m)*m) - b*n*Ei(log(f)/m + d/(e*m) + log(x))*e^(-d/(e*m))*log(f)/(e*f^(1/ m)*m^2) + a*Ei(log(f)/m + d/(e*m) + log(x))*e^(-d/(e*m))/(e*f^(1/m)*m) - b *d*n*Ei(log(f)/m + d/(e*m) + log(x))*e^(-d/(e*m))/(e^2*f^(1/m)*m^2)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{d+e \log \left (f x^m\right )} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{d+e\,\ln \left (f\,x^m\right )} \,d x \]