\(\int \frac {a+b \log (c x^n)}{x^3 (d+e \log (f x^m))} \, dx\) [181]

Optimal result

Integrand size = 26, antiderivative size = 141 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=-\frac {b n}{2 e m x^2}-\frac {b e^{\frac {2 d}{e m}} n \left (f x^m\right )^{2/m} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2 x^2}+\frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x^2} \] Output:

-1/2*b*n/e/m/x^2-b*exp(2*d/e/m)*n*(f*x^m)^(2/m)*Ei((-2*d-2*e*ln(f*x^m))/e/ 
m)*(d+e*ln(f*x^m))/e^2/m^2/x^2+exp(2*d/e/m)*(f*x^m)^(2/m)*Ei((-2*d-2*e*ln( 
f*x^m))/e/m)*(a+b*ln(c*x^n))/e/m/x^2
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.67 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=\frac {-b e m n+2 e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a e m-b d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right )}{2 e^2 m^2 x^2} \] Input:

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*Log[f*x^m])),x]
 

Output:

(-(b*e*m*n) + 2*E^((2*d)/(e*m))*(f*x^m)^(2/m)*ExpIntegralEi[(-2*(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]))/(2 
*e^2*m^2*x^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.66 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2813, 27, 31, 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.

\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx\)

\(\Big \downarrow \) 2813

\(\displaystyle \frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}-b n \int \frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}-\frac {b n e^{\frac {2 d}{e m}} \int \frac {\left (f x^m\right )^{2/m} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{x^3}dx}{e m}\)

\(\Big \downarrow \) 31

\(\displaystyle \frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}-\frac {b n e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{x}dx}{e m x^2}\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}-\frac {b n e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \int \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )d\log \left (f x^m\right )}{e m^2 x^2}\)

\(\Big \downarrow \) 7281

\(\displaystyle \frac {b n e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \int \operatorname {ExpIntegralEi}\left (-\frac {2 d}{e m}-\frac {2 \log \left (f x^m\right )}{m}\right )d\left (-\frac {2 d}{e m}-\frac {2 \log \left (f x^m\right )}{m}\right )}{2 e m x^2}+\frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}\)

\(\Big \downarrow \) 7036

\(\displaystyle \frac {e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (-\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}+\frac {b n e^{\frac {2 d}{e m}} \left (f x^m\right )^{2/m} \left (\left (-\frac {2 d}{e m}-\frac {2 \log \left (f x^m\right )}{m}\right ) \operatorname {ExpIntegralEi}\left (-\frac {2 d}{e m}-\frac {2 \log \left (f x^m\right )}{m}\right )-f x^m\right )}{2 e m x^2}\)

Input:

Int[(a + b*Log[c*x^n])/(x^3*(d + e*Log[f*x^m])),x]
 

Output:

(b*E^((2*d)/(e*m))*n*(f*x^m)^(2/m)*(-(f*x^m) + ExpIntegralEi[(-2*d)/(e*m) 
- (2*Log[f*x^m])/m]*((-2*d)/(e*m) - (2*Log[f*x^m])/m)))/(2*e*m*x^2) + (E^( 
(2*d)/(e*m))*(f*x^m)^(2/m)*ExpIntegralEi[(-2*(d + e*Log[f*x^m]))/(e*m)]*(a 
 + b*Log[c*x^n]))/(e*m*x^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 31

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[(b* 
x^i)^p/(a*x)^(i*p)   Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p} 
, x] &&  !IntegerQ[p]
 

rule 2813

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ 
.)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + 
b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r])   u, x] - Simp[e*r   Int[Simp 
lifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, 
x] &&  !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
 

rule 3039

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]
 

rule 7036

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]
 

rule 7281

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]]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.83 (sec) , antiderivative size = 2341, normalized size of antiderivative = 16.60

method	result	size
risch	\(\text {Expression too large to display}\)	\(2341\)

Input:

int((a+b*ln(c*x^n))/x^3/(d+e*ln(f*x^m)),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)* 
csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c 
)+2*a)/m/e/x^2*(x^m)^(2/m)*f^(2/m)*exp((-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)/m/e)*Ei(1,2*ln(x)-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)/ 
m/e)-b/m/e/x^2*(x^m)^(2/m)*f^(2/m)*exp((-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)/m/e)*Ei(1,2*ln(x)-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)/ 
m/e)*ln(x^n)-1/2*b*n/e/m/x^2-1/2*I*b*n/m^2/e/x^2*(x^m)^(2/m)*f^(2/m)*exp(( 
-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)/m/e)*Ei( 
1,2*ln(x)-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)/m/e)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x 
^m)+1/2*I*b*n/m^2/e/x^2*(x^m)^(2/m)*f^(2/m)*exp((-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)*cs...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=-\frac {b e m n - 2 \, {\left (b e m x^{2} \log \left (c\right ) - b e n x^{2} \log \left (f\right ) + {\left (a e m - b d n\right )} x^{2}\right )} e^{\left (\frac {2 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {2 \, {\left (e \log \left (f\right ) + d\right )}}{e m}\right )}}{x^{2}}\right )}{2 \, e^{2} m^{2} x^{2}} \] Input:

integrate((a+b*log(c*x^n))/x^3/(d+e*log(f*x^m)),x, algorithm="fricas")
 

Output:

-1/2*(b*e*m*n - 2*(b*e*m*x^2*log(c) - b*e*n*x^2*log(f) + (a*e*m - b*d*n)*x 
^2)*e^(2*(e*log(f) + d)/(e*m))*log_integral(e^(-2*(e*log(f) + d)/(e*m))/x^ 
2))/(e^2*m^2*x^2)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x^{3} \left (d + e \log {\left (f x^{m} \right )}\right )}\, dx \] Input:

integrate((a+b*ln(c*x**n))/x**3/(d+e*ln(f*x**m)),x)
 

Output:

Integral((a + b*log(c*x**n))/(x**3*(d + e*log(f*x**m))), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*x^n))/x^3/(d+e*log(f*x^m)),x, algorithm="maxima")
 

Output:

integrate((b*log(c*x^n) + a)/((e*log(f*x^m) + d)*x^3), x)
 

Giac [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*x^n))/x^3/(d+e*log(f*x^m)),x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)/((e*log(f*x^m) + d)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\left (d+e\,\ln \left (f\,x^m\right )\right )} \,d x \] Input:

int((a + b*log(c*x^n))/(x^3*(d + e*log(f*x^m))),x)
 

Output:

int((a + b*log(c*x^n))/(x^3*(d + e*log(f*x^m))), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx=\left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{\mathrm {log}\left (x^{m} f \right ) e \,x^{3}+d \,x^{3}}d x \right ) b +\left (\int \frac {1}{\mathrm {log}\left (x^{m} f \right ) e \,x^{3}+d \,x^{3}}d x \right ) a \] Input:

int((a+b*log(c*x^n))/x^3/(d+e*log(f*x^m)),x)
 

Output:

int(log(x**n*c)/(log(x**m*f)*e*x**3 + d*x**3),x)*b + int(1/(log(x**m*f)*e* 
x**3 + d*x**3),x)*a