\(\int \frac {1}{(a+b \log (c (d (e+f x)^m)^n))^2} \, dx\) [428]

Optimal result

Integrand size = 20, antiderivative size = 123 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b^2 f m^2 n^2}-\frac {e+f x}{b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=-\frac {e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \left (b e^{\frac {a}{b m n}} m n \left (c \left (d (e+f x)^m\right )^n\right )^{\frac {1}{m n}}-\operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )}{b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )} \] Input:

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-2),x]
 

Output:

-(((e + f*x)*(b*E^(a/(b*m*n))*m*n*(c*(d*(e + f*x)^m)^n)^(1/(m*n)) - ExpInt 
egralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)]*(a + b*Log[c*(d*(e + f*x 
)^m)^n])))/(b^2*E^(a/(b*m*n))*f*m^2*n^2*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(a 
 + b*Log[c*(d*(e + f*x)^m)^n])))
 

Rubi [A] (warning: unable to verify)

Time = 0.86 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2895, 2836, 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.

\(\displaystyle \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}dx\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \frac {1}{\left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^2}d(e+f x)}{f}\)

\(\Big \downarrow \) 2734

\(\displaystyle \frac {\frac {\int \frac {1}{a+b \log \left (c d^n (e+f x)^{m n}\right )}d(e+f x)}{b m n}-\frac {e+f x}{b m n \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )}}{f}\)

\(\Big \downarrow \) 2737

\(\displaystyle \frac {\frac {(e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \int \frac {\left (c d^n (e+f x)^{m n}\right )^{\frac {1}{m n}}}{a+b \log \left (c d^n (e+f x)^{m n}\right )}d\log \left (c d^n (e+f x)^{m n}\right )}{b m^2 n^2}-\frac {e+f x}{b m n \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )}}{f}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\frac {(e+f x) e^{-\frac {a}{b m n}} \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^n (e+f x)^{m n}\right )}{b m n}\right )}{b^2 m^2 n^2}-\frac {e+f x}{b m n \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )}}{f}\)

Input:

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-2),x]
 

Output:

(((e + f*x)*ExpIntegralEi[(a + b*Log[c*d^n*(e + f*x)^(m*n)])/(b*m*n)])/(b^ 
2*E^(a/(b*m*n))*m^2*n^2*(c*d^n*(e + f*x)^(m*n))^(1/(m*n))) - (e + f*x)/(b* 
m*n*(a + b*Log[c*d^n*(e + f*x)^(m*n)])))/f
 

Defintions of rubi rules used

rule 2609

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]
 

rule 2734

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]
 

rule 2737

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]
 

rule 2836

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]
 

rule 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

Maple [F]

\[\int \frac {1}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )}^{2}}d x\]

Input:

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^2,x)
 

Output:

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=-\frac {{\left ({\left (b f m n x + b e m n\right )} e^{\left (\frac {b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} - {\left (b m n \log \left (f x + e\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )} \operatorname {log\_integral}\left ({\left (f x + e\right )} e^{\left (\frac {b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}\right )\right )} e^{\left (-\frac {b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}}{b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}} \] Input:

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^2,x, algorithm="fricas")
 

Output:

-((b*f*m*n*x + b*e*m*n)*e^((b*n*log(d) + b*log(c) + a)/(b*m*n)) - (b*m*n*l 
og(f*x + e) + b*n*log(d) + b*log(c) + a)*log_integral((f*x + e)*e^((b*n*lo 
g(d) + b*log(c) + a)/(b*m*n))))*e^(-(b*n*log(d) + b*log(c) + a)/(b*m*n))/( 
b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n^2*log(c) + 
 a*b^2*f*m^2*n^2)
 

Sympy [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{2}}\, dx \] Input:

integrate(1/(a+b*ln(c*(d*(f*x+e)**m)**n))**2,x)
 

Output:

Integral((a + b*log(c*(d*(e + f*x)**m)**n))**(-2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^2,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 582 vs. \(2 (123) = 246\).

Time = 0.14 (sec) , antiderivative size = 582, normalized size of antiderivative = 4.73 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=-\frac {{\left (f x + e\right )} b m n}{b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}} + \frac {b m n {\rm Ei}\left (\frac {\log \left (d\right )}{m} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b m n}\right )} \log \left (f x + e\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac {1}{m n}} d^{\left (\frac {1}{m}\right )}} + \frac {b n {\rm Ei}\left (\frac {\log \left (d\right )}{m} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b m n}\right )} \log \left (d\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac {1}{m n}} d^{\left (\frac {1}{m}\right )}} + \frac {b {\rm Ei}\left (\frac {\log \left (d\right )}{m} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b m n}\right )} \log \left (c\right )}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac {1}{m n}} d^{\left (\frac {1}{m}\right )}} + \frac {a {\rm Ei}\left (\frac {\log \left (d\right )}{m} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b m n}\right )}}{{\left (b^{3} f m^{3} n^{3} \log \left (f x + e\right ) + b^{3} f m^{2} n^{3} \log \left (d\right ) + b^{3} f m^{2} n^{2} \log \left (c\right ) + a b^{2} f m^{2} n^{2}\right )} c^{\frac {1}{m n}} d^{\left (\frac {1}{m}\right )}} \] Input:

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-(f*x + e)*b*m*n/(b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*n^3*log(d) + b^3* 
f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2) + b*m*n*Ei(log(d)/m + log(c)/(m*n) + a 
/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)/((b^3*f*m^3*n^3*log(f 
*x + e) + b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2)*c 
^(1/(m*n))*d^(1/m)) + b*n*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x 
 + e))*e^(-a/(b*m*n))*log(d)/((b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*n^3* 
log(d) + b^3*f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2)*c^(1/(m*n))*d^(1/m)) + b* 
Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log( 
c)/((b^3*f*m^3*n^3*log(f*x + e) + b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n^2*log 
(c) + a*b^2*f*m^2*n^2)*c^(1/(m*n))*d^(1/m)) + a*Ei(log(d)/m + log(c)/(m*n) 
 + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))/((b^3*f*m^3*n^3*log(f*x + e) + 
 b^3*f*m^2*n^3*log(d) + b^3*f*m^2*n^2*log(c) + a*b^2*f*m^2*n^2)*c^(1/(m*n) 
)*d^(1/m))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \, dx=\frac {\left (\int \frac {x}{\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} b^{2} e +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} b^{2} f x +2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b e +2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b f x +a^{2} e +a^{2} f x}d x \right ) \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b \,f^{2} m n +\left (\int \frac {x}{\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} b^{2} e +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} b^{2} f x +2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b e +2 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a b f x +a^{2} e +a^{2} f x}d x \right ) a^{2} f^{2} m n +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) e}{a f m n \left (\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b +a \right )} \] Input:

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

Output:

(int(x/(log(d**n*(e + f*x)**(m*n)*c)**2*b**2*e + log(d**n*(e + f*x)**(m*n) 
*c)**2*b**2*f*x + 2*log(d**n*(e + f*x)**(m*n)*c)*a*b*e + 2*log(d**n*(e + f 
*x)**(m*n)*c)*a*b*f*x + a**2*e + a**2*f*x),x)*log(d**n*(e + f*x)**(m*n)*c) 
*a*b*f**2*m*n + int(x/(log(d**n*(e + f*x)**(m*n)*c)**2*b**2*e + log(d**n*( 
e + f*x)**(m*n)*c)**2*b**2*f*x + 2*log(d**n*(e + f*x)**(m*n)*c)*a*b*e + 2* 
log(d**n*(e + f*x)**(m*n)*c)*a*b*f*x + a**2*e + a**2*f*x),x)*a**2*f**2*m*n 
 + log(d**n*(e + f*x)**(m*n)*c)*e)/(a*f*m*n*(log(d**n*(e + f*x)**(m*n)*c)* 
b + a))