\(\int \frac {1}{(a+b \log (c x^n))^3} \, dx\) [84]

Optimal result

Integrand size = 12, antiderivative size = 98 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3}-\frac {x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )} \] Output:

1/2*x*Ei((a+b*ln(c*x^n))/b/n)/b^3/exp(a/b/n)/n^3/((c*x^n)^(1/n))-1/2*x/b/n 
/(a+b*ln(c*x^n))^2-1/2*x/b^2/n^2/(a+b*ln(c*x^n))
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {x \left (e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )-\frac {b n \left (a+b n+b \log \left (c x^n\right )\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \] Input:

Integrate[(a + b*Log[c*x^n])^(-3),x]
 

Output:

(x*(ExpIntegralEi[(a + b*Log[c*x^n])/(b*n)]/(E^(a/(b*n))*(c*x^n)^n^(-1)) - 
 (b*n*(a + b*n + b*Log[c*x^n]))/(a + b*Log[c*x^n])^2))/(2*b^3*n^3)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.

\(\displaystyle \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx\)

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2609

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

Input:

Int[(a + b*Log[c*x^n])^(-3),x]
 

Output:

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

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]
 

Maple [C] (warning: unable to verify)

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

Time = 0.80 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.68

method	result	size
risch	\(-\frac {2 b n x +i \pi b x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i \pi b x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i \pi b x \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right ) b x +2 b x \ln \left (x^{n}\right )+2 a x}{{\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (x^{n}\right ) b +2 b \ln \left (c \right )+2 a \right )}^{2} n^{2} b^{2}}-\frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 a}{2 n b}} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{2 n b}\right )}{2 b^{3} n^{3}}\)	\(459\)

Input:

int(1/(a+b*ln(c*x^n))^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (91) = 182\).

Time = 0.08 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {{\left ({\left (b^{2} n^{2} x \log \left (x\right ) + b^{2} n x \log \left (c\right ) + {\left (b^{2} n^{2} + a b n\right )} x\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b^{2} n^{2} \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \, {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{2 \, {\left (b^{5} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \, {\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \] Input:

integrate(1/(a+b*log(c*x^n))^3,x, algorithm="fricas")
 

Output:

-1/2*((b^2*n^2*x*log(x) + b^2*n*x*log(c) + (b^2*n^2 + a*b*n)*x)*e^((b*log( 
c) + a)/(b*n)) - (b^2*n^2*log(x)^2 + b^2*log(c)^2 + 2*a*b*log(c) + a^2 + 2 
*(b^2*n*log(c) + a*b*n)*log(x))*log_integral(x*e^((b*log(c) + a)/(b*n))))* 
e^(-(b*log(c) + a)/(b*n))/(b^5*n^5*log(x)^2 + b^5*n^3*log(c)^2 + 2*a*b^4*n 
^3*log(c) + a^2*b^3*n^3 + 2*(b^5*n^4*log(c) + a*b^4*n^4)*log(x))
 

Sympy [F]

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

integrate(1/(a+b*ln(c*x**n))**3,x)
 

Output:

Integral((a + b*log(c*x**n))**(-3), x)
 

Maxima [F]

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

integrate(1/(a+b*log(c*x^n))^3,x, algorithm="maxima")
 

Output:

-1/2*(b*x*log(x^n) + (b*(n + log(c)) + a)*x)/(b^4*n^2*log(c)^2 + b^4*n^2*l 
og(x^n)^2 + 2*a*b^3*n^2*log(c) + a^2*b^2*n^2 + 2*(b^4*n^2*log(c) + a*b^3*n 
^2)*log(x^n)) + integrate(1/2/(b^3*n^2*log(c) + b^3*n^2*log(x^n) + a*b^2*n 
^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 982 vs. \(2 (91) = 182\).

Time = 0.13 (sec) , antiderivative size = 982, normalized size of antiderivative = 10.02 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*log(c*x^n))^3,x, algorithm="giac")
 

Output:

1/2*b^2*n^2*Ei(log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))*log(x)^2/((b^5*n^ 
5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log( 
x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(1/n)) - 1/2*b^2*n^2*x*log(x)/(b^ 
5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4* 
log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) + b^2*n*Ei(log(c)/n + a/(b*n) + 
 log(x))*e^(-a/(b*n))*log(c)*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)* 
log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2* 
b^3*n^3)*c^(1/n)) - 1/2*b^2*n^2*x/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log 
(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3 
*n^3) - 1/2*b^2*n*x*log(c)/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b 
^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) + 
 1/2*b^2*Ei(log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))*log(c)^2/((b^5*n^5*l 
og(x)^2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) 
+ 2*a*b^4*n^3*log(c) + a^2*b^3*n^3)*c^(1/n)) + a*b*n*Ei(log(c)/n + a/(b*n) 
 + log(x))*e^(-a/(b*n))*log(x)/((b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x 
) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n 
^3)*c^(1/n)) - 1/2*a*b*n*x/(b^5*n^5*log(x)^2 + 2*b^5*n^4*log(c)*log(x) + b 
^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2*a*b^4*n^3*log(c) + a^2*b^3*n^3) + 
 a*b*Ei(log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))*log(c)/((b^5*n^5*log(x)^ 
2 + 2*b^5*n^4*log(c)*log(x) + b^5*n^3*log(c)^2 + 2*a*b^4*n^4*log(x) + 2...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \] Input:

int(1/(a + b*log(c*x^n))^3,x)
 

Output:

int(1/(a + b*log(c*x^n))^3, x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx=\int \frac {1}{\mathrm {log}\left (x^{n} c \right )^{3} b^{3}+3 \mathrm {log}\left (x^{n} c \right )^{2} a \,b^{2}+3 \,\mathrm {log}\left (x^{n} c \right ) a^{2} b +a^{3}}d x \] Input:

int(1/(a+b*log(c*x^n))^3,x)
 

Output:

int(1/(log(x**n*c)**3*b**3 + 3*log(x**n*c)**2*a*b**2 + 3*log(x**n*c)*a**2* 
b + a**3),x)