\(\int \frac {1}{x^2 (a+b \log (c x^n))^3} \, dx\) [86]

Optimal result

Integrand size = 16, antiderivative size = 102 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3 x}-\frac {1}{2 b n x \left (a+b \log \left (c x^n\right )\right )^2}+\frac {1}{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^2+b n \left (a-b n+b \log \left (c x^n\right )\right )}{2 b^3 n^3 x \left (a+b \log \left (c x^n\right )\right )^2} \] Input:

Integrate[1/(x^2*(a + b*Log[c*x^n])^3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2743, 2743, 2747, 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.

\(\displaystyle \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx\)

\(\Big \downarrow \) 2743

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

\(\Big \downarrow \) 2743

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

Int[1/(x^2*(a + b*Log[c*x^n])^3),x]
 

Output:

-1/2*1/(b*n*x*(a + b*Log[c*x^n])^2) - (-((E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIn 
tegralEi[-((a + b*Log[c*x^n])/(b*n))])/(b^2*n^2*x)) - 1/(b*n*x*(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 2743

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^(p + 1)/(b*d*n*(p + 1))), x] - 
Simp[(m + 1)/(b*n*(p + 1))   Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x], x] 
 /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]
 

rule 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n))   Subst[Int[E^(((m + 1)/n 
)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
 

Maple [C] (warning: unable to verify)

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

Time = 1.10 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.40

method	result	size
risch	\(\frac {-2 n b +2 a +2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b +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 )}{{\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} x}-\frac {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} x}\)	\(449\)

Input:

int(1/x^2/(a+b*ln(c*x^n))^3,x,method=_RETURNVERBOSE)
 

Output:

(-2*n*b+2*a+2*b*ln(c)+2*ln(x^n)*b+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))/(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*cs 
gn(I*c)+2*ln(x^n)*b+2*b*ln(c)+2*a)^2/n^2/b^2/x-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. 192 vs. \(2 (95) = 190\).

Time = 0.07 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.88 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\frac {b^{2} n^{2} \log \left (x\right ) - b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n + {\left (b^{2} n^{2} x \log \left (x\right )^{2} + b^{2} x \log \left (c\right )^{2} + 2 \, a b x \log \left (c\right ) + a^{2} x + 2 \, {\left (b^{2} n x \log \left (c\right ) + a b n x\right )} \log \left (x\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{x}\right )}{2 \, {\left (b^{5} n^{5} x \log \left (x\right )^{2} + b^{5} n^{3} x \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} x \log \left (c\right ) + a^{2} b^{3} n^{3} x + 2 \, {\left (b^{5} n^{4} x \log \left (c\right ) + a b^{4} n^{4} x\right )} \log \left (x\right )\right )}} \] Input:

integrate(1/x^2/(a+b*log(c*x^n))^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(1/x**2/(a+b*ln(c*x**n))**3,x)
 

Output:

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

Maxima [F]

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

integrate(1/x^2/(a+b*log(c*x^n))^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/x^2/(a+b*log(c*x^n))^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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