\(\int \frac {1}{x^2 (a+b \log (c x^n))^2} \, dx\) [78]

Optimal result

Integrand size = 16, antiderivative size = 73 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2 x}-\frac {1}{b n x \left (a+b \log \left (c x^n\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {b n+e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {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.

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

\(\Big \downarrow \) 2743

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

-((E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))])/ 
(b^2*n^2*x)) - 1/(b*n*x*(a + b*Log[c*x^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 = 0.82 (sec) , antiderivative size = 347, normalized size of antiderivative = 4.75

method	result	size
risch	\(-\frac {2}{x \left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (x^{n}\right ) b +2 b \ln \left (c \right )+2 a \right ) b n}+\frac {\left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 a}{2 n b}} \operatorname {expIntegral}_{1}\left (\ln \left (x \right )+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{2 n b}\right )}{n^{2} b^{2} x}\)	\(347\)

Input:

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

Output:

-2/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*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(x^n 
)*b+2*b*ln(c)+2*a)/b/n+1/n^2/b^2/x*(x^n)^(1/n)*c^(1/n)*exp(1/2*(I*Pi*b*csg 
n(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 [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {{\left (b n x \log \left (x\right ) + b x \log \left (c\right ) + a x\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{x}\right ) + b n}{b^{3} n^{3} x \log \left (x\right ) + b^{3} n^{2} x \log \left (c\right ) + a b^{2} n^{2} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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