Integrand size = 12, antiderivative size = 70 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2}-\frac {x}{b n \left (a+b \log \left (c x^n\right )\right )} \] Output:
x*Ei((a+b*ln(c*x^n))/b/n)/b^2/exp(a/b/n)/n^2/((c*x^n)^(1/n))-x/b/n/(a+b*ln (c*x^n))
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x \left (e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )-\frac {b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \] Input:
Integrate[(a + b*Log[c*x^n])^(-2),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*Log[c*x^n])))/(b^2*n^2)
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.
\(\displaystyle \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 2734 |
\(\displaystyle \frac {\int \frac {1}{a+b \log \left (c x^n\right )}dx}{b n}-\frac {x}{b n \left (a+b \log \left (c x^n\right )\right )}\) |
\(\Big \downarrow \) 2737 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{b n^2}-\frac {x}{b n \left (a+b \log \left (c x^n\right )\right )}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2}-\frac {x}{b n \left (a+b \log \left (c x^n\right )\right )}\) |
Input:
Int[(a + b*Log[c*x^n])^(-2),x]
Output:
(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]))
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.67 (sec) , antiderivative size = 350, normalized size of antiderivative = 5.00
method | result | size |
risch | \(-\frac {2 x}{\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (x^{n}\right ) b +2 b \ln \left (c \right )+2 a \right ) b n}-\frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 a}{2 n b}} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{2 n b}\right )}{n^{2} b^{2}}\) | \(350\) |
Input:
int(1/(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*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*P i*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)
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {{\left (b n x e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} \] Input:
integrate(1/(a+b*log(c*x^n))^2,x, algorithm="fricas")
Output:
-(b*n*x*e^((b*log(c) + a)/(b*n)) - (b*n*log(x) + b*log(c) + a)*log_integra l(x*e^((b*log(c) + a)/(b*n))))*e^(-(b*log(c) + a)/(b*n))/(b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2)
\[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \] Input:
integrate(1/(a+b*ln(c*x**n))**2,x)
Output:
Integral((a + b*log(c*x**n))**(-2), x)
\[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a+b*log(c*x^n))^2,x, algorithm="maxima")
Output:
-x/(b^2*n*log(c) + b^2*n*log(x^n) + a*b*n) + integrate(1/(b^2*n*log(c) + b ^2*n*log(x^n) + a*b*n), x)
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (69) = 138\).
Time = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.40 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {b n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (x\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b n x}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} + \frac {b {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )}}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} \] Input:
integrate(1/(a+b*log(c*x^n))^2,x, algorithm="giac")
Output:
b*n*Ei(log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))*log(x)/((b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2)*c^(1/n)) - b*n*x/(b^3*n^3*log(x) + b^3*n^2*lo g(c) + a*b^2*n^2) + b*Ei(log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))*log(c)/ ((b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n^2)*c^(1/n)) + a*Ei(log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))/((b^3*n^3*log(x) + b^3*n^2*log(c) + a*b^2*n ^2)*c^(1/n))
Timed out. \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \] Input:
int(1/(a + b*log(c*x^n))^2,x)
Output:
int(1/(a + b*log(c*x^n))^2, x)
\[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\mathrm {log}\left (x^{n} c \right )^{2} b^{2}+2 \,\mathrm {log}\left (x^{n} c \right ) a b +a^{2}}d x \] Input:
int(1/(a+b*log(c*x^n))^2,x)
Output:
int(1/(log(x**n*c)**2*b**2 + 2*log(x**n*c)*a*b + a**2),x)