Integrand size = 16, antiderivative size = 76 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {2 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac {1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )} \] Output:
-2*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei((-2*a-2*b*ln(c*x^n))/b/n)/b^2/n^2/x^2-1/b /n/x^2/(a+b*ln(c*x^n))
Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {b n+2 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )} \] Input:
Integrate[1/(x^3*(a + b*Log[c*x^n])^2),x]
Output:
-((b*n + 2*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[(-2*(a + b*Log[c*x^ n]))/(b*n)]*(a + b*Log[c*x^n]))/(b^2*n^2*x^2*(a + b*Log[c*x^n])))
Time = 0.32 (sec) , antiderivative size = 76, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 2743 |
\(\displaystyle -\frac {2 \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )}dx}{b n}-\frac {1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle -\frac {2 \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-2/n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{b n^2 x^2}-\frac {1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle -\frac {2 e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac {1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}\) |
Input:
Int[1/(x^3*(a + b*Log[c*x^n])^2),x]
Output:
(-2*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[(-2*(a + b*Log[c*x^n]))/(b *n)])/(b^2*n^2*x^2) - 1/(b*n*x^2*(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_)*((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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.89 (sec) , antiderivative size = 349, normalized size of antiderivative = 4.59
method | result | size |
risch | \(\frac {2 i}{b n \,x^{2} \left (b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-\pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-2 i b \ln \left (c \right )-2 i \ln \left (x^{n}\right ) b -2 i a \right )}+\frac {2 c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 a}{n b}} \operatorname {expIntegral}_{1}\left (2 \ln \left (x \right )+\frac {i \left (b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-\pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-2 i b \ln \left (c \right )-2 i b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )-2 i a \right )}{n b}\right )}{b^{2} n^{2} x^{2}}\) | \(349\) |
Input:
int(1/x^3/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
Output:
2*I/b/n/x^2/(b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-Pi*b*csgn(I*x^n)*csgn(I*c*x^ n)*csgn(I*c)-b*Pi*csgn(I*c*x^n)^3+b*Pi*csgn(I*c*x^n)^2*csgn(I*c)-2*I*b*ln( c)-2*I*ln(x^n)*b-2*I*a)+2/b^2/n^2/x^2*c^(2/n)*(x^n)^(2/n)*exp((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(1,2*ln(x)+I* (b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c) -b*Pi*csgn(I*c*x^n)^3+b*Pi*csgn(I*c*x^n)^2*csgn(I*c)-2*I*b*ln(c)-2*I*b*(ln (x^n)-n*ln(x))-2*I*a)/n/b)
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {2 \, {\left (b n x^{2} \log \left (x\right ) + b x^{2} \log \left (c\right ) + a x^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{2}}\right ) + b n}{b^{3} n^{3} x^{2} \log \left (x\right ) + b^{3} n^{2} x^{2} \log \left (c\right ) + a b^{2} n^{2} x^{2}} \] Input:
integrate(1/x^3/(a+b*log(c*x^n))^2,x, algorithm="fricas")
Output:
-(2*(b*n*x^2*log(x) + b*x^2*log(c) + a*x^2)*e^(2*(b*log(c) + a)/(b*n))*log _integral(e^(-2*(b*log(c) + a)/(b*n))/x^2) + b*n)/(b^3*n^3*x^2*log(x) + b^ 3*n^2*x^2*log(c) + a*b^2*n^2*x^2)
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \] Input:
integrate(1/x**3/(a+b*ln(c*x**n))**2,x)
Output:
Integral(1/(x**3*(a + b*log(c*x**n))**2), x)
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a+b*log(c*x^n))^2,x, algorithm="maxima")
Output:
-1/(b^2*n*x^2*log(x^n) + (b^2*n*log(c) + a*b*n)*x^2) - 2*integrate(1/(b^2* n*x^3*log(x^n) + (b^2*n*log(c) + a*b*n)*x^3), x)
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a+b*log(c*x^n))^2,x, algorithm="giac")
Output:
integrate(1/((b*log(c*x^n) + a)^2*x^3), x)
Timed out. \[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \] Input:
int(1/(x^3*(a + b*log(c*x^n))^2),x)
Output:
int(1/(x^3*(a + b*log(c*x^n))^2), x)
\[ \int \frac {1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\mathrm {log}\left (x^{n} c \right )^{2} b^{2} x^{3}+2 \,\mathrm {log}\left (x^{n} c \right ) a b \,x^{3}+a^{2} x^{3}}d x \] Input:
int(1/x^3/(a+b*log(c*x^n))^2,x)
Output:
int(1/(log(x**n*c)**2*b**2*x**3 + 2*log(x**n*c)*a*b*x**3 + a**2*x**3),x)