\(\int \frac {1}{x^4 (a+b \log (c x^n))^2} \, dx\) [80]

Optimal result

Integrand size = 16, antiderivative size = 76 \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {3 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^3}-\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2343, 2347, 2209} \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {3 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^3}-\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[1/(x^4*(a + b*Log[c*x^n])^2),x]

[Out]

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2343

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] - Dist[(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 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )}-\frac {3 \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )} \, dx}{b n} \\ & = -\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )}-\frac {\left (3 \left (c x^n\right )^{3/n}\right ) \text {Subst}\left (\int \frac {e^{-\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2 x^3} \\ & = -\frac {3 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \text {Ei}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^3}-\frac {1}{b n x^3 \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {b n+3 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \operatorname {ExpIntegralEi}\left (-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )} \]

[In]

Integrate[1/(x^4*(a + b*Log[c*x^n])^2),x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.66 (sec) , antiderivative size = 354, normalized size of antiderivative = 4.66

method	result	size
risch	\(-\frac {2}{x^{3} \left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b +2 a \right ) b n}+\frac {3 \left (x^{n}\right )^{\frac {3}{n}} c^{\frac {3}{n}} {\mathrm e}^{\frac {-\frac {3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+3 a}{b n}} \operatorname {Ei}_{1}\left (3 \ln \left (x \right )+\frac {-\frac {3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {3 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+3 b \ln \left (c \right )+3 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+3 a}{b n}\right )}{b^{2} n^{2} x^{3}}\)	\(354\)

[In]

int(1/x^4/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {3 \, {\left (b n x^{3} \log \left (x\right ) + b x^{3} \log \left (c\right ) + a x^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{3}}\right ) + b n}{b^{3} n^{3} x^{3} \log \left (x\right ) + b^{3} n^{2} x^{3} \log \left (c\right ) + a b^{2} n^{2} x^{3}} \]

[In]

integrate(1/x^4/(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]

[In]

integrate(1/x**4/(a+b*ln(c*x**n))**2,x)

[Out]

Integral(1/(x**4*(a + b*log(c*x**n))**2), x)

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{x^4\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]

[In]

int(1/(x^4*(a + b*log(c*x^n))^2),x)

[Out]

int(1/(x^4*(a + b*log(c*x^n))^2), x)