\(\int \frac {1}{(a+b \log (c x^n))^2} \, dx\) [76]

Optimal result

Integrand size = 12, antiderivative size = 70 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2}-\frac {x}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2334, 2337, 2209} \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b^2 n^2}-\frac {x}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[(a + b*Log[c*x^n])^(-2),x]

[Out]

(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]))

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 2334

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] - Dist[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] &&
IntegerQ[2*p]

Rule 2337

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x \left (e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )-\frac {b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]

[In]

Integrate[(a + b*Log[c*x^n])^(-2),x]

[Out]

(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
)

Maple [C] (warning: unable to verify)

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

Time = 0.34 (sec) , antiderivative size = 350, normalized size of antiderivative = 5.00

method	result	size
risch	\(-\frac {2 x}{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b +2 a \right ) b n}-\frac {x \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} {\mathrm e}^{-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{2 b n}} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{2 b n}\right )}{b^{2} n^{2}}\)	\(350\)

[In]

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

[Out]

-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*ln(x^n)*b+2*a)/b/n*x-1/b^2/n^2*x*(x^n)^(-1/n)*c^(-1/n)*exp(-1/2*(-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,-ln(x)-1/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*c
sgn(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 = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {{\left (b n x e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (69) = 138\).

Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.40 \[ \int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {b n {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (x\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} - \frac {b n x}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} + \frac {b {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} + \frac {a {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )}}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

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*log(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))

Mupad [F(-1)]

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

[In]

int(1/(a + b*log(c*x^n))^2,x)

[Out]

int(1/(a + b*log(c*x^n))^2, x)