\(\int \frac {1}{a+b \log (c x^n)} \, dx\) [68]

Optimal result

Integrand size = 12, antiderivative size = 48 \[ \int \frac {1}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2337, 2209} \[ \int \frac {1}{a+b \log \left (c x^n\right )} \, dx=\frac {x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]

[In]

Int[(a + b*Log[c*x^n])^(-1),x]

[Out]

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

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 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 {\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 )}{n} \\ & = \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 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]

[In]

Integrate[(a + b*Log[c*x^n])^(-1),x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 5.00

method	result	size
risch	\(-\frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{2 b n}} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{2 b n}\right )}{b n}\)	\(240\)

[In]

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

[Out]

-1/b/n*x*c^(-1/n)*(x^n)^(-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)*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 = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {1}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )}{b n} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/(a + b*log(c*x**n)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/(b*log(c*x^n) + a), x)

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int \frac {1}{a+b \log \left (c x^n\right )} \, dx=\frac {{\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )}}{b c^{\left (\frac {1}{n}\right )} n} \]

[In]

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

[Out]

Ei(log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))/(b*c^(1/n)*n)

Mupad [F(-1)]

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

[In]

int(1/(a + b*log(c*x^n)),x)

[Out]

int(1/(a + b*log(c*x^n)), x)