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

Optimal. Leaf size=48 \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0358402, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2300, 2178} \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]

Antiderivative was successfully verified.

[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 2300

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]

Rule 2178

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

Rubi steps

\begin{align*} \int \frac{1}{a+b \log \left (c x^n\right )} \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{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]  time = 0.0410216, size = 48, normalized size = 1. \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]

Antiderivative was successfully verified.

[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))

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Maple [C]  time = 0.267, size = 241, normalized size = 5. \begin{align*} -{\frac{1}{bn}{{\rm e}^{-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\ln \left ( x \right ) bn+2\,b\ln \left ( c \right ) +2\,b\ln \left ({x}^{n} \right ) +2\,a}{2\,bn}}}}{\it Ei} \left ( 1,-\ln \left ( x \right ) -{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +2\,b \left ( \ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) \right ) +2\,a}{2\,bn}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*log(c*x^n) + a), x)

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Fricas [A]  time = 0.836507, size = 100, normalized size = 2.08 \begin{align*} \frac{e^{\left (-\frac{b \log \left (c\right ) + a}{b n}\right )} \logintegral \left (x e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \log{\left (c x^{n} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a + b*log(c*x**n)), x)

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Giac [A]  time = 1.21367, size = 57, normalized size = 1.19 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)