3.5 \(\int \frac{1}{\log (c (d+e x))} \, dx\)

Optimal. Leaf size=15 \[ \frac{\text{li}(c (d+e x))}{c e} \]

[Out]

LogIntegral[c*(d + e*x)]/(c*e)

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Rubi [A]  time = 0.0092104, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2298} \[ \frac{\text{li}(c (d+e x))}{c e} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^(-1),x]

[Out]

LogIntegral[c*(d + e*x)]/(c*e)

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rubi steps

\begin{align*} \int \frac{1}{\log (c (d+e x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\log (c x)} \, dx,x,d+e x\right )}{e}\\ &=\frac{\text{li}(c (d+e x))}{c e}\\ \end{align*}

Mathematica [A]  time = 0.0076994, size = 15, normalized size = 1. \[ \frac{\text{li}(c (d+e x))}{c e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^(-1),x]

[Out]

LogIntegral[c*(d + e*x)]/(c*e)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(c*(e*x+d)),x)

[Out]

int(1/ln(c*(e*x+d)),x)

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Maxima [A]  time = 1.11473, size = 23, normalized size = 1.53 \begin{align*} \frac{{\rm Ei}\left (\log \left (c e x + c d\right )\right )}{c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d)),x, algorithm="maxima")

[Out]

Ei(log(c*e*x + c*d))/(c*e)

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Fricas [A]  time = 2.0023, size = 45, normalized size = 3. \begin{align*} \frac{\logintegral \left (c e x + c d\right )}{c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d)),x, algorithm="fricas")

[Out]

log_integral(c*e*x + c*d)/(c*e)

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Sympy [A]  time = 0.765209, size = 12, normalized size = 0.8 \begin{align*} \frac{\operatorname{li}{\left (c d + c e x \right )}}{c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(c*(e*x+d)),x)

[Out]

li(c*d + c*e*x)/(c*e)

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Giac [A]  time = 1.27484, size = 22, normalized size = 1.47 \begin{align*} \frac{{\rm Ei}\left (\log \left ({\left (x e + d\right )} c\right )\right ) e^{\left (-1\right )}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d)),x, algorithm="giac")

[Out]

Ei(log((x*e + d)*c))*e^(-1)/c