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]

Li(c*(e*x+d))/c/e

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2298

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

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]

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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\text {li}(c (d+e x))}{c e} \]

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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fricas [A]  time = 0.64, size = 16, normalized size = 1.07 \[ \frac {\operatorname {log\_integral}\left (c e x + c d\right )}{c e} \]

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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giac [A]  time = 0.20, size = 16, normalized size = 1.07 \[ \frac {{\rm Ei}\left (\log \left ({\left (x e + d\right )} c\right )\right ) e^{\left (-1\right )}}{c} \]

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

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.09, size = 17, normalized size = 1.13 \[ \frac {{\rm Ei}\left (\log \left (c e x + c d\right )\right )}{c e} \]

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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mupad [B]  time = 0.21, size = 15, normalized size = 1.00 \[ \frac {\mathrm {logint}\left (c\,\left (d+e\,x\right )\right )}{c\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(c*(d + e*x)),x)

[Out]

logint(c*(d + e*x))/(c*e)

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sympy [A]  time = 0.75, size = 12, normalized size = 0.80 \[ \frac {\operatorname {li}{\left (c d + c e x \right )}}{c e} \]

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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