3.1.5 \(\int \frac {1}{\log (c (d+e x))} \, dx\) [5]

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

[Out]

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

________________________________________________________________________________________

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 = {2436, 2335} \begin {gather*} \frac {\text {li}(c (d+e x))}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^(-1),x]

[Out]

LogIntegral[c*(d + e*x)]/(c*e)

Rule 2335

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

Rule 2436

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 {\text {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 = 0.02, size = 17, normalized size = 1.13 \begin {gather*} \frac {\text {Ei}(\log (c d+c e x))}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^(-1),x]

[Out]

ExpIntegralEi[Log[c*d + c*e*x]]/(c*e)

________________________________________________________________________________________

Maple [A]
time = 0.21, size = 22, normalized size = 1.47

method	result	size
derivativedivides	\(-\frac {\expIntegral \left (1, -\ln \left (c e x +c d \right )\right )}{c e}\)	\(22\)
default	\(-\frac {\expIntegral \left (1, -\ln \left (c e x +c d \right )\right )}{c e}\)	\(22\)
risch	\(-\frac {\expIntegral \left (1, -\ln \left (c e x +c d \right )\right )}{c e}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.33, size = 17, normalized size = 1.13 \begin {gather*} \frac {{\rm Ei}\left (\log \left (c x e + c d\right )\right ) e^{\left (-1\right )}}{c} \end {gather*}

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*x*e + c*d))*e^(-1)/c

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 16, normalized size = 1.07 \begin {gather*} \frac {e^{\left (-1\right )} \operatorname {log\_integral}\left (c x e + c d\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1)*log_integral(c*x*e + c*d)/c

________________________________________________________________________________________

Sympy [A]
time = 0.34, size = 12, normalized size = 0.80 \begin {gather*} \frac {\operatorname {li}{\left (c d + c e x \right )}}{c e} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 4.10, size = 16, normalized size = 1.07 \begin {gather*} \frac {{\rm Ei}\left (\log \left ({\left (x e + d\right )} c\right )\right ) e^{\left (-1\right )}}{c} \end {gather*}

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________