3.8 \(\int \frac{1}{\log ^4(c (d+e x))} \, dx\)

Optimal. Leaf size=85 \[ \frac{\text{li}(c (d+e x))}{6 c e}-\frac{d+e x}{6 e \log ^2(c (d+e x))}-\frac{d+e x}{3 e \log ^3(c (d+e x))}-\frac{d+e x}{6 e \log (c (d+e x))} \]

[Out]

-(d + e*x)/(3*e*Log[c*(d + e*x)]^3) - (d + e*x)/(6*e*Log[c*(d + e*x)]^2) - (d + e*x)/(6*e*Log[c*(d + e*x)]) +
LogIntegral[c*(d + e*x)]/(6*c*e)

________________________________________________________________________________________

Rubi [A]  time = 0.032716, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2297, 2298} \[ \frac{\text{li}(c (d+e x))}{6 c e}-\frac{d+e x}{6 e \log ^2(c (d+e x))}-\frac{d+e x}{3 e \log ^3(c (d+e x))}-\frac{d+e x}{6 e \log (c (d+e x))} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^(-4),x]

[Out]

-(d + e*x)/(3*e*Log[c*(d + e*x)]^3) - (d + e*x)/(6*e*Log[c*(d + e*x)]^2) - (d + e*x)/(6*e*Log[c*(d + e*x)]) +
LogIntegral[c*(d + e*x)]/(6*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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

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 ^4(c (d+e x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^4(c x)} \, dx,x,d+e x\right )}{e}\\ &=-\frac{d+e x}{3 e \log ^3(c (d+e x))}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^3(c x)} \, dx,x,d+e x\right )}{3 e}\\ &=-\frac{d+e x}{3 e \log ^3(c (d+e x))}-\frac{d+e x}{6 e \log ^2(c (d+e x))}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^2(c x)} \, dx,x,d+e x\right )}{6 e}\\ &=-\frac{d+e x}{3 e \log ^3(c (d+e x))}-\frac{d+e x}{6 e \log ^2(c (d+e x))}-\frac{d+e x}{6 e \log (c (d+e x))}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log (c x)} \, dx,x,d+e x\right )}{6 e}\\ &=-\frac{d+e x}{3 e \log ^3(c (d+e x))}-\frac{d+e x}{6 e \log ^2(c (d+e x))}-\frac{d+e x}{6 e \log (c (d+e x))}+\frac{\text{li}(c (d+e x))}{6 c e}\\ \end{align*}

Mathematica [A]  time = 0.0186455, size = 57, normalized size = 0.67 \[ \frac{\frac{\text{li}(c (d+e x))}{c}-\frac{(d+e x) \left (\log ^2(c (d+e x))+\log (c (d+e x))+2\right )}{\log ^3(c (d+e x))}}{6 e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^(-4),x]

[Out]

(-(((d + e*x)*(2 + Log[c*(d + e*x)] + Log[c*(d + e*x)]^2))/Log[c*(d + e*x)]^3) + LogIntegral[c*(d + e*x)]/c)/(
6*e)

________________________________________________________________________________________

Maple [A]  time = 0.059, size = 116, normalized size = 1.4 \begin{align*} -{\frac{x}{3\, \left ( \ln \left ( cex+cd \right ) \right ) ^{3}}}-{\frac{d}{3\,e \left ( \ln \left ( cex+cd \right ) \right ) ^{3}}}-{\frac{x}{6\, \left ( \ln \left ( cex+cd \right ) \right ) ^{2}}}-{\frac{d}{6\,e \left ( \ln \left ( cex+cd \right ) \right ) ^{2}}}-{\frac{x}{6\,\ln \left ( cex+cd \right ) }}-{\frac{d}{6\,e\ln \left ( cex+cd \right ) }}-{\frac{{\it Ei} \left ( 1,-\ln \left ( cex+cd \right ) \right ) }{6\,ce}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(c*(e*x+d))^4,x)

[Out]

-1/3/ln(c*e*x+c*d)^3*x-1/3/e/ln(c*e*x+c*d)^3*d-1/6/ln(c*e*x+c*d)^2*x-1/6/e/ln(c*e*x+c*d)^2*d-1/6/ln(c*e*x+c*d)
*x-1/6/e/ln(c*e*x+c*d)*d-1/6/c/e*Ei(1,-ln(c*e*x+c*d))

________________________________________________________________________________________

Maxima [A]  time = 1.14629, size = 27, normalized size = 0.32 \begin{align*} \frac{\Gamma \left (-3, -\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))^4,x, algorithm="maxima")

[Out]

gamma(-3, -log(c*e*x + c*d))/(c*e)

________________________________________________________________________________________

Fricas [A]  time = 1.92004, size = 220, normalized size = 2.59 \begin{align*} \frac{\log \left (c e x + c d\right )^{3} \logintegral \left (c e x + c d\right ) - 2 \, c e x -{\left (c e x + c d\right )} \log \left (c e x + c d\right )^{2} - 2 \, c d -{\left (c e x + c d\right )} \log \left (c e x + c d\right )}{6 \, c e \log \left (c e x + c d\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d))^4,x, algorithm="fricas")

[Out]

1/6*(log(c*e*x + c*d)^3*log_integral(c*e*x + c*d) - 2*c*e*x - (c*e*x + c*d)*log(c*e*x + c*d)^2 - 2*c*d - (c*e*
x + c*d)*log(c*e*x + c*d))/(c*e*log(c*e*x + c*d)^3)

________________________________________________________________________________________

Sympy [A]  time = 0.860454, size = 71, normalized size = 0.84 \begin{align*} \frac{- d - e x + \left (- \frac{d}{2} - \frac{e x}{2}\right ) \log{\left (c \left (d + e x\right ) \right )}^{2} + \left (- \frac{d}{2} - \frac{e x}{2}\right ) \log{\left (c \left (d + e x\right ) \right )}}{3 e \log{\left (c \left (d + e x\right ) \right )}^{3}} + \frac{\operatorname{li}{\left (c d + c e x \right )}}{6 c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(c*(e*x+d))**4,x)

[Out]

(-d - e*x + (-d/2 - e*x/2)*log(c*(d + e*x))**2 + (-d/2 - e*x/2)*log(c*(d + e*x)))/(3*e*log(c*(d + e*x))**3) +
li(c*d + c*e*x)/(6*c*e)

________________________________________________________________________________________

Giac [A]  time = 1.21198, size = 109, normalized size = 1.28 \begin{align*} \frac{{\rm Ei}\left (\log \left ({\left (x e + d\right )} c\right )\right ) e^{\left (-1\right )}}{6 \, c} - \frac{{\left (x e + d\right )} e^{\left (-1\right )}}{6 \, \log \left ({\left (x e + d\right )} c\right )} - \frac{{\left (x e + d\right )} e^{\left (-1\right )}}{6 \, \log \left ({\left (x e + d\right )} c\right )^{2}} - \frac{{\left (x e + d\right )} e^{\left (-1\right )}}{3 \, \log \left ({\left (x e + d\right )} c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d))^4,x, algorithm="giac")

[Out]

1/6*Ei(log((x*e + d)*c))*e^(-1)/c - 1/6*(x*e + d)*e^(-1)/log((x*e + d)*c) - 1/6*(x*e + d)*e^(-1)/log((x*e + d)
*c)^2 - 1/3*(x*e + d)*e^(-1)/log((x*e + d)*c)^3