3.23 \(\int \frac{1}{(a+b \log (c (d+e x)^n))^3} \, dx\)

Optimal. Leaf size=135 \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

[Out]

((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(2*b^3*e*E^(a/(b*n))*n^3*(c*(d + e*x)^n)^n^(-1)) -
(d + e*x)/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) - (d + e*x)/(2*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))

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Rubi [A]  time = 0.0812606, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2389, 2297, 2300, 2178} \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^(-3),x]

[Out]

((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(2*b^3*e*E^(a/(b*n))*n^3*(c*(d + e*x)^n)^n^(-1)) -
(d + e*x)/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) - (d + e*x)/(2*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))

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 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}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx,x,d+e x\right )}{e}\\ &=-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{2 b e n}\\ &=-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e n^2}\\ &=-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e n^3}\\ &=\frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.0950437, size = 144, normalized size = 1.07 \[ -\frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (b n e^{\frac{a}{b n}} \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (a+b \log \left (c (d+e x)^n\right )+b n\right )-\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )\right )}{2 b^3 e n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^(-3),x]

[Out]

-((d + e*x)*(-(ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2) + b*E^(a/(b*n))*n
*(c*(d + e*x)^n)^n^(-1)*(a + b*n + b*Log[c*(d + e*x)^n])))/(2*b^3*e*E^(a/(b*n))*n^3*(c*(d + e*x)^n)^n^(-1)*(a
+ b*Log[c*(d + e*x)^n])^2)

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Maple [C]  time = 0.543, size = 735, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*ln(c*(e*x+d)^n))^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (d n + d \log \left (c\right )\right )} b + a d +{\left ({\left (e n + e \log \left (c\right )\right )} b + a e\right )} x +{\left (b e x + b d\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{2 \,{\left (b^{4} e n^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{4} e n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} e n^{2} \log \left (c\right ) + a^{2} b^{2} e n^{2} + 2 \,{\left (b^{4} e n^{2} \log \left (c\right ) + a b^{3} e n^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}} + \int \frac{1}{2 \,{\left (b^{3} n^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

-1/2*((d*n + d*log(c))*b + a*d + ((e*n + e*log(c))*b + a*e)*x + (b*e*x + b*d)*log((e*x + d)^n))/(b^4*e*n^2*log
((e*x + d)^n)^2 + b^4*e*n^2*log(c)^2 + 2*a*b^3*e*n^2*log(c) + a^2*b^2*e*n^2 + 2*(b^4*e*n^2*log(c) + a*b^3*e*n^
2)*log((e*x + d)^n)) + integrate(1/2/(b^3*n^2*log((e*x + d)^n) + b^3*n^2*log(c) + a*b^2*n^2), x)

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Fricas [B]  time = 2.04376, size = 632, normalized size = 4.68 \begin{align*} -\frac{{\left ({\left (b^{2} d n^{2} + a b d n +{\left (b^{2} e n^{2} + a b e n\right )} x +{\left (b^{2} e n^{2} x + b^{2} d n^{2}\right )} \log \left (e x + d\right ) +{\left (b^{2} e n x + b^{2} d n\right )} \log \left (c\right )\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} -{\left (b^{2} n^{2} \log \left (e x + d\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (e x + d\right )\right )} \logintegral \left ({\left (e x + d\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac{b \log \left (c\right ) + a}{b n}\right )}}{2 \,{\left (b^{5} e n^{5} \log \left (e x + d\right )^{2} + b^{5} e n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} e n^{3} \log \left (c\right ) + a^{2} b^{3} e n^{3} + 2 \,{\left (b^{5} e n^{4} \log \left (c\right ) + a b^{4} e n^{4}\right )} \log \left (e x + d\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

-1/2*((b^2*d*n^2 + a*b*d*n + (b^2*e*n^2 + a*b*e*n)*x + (b^2*e*n^2*x + b^2*d*n^2)*log(e*x + d) + (b^2*e*n*x + b
^2*d*n)*log(c))*e^((b*log(c) + a)/(b*n)) - (b^2*n^2*log(e*x + d)^2 + b^2*log(c)^2 + 2*a*b*log(c) + a^2 + 2*(b^
2*n*log(c) + a*b*n)*log(e*x + d))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))))*e^(-(b*log(c) + a)/(b*n))/
(b^5*e*n^5*log(e*x + d)^2 + b^5*e*n^3*log(c)^2 + 2*a*b^4*e*n^3*log(c) + a^2*b^3*e*n^3 + 2*(b^5*e*n^4*log(c) +
a*b^4*e*n^4)*log(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))**(-3), x)

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Giac [B]  time = 1.32825, size = 1785, normalized size = 13.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

1/2*b^2*n^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n))*log(x*e + d)^2/((b^5*n^5*e*log(x*e + d)^2 + 2*b
^5*n^4*e*log(x*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^
3*n^3*e)*c^(1/n)) - 1/2*(x*e + d)*b^2*n^2*log(x*e + d)/(b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log(x*e + d)*lo
g(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e) + b^2*n*Ei(log(
c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n))*log(x*e + d)*log(c)/((b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log(x
*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e)*c^(1/
n)) - 1/2*(x*e + d)*b^2*n^2/(b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log(x*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*
e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e) + a*b*n*Ei(log(c)/n + a/(b*n) + log(x*e +
d))*e^(-a/(b*n))*log(x*e + d)/((b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log(x*e + d)*log(c) + 2*a*b^4*n^4*e*log
(x*e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e)*c^(1/n)) - 1/2*(x*e + d)*b^2*n*log(c)/(
b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log(x*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2 +
2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e) + 1/2*b^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n))*log(c)^2/(
(b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log(x*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2
+ 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e)*c^(1/n)) - 1/2*(x*e + d)*a*b*n/(b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e
*log(x*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e)
+ a*b*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n))*log(c)/((b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log(
x*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e)*c^(1
/n)) + 1/2*a^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n))/((b^5*n^5*e*log(x*e + d)^2 + 2*b^5*n^4*e*log
(x*e + d)*log(c) + 2*a*b^4*n^4*e*log(x*e + d) + b^5*n^3*e*log(c)^2 + 2*a*b^4*n^3*e*log(c) + a^2*b^3*n^3*e)*c^(
1/n))