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

Optimal. Leaf size=98 \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3}-\frac{x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0524806, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2297, 2300, 2178} \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{2 b^3 n^3}-\frac{x}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{x}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^(-3),x]

[Out]

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

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

Mathematica [A]  time = 0.113674, size = 82, normalized size = 0.84 \[ \frac{x \left (e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )-\frac{b n \left (a+b \log \left (c x^n\right )+b n\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^(-3),x]

[Out]

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

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Maple [C]  time = 0.268, size = 460, normalized size = 4.7 \begin{align*} -{\frac{2\,bnx+i\pi \,bx{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,bx{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \,bx \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,bx \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( c \right ) bx+2\,bx\ln \left ({x}^{n} \right ) +2\,ax}{ \left ( 2\,a+2\,b\ln \left ( c \right ) +2\,b\ln \left ({x}^{n} \right ) +ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \right ) ^{2}{b}^{2}{n}^{2}}}-{\frac{1}{2\,{b}^{3}{n}^{3}}{{\rm e}^{-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\ln \left ( x \right ) bn+2\,b\ln \left ( c \right ) +2\,b\ln \left ({x}^{n} \right ) +2\,a}{2\,bn}}}}{\it Ei} \left ( 1,-\ln \left ( x \right ) -{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +2\,b \left ( \ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) \right ) +2\,a}{2\,bn}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B]  time = 0.761523, size = 489, normalized size = 4.99 \begin{align*} -\frac{{\left ({\left (b^{2} n^{2} x \log \left (x\right ) + b^{2} n x \log \left (c\right ) +{\left (b^{2} n^{2} + a b n\right )} x\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} -{\left (b^{2} n^{2} \log \left (x\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 (x\right )\right )} \logintegral \left (x 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} n^{5} \log \left (x\right )^{2} + b^{5} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} n^{3} \log \left (c\right ) + a^{2} b^{3} n^{3} + 2 \,{\left (b^{5} n^{4} \log \left (c\right ) + a b^{4} n^{4}\right )} \log \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.32889, size = 1326, normalized size = 13.53 \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*x^n))^3,x, algorithm="giac")

[Out]

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