3.86 \(\int \frac{\log ^2(c (a+b x)^n)}{x^4} \, dx\)

Optimal. Leaf size=177 \[ -\frac{2 b^3 n^2 \text{PolyLog}\left (2,\frac{a}{a+b x}\right )}{3 a^3}+\frac{2 b^3 n \log \left (1-\frac{a}{a+b x}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}+\frac{2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac{b^2 n^2}{3 a^2 x}-\frac{b^3 n^2 \log (x)}{a^3}+\frac{b^3 n^2 \log (a+b x)}{3 a^3}-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac{b n \log \left (c (a+b x)^n\right )}{3 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.306162, antiderivative size = 193, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac{2 b^3 n^2 \text{PolyLog}\left (2,\frac{b x}{a}+1\right )}{3 a^3}-\frac{b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}+\frac{2 b^3 n \log \left (-\frac{b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}+\frac{2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac{b^2 n^2}{3 a^2 x}-\frac{b^3 n^2 \log (x)}{a^3}+\frac{b^3 n^2 \log (a+b x)}{3 a^3}-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac{b n \log \left (c (a+b x)^n\right )}{3 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^n]^2/x^4,x]

[Out]

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

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx &=-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac{1}{3} (2 b n) \int \frac{\log \left (c (a+b x)^n\right )}{x^3 (a+b x)} \, dx\\ &=-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac{1}{3} (2 n) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x\right )\\ &=-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac{(2 n) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{\left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x\right )}{3 a}-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x\right )}{3 a}\\ &=-\frac{b n \log \left (c (a+b x)^n\right )}{3 a x^2}-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x\right )}{3 a^2}+\frac{\left (2 b^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )} \, dx,x,a+b x\right )}{3 a^2}+\frac{\left (b n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x\right )}{3 a}\\ &=-\frac{b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac{2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac{\left (2 b^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x\right )}{3 a^3}-\frac{\left (2 b^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{x} \, dx,x,a+b x\right )}{3 a^3}+\frac{\left (b n^2\right ) \operatorname{Subst}\left (\int \left (\frac{b^2}{a (a-x)^2}+\frac{b^2}{a^2 (a-x)}+\frac{b^2}{a^2 x}\right ) \, dx,x,a+b x\right )}{3 a}-\frac{\left (2 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x\right )}{3 a^3}\\ &=-\frac{b^2 n^2}{3 a^2 x}-\frac{b^3 n^2 \log (x)}{a^3}+\frac{b^3 n^2 \log (a+b x)}{3 a^3}-\frac{b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac{2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}+\frac{2 b^3 n \log \left (-\frac{b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac{b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac{\left (2 b^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{a}\right )}{x} \, dx,x,a+b x\right )}{3 a^3}\\ &=-\frac{b^2 n^2}{3 a^2 x}-\frac{b^3 n^2 \log (x)}{a^3}+\frac{b^3 n^2 \log (a+b x)}{3 a^3}-\frac{b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac{2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}+\frac{2 b^3 n \log \left (-\frac{b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac{b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac{2 b^3 n^2 \text{Li}_2\left (1+\frac{b x}{a}\right )}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0566782, size = 186, normalized size = 1.05 \[ \frac{2 b^3 n^2 \text{PolyLog}\left (2,\frac{a+b x}{a}\right )}{3 a^3}-\frac{b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}+\frac{2 b^3 n \log \left (-\frac{b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}+\frac{2 b^2 n \log \left (c (a+b x)^n\right )}{3 a^2 x}-\frac{b^2 n^2}{3 a^2 x}-\frac{b^3 n^2 \log (x)}{a^3}+\frac{b^3 n^2 \log (a+b x)}{a^3}-\frac{\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac{b n \log \left (c (a+b x)^n\right )}{3 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x)^n]^2/x^4,x]

[Out]

-(b^2*n^2)/(3*a^2*x) - (b^3*n^2*Log[x])/a^3 + (b^3*n^2*Log[a + b*x])/a^3 - (b*n*Log[c*(a + b*x)^n])/(3*a*x^2)
+ (2*b^2*n*Log[c*(a + b*x)^n])/(3*a^2*x) + (2*b^3*n*Log[-((b*x)/a)]*Log[c*(a + b*x)^n])/(3*a^3) - (b^3*Log[c*(
a + b*x)^n]^2)/(3*a^3) - Log[c*(a + b*x)^n]^2/(3*x^3) + (2*b^3*n^2*PolyLog[2, (a + b*x)/a])/(3*a^3)

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Maple [C]  time = 0.546, size = 1102, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x+a)^n)^2/x^4,x)

[Out]

-2/3*b^3*n^2/a^3*dilog(1/a*(b*x+a))+1/3*b^3*n^2/a^3*ln(b*x+a)^2+1/3*I*b^3*n/a^3*ln(b*x+a)*Pi*csgn(I*(b*x+a)^n)
*csgn(I*c*(b*x+a)^n)*csgn(I*c)+2/3*b^2*n/a^2/x*ln(c)+2/3*b^3*n/a^3*ln(x)*ln(c)-2/3*b^3*n/a^3*ln(b*x+a)*ln(c)-2
/3*b^3*n^2/a^3*ln(x)*ln(1/a*(b*x+a))-1/3*b*n/a/x^2*ln(c)-1/3*I/x^3*ln((b*x+a)^n)*Pi*csgn(I*c*(b*x+a)^n)^2*csgn
(I*c)+1/3*I/x^3*ln((b*x+a)^n)*Pi*csgn(I*c*(b*x+a)^n)^3+1/6*I*b*n/a/x^2*Pi*csgn(I*c*(b*x+a)^n)^3-1/3*I*b^3*n/a^
3*ln(x)*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)*csgn(I*c)+1/6*I*b*n/a/x^2*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+
a)^n)*csgn(I*c)+1/3*I/x^3*ln((b*x+a)^n)*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)*csgn(I*c)+2/3*b^2*n*ln((b*x+a
)^n)/a^2/x-1/3*I*b^2*n/a^2/x*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)*csgn(I*c)-1/3*I/x^3*ln((b*x+a)^n)*Pi*csg
n(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)^2-2/3*b^3*n*ln((b*x+a)^n)/a^3*ln(b*x+a)-1/3*b*n*ln((b*x+a)^n)/a/x^2+2/3*b^3
*n*ln((b*x+a)^n)/a^3*ln(x)-1/6*I*b*n/a/x^2*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)+1/3*I*b^3*n/a^3*ln(x)*Pi*csgn(I*
c*(b*x+a)^n)^2*csgn(I*c)-1/3*I*b^3*n/a^3*ln(b*x+a)*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)+1/3*I*b^2*n/a^2/x*Pi*csg
n(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)^2+1/3*I*b^3*n/a^3*ln(x)*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)^2-1/6*I*b*
n/a/x^2*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)^2-1/3*I*b^3*n/a^3*ln(b*x+a)*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*
x+a)^n)^2+1/3*I*b^2*n/a^2/x*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)+1/3*I*b^3*n/a^3*ln(b*x+a)*Pi*csgn(I*c*(b*x+a)^n
)^3-1/3*I*b^3*n/a^3*ln(x)*Pi*csgn(I*c*(b*x+a)^n)^3-2/3/x^3*ln((b*x+a)^n)*ln(c)-1/12*(I*Pi*csgn(I*(b*x+a)^n)*cs
gn(I*c*(b*x+a)^n)^2-I*Pi*csgn(I*(b*x+a)^n)*csgn(I*c*(b*x+a)^n)*csgn(I*c)-I*Pi*csgn(I*c*(b*x+a)^n)^3+I*Pi*csgn(
I*c*(b*x+a)^n)^2*csgn(I*c)+2*ln(c))^2/x^3-1/3/x^3*ln((b*x+a)^n)^2-1/3*I*b^2*n/a^2/x*Pi*csgn(I*c*(b*x+a)^n)^3-1
/3*b^2*n^2/x/a^2-b^3*n^2*ln(x)/a^3+b^3*n^2*ln(b*x+a)/a^3

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Maxima [A]  time = 1.2232, size = 203, normalized size = 1.15 \begin{align*} -\frac{1}{3} \, b^{2} n^{2}{\left (\frac{2 \,{\left (\log \left (\frac{b x}{a} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{b x}{a}\right )\right )} b}{a^{3}} - \frac{3 \, b \log \left (b x + a\right )}{a^{3}} - \frac{b x \log \left (b x + a\right )^{2} - 3 \, b x \log \left (x\right ) - a}{a^{3} x}\right )} - \frac{1}{3} \, b n{\left (\frac{2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac{2 \, b^{2} \log \left (x\right )}{a^{3}} - \frac{2 \, b x - a}{a^{2} x^{2}}\right )} \log \left ({\left (b x + a\right )}^{n} c\right ) - \frac{\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^n)^2/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^n)^2/x^4,x, algorithm="fricas")

[Out]

integral(log((b*x + a)^n*c)^2/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x+a)**n)**2/x**4,x)

[Out]

Integral(log(c*(a + b*x)**n)**2/x**4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^n)^2/x^4,x, algorithm="giac")

[Out]

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