3.382 \(\int \frac{(a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n))}{x} \, dx\)

Optimal. Leaf size=158 \[ n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )-2 b g n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )-\frac{\log (x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right ) \]

[Out]

Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) - (Log[x]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])^
2)/(4*b*g) + (Log[-((e*x)/d)]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])^2)/(4*b*g) + n*(b*f + a*g + 2*b*g*Log[c*(
d + e*x)^n])*PolyLog[2, 1 + (e*x)/d] - 2*b*g*n^2*PolyLog[3, 1 + (e*x)/d]

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Rubi [A]  time = 0.325995, antiderivative size = 219, normalized size of antiderivative = 1.39, number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2434, 2433, 2375, 2317, 2374, 6589} \[ g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-2 b g n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-\frac{g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}-\frac{b \log (x) \left (g \log \left (c (d+e x)^n\right )+f\right )^2}{2 g}+\frac{b \log \left (-\frac{e x}{d}\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )^2}{2 g} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x,x]

[Out]

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

Rule 2434

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.)
))/(x_), x_Symbol] :> Simp[Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[e*g*m, In
t[(Log[x]*(a + b*Log[c*(d + e*x)^n]))/(d + e*x), x], x] - Dist[b*j*n, Int[(Log[x]*(f + g*Log[h*(i + j*x)^m]))/
(i + j*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2375

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

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx &=\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-(b e n) \int \frac{\log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(e g n) \int \frac{\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx\\ &=\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-(b n) \operatorname{Subst}\left (\int \frac{\left (f+g \log \left (c x^n\right )\right ) \log \left (-\frac{d}{e}+\frac{x}{e}\right )}{x} \, dx,x,d+e x\right )-(g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac{d}{e}+\frac{x}{e}\right )}{x} \, dx,x,d+e x\right )\\ &=-\frac{g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac{b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac{b \operatorname{Subst}\left (\int \frac{\left (f+g \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{2 e g}+\frac{g \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{2 b e}\\ &=-\frac{g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac{b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac{b \log \left (-\frac{e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}-(b n) \operatorname{Subst}\left (\int \frac{\left (f+g \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )-(g n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )\\ &=-\frac{g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac{b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac{b \log \left (-\frac{e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )+b n \left (f+g \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )-2 \left (\left (b g n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )\right )\\ &=-\frac{g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\frac{g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac{b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac{b \log \left (-\frac{e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )+b n \left (f+g \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )-2 b g n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )\\ \end{align*}

Mathematica [A]  time = 0.0629501, size = 227, normalized size = 1.44 \[ a g n \text{PolyLog}\left (2,\frac{d+e x}{d}\right )+2 b g n \left (\log (x) \left (\log (d+e x)-\log \left (\frac{e x}{d}+1\right )\right )-\text{PolyLog}\left (2,-\frac{e x}{d}\right )\right ) \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )+b f n \text{PolyLog}\left (2,\frac{d+e x}{d}\right )+2 b g n^2 \left (-\text{PolyLog}\left (3,\frac{d+e x}{d}\right )+\log (d+e x) \text{PolyLog}\left (2,\frac{d+e x}{d}\right )+\frac{1}{2} \log \left (1-\frac{d+e x}{d}\right ) \log ^2(d+e x)\right )+a g \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )+a f \log (x)+b f \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )+b g \log (x) \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )^2 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x,x]

[Out]

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

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Maple [C]  time = 0.6, size = 1534, normalized size = 9.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*ln(x)*Pi*ln((e*x+d)^n)*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-dilog((e*x+d)/d)*a*g*n-dilog((e*
x+d)/d)*b*f*n-I*ln(x)*ln(c)*Pi*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/2*I*ln(x)*Pi*a*g*csgn(I*c
*(e*x+d)^n)^3-2*ln(x)*ln(c)*ln((e*x+d)/d)*b*g*n+2*dilog(-e*x/d)*ln((e*x+d)^n)*b*g*n+2*ln(x)*ln(c)*ln((e*x+d)^n
)*b*g+1/2*ln(x)*Pi^2*b*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5-1/4*ln(x)*Pi^2*b*g*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+
d)^n)^4+1/2*ln(x)*Pi^2*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-I*Pi*dilog((e*x+d)/d)*b*g*n*csgn(I*c)*csgn(
I*c*(e*x+d)^n)^2-I*Pi*dilog((e*x+d)/d)*b*g*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*I*ln(x)*Pi*b*f*csgn(I
*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*ln(x)*ln(c)*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*ln(x)
*Pi*ln((e*x+d)^n)*b*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/2*I*ln(x)*Pi*a*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*
(e*x+d)^n)+I*ln(x)*Pi*ln((e*x+d)/d)*b*g*n*csgn(I*c*(e*x+d)^n)^3+2*ln(-e*x/d)*ln((e*x+d)^n)*ln(e*x+d)*b*g*n-2*l
n(e*x)*ln((e*x+d)^n)*ln(e*x+d)*b*g*n+I*ln(x)*Pi*ln((e*x+d)/d)*b*g*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+
d)^n)+I*ln(x)*Pi*ln((e*x+d)^n)*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*ln(x)*ln(c)*Pi*b*g*csgn(I*c)*csgn
(I*c*(e*x+d)^n)^2-ln(x)*ln((e*x+d)/d)*a*g*n-ln(x)*ln((e*x+d)/d)*b*f*n-2*ln(-e*x/d)*ln(e*x+d)^2*b*g*n^2+ln(e*x)
*ln(e*x+d)^2*b*g*n^2+2*polylog(2,(e*x+d)/d)*ln(e*x+d)*b*g*n^2+ln(1-(e*x+d)/d)*ln(e*x+d)^2*b*g*n^2-1/4*ln(x)*Pi
^2*b*g*csgn(I*c*(e*x+d)^n)^6+ln(x)*ln(c)*b*f+ln(x)*ln(c)^2*b*g+ln(x)*ln(c)*a*g+ln(e*x)*ln((e*x+d)^n)^2*b*g-2*p
olylog(3,(e*x+d)/d)*b*g*n^2-2*dilog(-e*x/d)*ln(e*x+d)*b*g*n^2-2*ln(c)*dilog((e*x+d)/d)*b*g*n-I*ln(x)*Pi*ln((e*
x+d)/d)*b*g*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+ln(x)*ln((e*x+d)^n)*a*g+ln(x)*ln((e*x+d)^n)*b*f+a*f*ln(x
)-1/2*I*ln(x)*Pi*b*f*csgn(I*c*(e*x+d)^n)^3-1/4*ln(x)*Pi^2*b*g*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4-1/4*ln(x)*Pi^2
*b*g*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+1/2*ln(x)*Pi^2*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)^2*cs
gn(I*c*(e*x+d)^n)^3+I*Pi*dilog((e*x+d)/d)*b*g*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-I*ln(x)*Pi*ln(
(e*x+d)/d)*b*g*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/2*I*ln(x)*Pi*b*f*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-ln
(x)*Pi^2*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4+1/2*ln(x)*Pi^2*b*g*csgn(I*c)^2*csgn(I*(e*x+d)^n
)*csgn(I*c*(e*x+d)^n)^3+1/2*I*ln(x)*Pi*a*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/2*I*ln(x)*Pi*b*f*csgn(I*c
)*csgn(I*c*(e*x+d)^n)^2+1/2*I*ln(x)*Pi*a*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*ln(x)*ln(c)*Pi*b*g*csgn(I*c*(e*x+
d)^n)^3+I*Pi*dilog((e*x+d)/d)*b*g*n*csgn(I*c*(e*x+d)^n)^3-I*ln(x)*Pi*ln((e*x+d)^n)*b*g*csgn(I*c*(e*x+d)^n)^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*f*log(x) + integrate((b*g*log((e*x + d)^n)^2 + a*g*log(c) + (g*log(c)^2 + f*log(c))*b + ((2*g*log(c) + f)*b
+ a*g)*log((e*x + d)^n))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*g*log((e*x + d)^n*c)^2 + a*f + (b*f + a*g)*log((e*x + d)^n*c))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))*(f+g*ln(c*(e*x+d)**n))/x,x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))*(f + g*log(c*(d + e*x)**n))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log((e*x + d)^n*c) + a)*(g*log((e*x + d)^n*c) + f)/x, x)