\(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n))}{x} \, dx\) [382]

Optimal result

Integrand size = 32, antiderivative size = 158 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {\log (x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 b g n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right ) \]

[Out]

ln(x)*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(c*(e*x+d)^n))-1/4*ln(x)*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))^2/b/g+1/4*ln(-e*x/
d)*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))^2/b/g+n*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))*polylog(2,1+e*x/d)-2*b*g*n^2*polylo
g(3,1+e*x/d)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2482, 2481, 2422, 2354, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )-\frac {\log (x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-2 b g n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right ) \]

[In]

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

[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]

Rule 2354

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2422

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 2481

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 2482

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(g_.))
)/(x_), x_Symbol] :> Simp[Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]), x] - Dist[e*n, Int[(Lo
g[x]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x]

Rule 6724

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*} \text {integral}& = \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 )-(e n) \int \frac {\log (x) \left (b f+a g+2 b g \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 )-n \text {Subst}\left (\int \frac {\left (b f+a g+2 b g \log \left (c x^n\right )\right ) \log \left (-\frac {d}{e}+\frac {x}{e}\right )}{x} \, dx,x,d+e x\right ) \\ & = \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 {\log (x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+\frac {\text {Subst}\left (\int \frac {\left (b f+a g+2 b g \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{4 b e g} \\ & = \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 {\log (x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}-n \text {Subst}\left (\int \frac {\left (b f+a g+2 b g \log \left (c x^n\right )\right ) \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right ) \\ & = \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 {\log (x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )-\left (2 b g n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right ) \\ & = \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 {\log (x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+n \left (b f+a g+2 b 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] (verified)

Time = 0.07 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=a f \log (x)+b f \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+a g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+b g \log (x) \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )^2+2 b g n \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+b f n \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )+a g n \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )+2 b g n^2 \left (\frac {1}{2} \log ^2(d+e x) \log \left (1-\frac {d+e x}{d}\right )+\log (d+e x) \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )-\operatorname {PolyLog}\left (3,\frac {d+e x}{d}\right )\right ) \]

[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])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.76 (sec) , antiderivative size = 626, normalized size of antiderivative = 3.96

method	result	size
risch	\(\left (-i \pi b g \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b g \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b g \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b g \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b \ln \left (c \right ) g +a g +b f \right ) \left (\ln \left (\left (e x +d \right )^{n}\right ) \ln \left (x \right )-e n \left (\frac {\operatorname {dilog}\left (\frac {e x +d}{d}\right )}{e}+\frac {\ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{e}\right )\right )+\ln \left (e x +d \right )^{2} \ln \left (e x \right ) b g \,n^{2}+\ln \left (e x +d \right )^{2} \ln \left (1-\frac {e x +d}{d}\right ) b g \,n^{2}-2 \ln \left (e x +d \right )^{2} \ln \left (-\frac {e x}{d}\right ) b g \,n^{2}-2 \ln \left (e x +d \right ) \ln \left (e x \right ) \ln \left (\left (e x +d \right )^{n}\right ) b g n +2 \ln \left (e x +d \right ) \operatorname {Li}_{2}\left (\frac {e x +d}{d}\right ) b g \,n^{2}-2 \ln \left (e x +d \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) b g \,n^{2}+2 \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) \ln \left (\left (e x +d \right )^{n}\right ) b g n +\ln \left (e x \right ) \ln \left (\left (e x +d \right )^{n}\right )^{2} b g -2 \,\operatorname {Li}_{3}\left (\frac {e x +d}{d}\right ) b g \,n^{2}+2 \operatorname {dilog}\left (-\frac {e x}{d}\right ) \ln \left (\left (e x +d \right )^{n}\right ) b g n +\frac {\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 a \right ) \left (-i g \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+i g \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i g \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i g \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 g \ln \left (c \right )+2 f \right ) \ln \left (x \right )}{4}\)	\(626\)

[In]

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

[Out]

(-I*Pi*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*Pi*b*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*g*c
sgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*g*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)*g+a*g+b*f)*(ln((e*x+d)^n)*ln(x
)-e*n*(dilog((e*x+d)/d)/e+ln(x)*ln((e*x+d)/d)/e))+ln(e*x+d)^2*ln(e*x)*b*g*n^2+ln(e*x+d)^2*ln(1-(e*x+d)/d)*b*g*
n^2-2*ln(e*x+d)^2*ln(-e*x/d)*b*g*n^2-2*ln(e*x+d)*ln(e*x)*ln((e*x+d)^n)*b*g*n+2*ln(e*x+d)*polylog(2,(e*x+d)/d)*
b*g*n^2-2*ln(e*x+d)*dilog(-e*x/d)*b*g*n^2+2*ln(e*x+d)*ln(-e*x/d)*ln((e*x+d)^n)*b*g*n+ln(e*x)*ln((e*x+d)^n)^2*b
*g-2*polylog(3,(e*x+d)/d)*b*g*n^2+2*dilog(-e*x/d)*ln((e*x+d)^n)*b*g*n+1/4*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*
c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*P
i*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*a)*(-I*g*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*g*Pi*csgn(
I*c)*csgn(I*c*(e*x+d)^n)^2+I*g*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*g*Pi*csgn(I*c*(e*x+d)^n)^3+2*g*ln(
c)+2*f)*ln(x)

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (e x + d\right )}^{n} c\right ) + f\right )}}{x} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{x}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (e x + d\right )}^{n} c\right ) + f\right )}}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (e x + d\right )}^{n} c\right ) + f\right )}}{x} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x} \,d x \]

[In]

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

[Out]

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