Integrand size = 26, antiderivative size = 411 \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\frac {3}{2} b^3 n^3 x^2-\frac {9}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \left (1+d f x^2\right ) \log \left (1+d f x^2\right )}{8 d f}+\frac {3 b^2 n^2 \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d f}-\frac {3 b n \left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d f}+\frac {\left (1+d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 d f}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (2,-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f x^2\right )}{4 d f}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,-d f x^2\right )}{8 d f}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^2\right )}{4 d f}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,-d f x^2\right )}{8 d f} \]
3/2*b^3*n^3*x^2-9/4*b^2*n^2*x^2*(a+b*ln(c*x^n))+3/2*b*n*x^2*(a+b*ln(c*x^n) )^2-1/2*x^2*(a+b*ln(c*x^n))^3-3/8*b^3*n^3*(d*f*x^2+1)*ln(d*f*x^2+1)/d/f+3/ 4*b^2*n^2*(d*f*x^2+1)*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/d/f-3/4*b*n*(d*f*x^2+1 )*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)/d/f+1/2*(d*f*x^2+1)*(a+b*ln(c*x^n))^3*ln (d*f*x^2+1)/d/f+3/8*b^3*n^3*polylog(2,-d*f*x^2)/d/f-3/4*b^2*n^2*(a+b*ln(c* x^n))*polylog(2,-d*f*x^2)/d/f+3/4*b*n*(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^2 )/d/f+3/8*b^3*n^3*polylog(3,-d*f*x^2)/d/f-3/4*b^2*n^2*(a+b*ln(c*x^n))*poly log(3,-d*f*x^2)/d/f+3/8*b^3*n^3*polylog(4,-d*f*x^2)/d/f
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 1004, normalized size of antiderivative = 2.44 \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\frac {-d f x^2 \left (4 a^3-6 a^2 b n+6 a b^2 n^2-3 b^3 n^3+12 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+d f x^2 \left (4 a^3-6 a^2 b n+6 a b^2 n^2-3 b^3 n^3+6 b \left (2 a^2-2 a b n+b^2 n^2\right ) \log \left (c x^n\right )-6 b^2 (-2 a+b n) \log ^2\left (c x^n\right )+4 b^3 \log ^3\left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\left (4 a^3-6 a^2 b n+6 a b^2 n^2-3 b^3 n^3+12 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right ) \log \left (1+d f x^2\right )+6 b n \left (2 a^2-2 a b n+b^2 n^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+4 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (\frac {1}{2} d f x^2-d f x^2 \log (x)+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )+3 b^2 n^2 \left (-2 a+b n+2 b n \log (x)-2 b \log \left (c x^n\right )\right ) \left (d f x^2-2 d f x^2 \log (x)+2 d f x^2 \log ^2(x)-2 \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-2 \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-4 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-4 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+4 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )+4 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )-b^3 n^3 \left (-3 d f x^2+6 d f x^2 \log (x)-6 d f x^2 \log ^2(x)+4 d f x^2 \log ^3(x)-4 \log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-4 \log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-12 \log ^2(x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-12 \log ^2(x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+24 \log (x) \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )+24 \log (x) \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )-24 \operatorname {PolyLog}\left (4,-i \sqrt {d} \sqrt {f} x\right )-24 \operatorname {PolyLog}\left (4,i \sqrt {d} \sqrt {f} x\right )\right )}{8 d f} \]
(-(d*f*x^2*(4*a^3 - 6*a^2*b*n + 6*a*b^2*n^2 - 3*b^3*n^3 + 12*a*b^2*n*(n*Lo g[x] - Log[c*x^n]) + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n* Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 6*b^3*n*(- (n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3)) + d*f*x^ 2*(4*a^3 - 6*a^2*b*n + 6*a*b^2*n^2 - 3*b^3*n^3 + 6*b*(2*a^2 - 2*a*b*n + b^ 2*n^2)*Log[c*x^n] - 6*b^2*(-2*a + b*n)*Log[c*x^n]^2 + 4*b^3*Log[c*x^n]^3)* Log[1 + d*f*x^2] + (4*a^3 - 6*a^2*b*n + 6*a*b^2*n^2 - 3*b^3*n^3 + 12*a*b^2 *n*(n*Log[x] - Log[c*x^n]) + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n ^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 6* b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3)*L og[1 + d*f*x^2] + 6*b*n*(2*a^2 - 2*a*b*n + b^2*n^2 + 2*b^2*n*(n*Log[x] - L og[c*x^n]) + 4*a*b*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*(-(n*Log[x]) + Log[c *x^n])^2)*((d*f*x^2)/2 - d*f*x^2*Log[x] + Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f] *x] + Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f ]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) + 3*b^2*n^2*(-2*a + b*n + 2*b*n*Lo g[x] - 2*b*Log[c*x^n])*(d*f*x^2 - 2*d*f*x^2*Log[x] + 2*d*f*x^2*Log[x]^2 - 2*Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] - 2*Log[x]^2*Log[1 + I*Sqrt[d]*Sqr t[f]*x] - 4*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - 4*Log[x]*PolyLog[2 , I*Sqrt[d]*Sqrt[f]*x] + 4*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] + 4*PolyLog[ 3, I*Sqrt[d]*Sqrt[f]*x]) - b^3*n^3*(-3*d*f*x^2 + 6*d*f*x^2*Log[x] - 6*d...
Time = 1.13 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2824, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -3 b n \int \left (\frac {\left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right )}{2 d f x}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2\right )dx+\frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d f}-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 b n \left (\frac {b n \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}+\frac {b n \operatorname {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {\operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {b n \left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d f}+\frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-d f x^2\right )}{8 d f}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-d f x^2\right )}{8 d f}-\frac {b^2 n^2 \operatorname {PolyLog}\left (4,-d f x^2\right )}{8 d f}+\frac {b^2 n^2 \left (d f x^2+1\right ) \log \left (d f x^2+1\right )}{8 d f}-\frac {1}{2} b^2 n^2 x^2\right )+\frac {\left (d f x^2+1\right ) \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d f}-\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3\) |
-1/2*(x^2*(a + b*Log[c*x^n])^3) + ((1 + d*f*x^2)*(a + b*Log[c*x^n])^3*Log[ 1 + d*f*x^2])/(2*d*f) - 3*b*n*(-1/2*(b^2*n^2*x^2) + (3*b*n*x^2*(a + b*Log[ c*x^n]))/4 - (x^2*(a + b*Log[c*x^n])^2)/2 + (b^2*n^2*(1 + d*f*x^2)*Log[1 + d*f*x^2])/(8*d*f) - (b*n*(1 + d*f*x^2)*(a + b*Log[c*x^n])*Log[1 + d*f*x^2 ])/(4*d*f) + ((1 + d*f*x^2)*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*d*f) - (b^2*n^2*PolyLog[2, -(d*f*x^2)])/(8*d*f) + (b*n*(a + b*Log[c*x^n])*Poly Log[2, -(d*f*x^2)])/(4*d*f) - ((a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^2)] )/(4*d*f) - (b^2*n^2*PolyLog[3, -(d*f*x^2)])/(8*d*f) + (b*n*(a + b*Log[c*x ^n])*PolyLog[3, -(d*f*x^2)])/(4*d*f) - (b^2*n^2*PolyLog[4, -(d*f*x^2)])/(8 *d*f))
3.1.41.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ .))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)], x]}, Simp[(a + b*Log[c*x^n])^p u, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1)/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ [p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[ (q + 1)/m] && EqQ[d*e, 1]))
\[\int x {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )d x\]
\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
integral(b^3*x*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*x*log(d*f*x^2 + 1)* log(c*x^n)^2 + 3*a^2*b*x*log(d*f*x^2 + 1)*log(c*x^n) + a^3*x*log(d*f*x^2 + 1), x)
Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\text {Timed out} \]
\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
1/8*(4*b^3*x^2*log(x^n)^3 - 6*(b^3*(n - 2*log(c)) - 2*a*b^2)*x^2*log(x^n)^ 2 + 6*((n^2 - 2*n*log(c) + 2*log(c)^2)*b^3 - 2*a*b^2*(n - 2*log(c)) + 2*a^ 2*b)*x^2*log(x^n) + (6*(n^2 - 2*n*log(c) + 2*log(c)^2)*a*b^2 - (3*n^3 - 6* n^2*log(c) + 6*n*log(c)^2 - 4*log(c)^3)*b^3 - 6*a^2*b*(n - 2*log(c)) + 4*a ^3)*x^2)*log(d*f*x^2 + 1) - integrate(1/4*(4*b^3*d*f*x^3*log(x^n)^3 + 6*(2 *a*b^2*d*f - (d*f*n - 2*d*f*log(c))*b^3)*x^3*log(x^n)^2 + 6*(2*a^2*b*d*f - 2*(d*f*n - 2*d*f*log(c))*a*b^2 + (d*f*n^2 - 2*d*f*n*log(c) + 2*d*f*log(c) ^2)*b^3)*x^3*log(x^n) + (4*a^3*d*f - 6*(d*f*n - 2*d*f*log(c))*a^2*b + 6*(d *f*n^2 - 2*d*f*n*log(c) + 2*d*f*log(c)^2)*a*b^2 - (3*d*f*n^3 - 6*d*f*n^2*l og(c) + 6*d*f*n*log(c)^2 - 4*d*f*log(c)^3)*b^3)*x^3)/(d*f*x^2 + 1), x)
\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int x\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]