Integrand size = 28, antiderivative size = 425 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\frac {3}{4} b^3 d f n^3 \log (x)-\frac {3}{4} b^2 d f n^2 \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{8} b^3 d f n^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}+\frac {3}{8} b^3 d f n^3 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {3}{4} b d f n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \operatorname {PolyLog}\left (4,-\frac {1}{d f x^2}\right ) \]
3/4*b^3*d*f*n^3*ln(x)-3/4*b^2*d*f*n^2*ln(1+1/d/f/x^2)*(a+b*ln(c*x^n))-3/4* b*d*f*n*ln(1+1/d/f/x^2)*(a+b*ln(c*x^n))^2-1/2*d*f*ln(1+1/d/f/x^2)*(a+b*ln( c*x^n))^3-3/8*b^3*d*f*n^3*ln(d*f*x^2+1)-3/8*b^3*n^3*ln(d*f*x^2+1)/x^2-3/4* b^2*n^2*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/x^2-3/4*b*n*(a+b*ln(c*x^n))^2*ln(d*f *x^2+1)/x^2-1/2*(a+b*ln(c*x^n))^3*ln(d*f*x^2+1)/x^2+3/8*b^3*d*f*n^3*polylo g(2,-1/d/f/x^2)+3/4*b^2*d*f*n^2*(a+b*ln(c*x^n))*polylog(2,-1/d/f/x^2)+3/4* b*d*f*n*(a+b*ln(c*x^n))^2*polylog(2,-1/d/f/x^2)+3/8*b^3*d*f*n^3*polylog(3, -1/d/f/x^2)+3/4*b^2*d*f*n^2*(a+b*ln(c*x^n))*polylog(3,-1/d/f/x^2)+3/8*b^3* d*f*n^3*polylog(4,-1/d/f/x^2)
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 940, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\frac {1}{8} \left (2 d f \log (x) \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 n \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 )-\frac {\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 )}{x^2}-d f \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 n \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 d f n \left (2 a^2+2 a b n+b^2 n^2+4 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \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 (\log (x) \left (\log (x)-\log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )-\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )+12 b^2 d f n^2 \left (2 a+b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right ) \left (\frac {\log ^3(x)}{3}-\frac {1}{2} \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{2} \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-\log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-\log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )+2 b^3 d f n^3 \left (\log ^4(x)-2 \log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-2 \log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-6 \log ^2(x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-6 \log ^2(x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+12 \log (x) \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )+12 \log (x) \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )-12 \operatorname {PolyLog}\left (4,-i \sqrt {d} \sqrt {f} x\right )-12 \operatorname {PolyLog}\left (4,i \sqrt {d} \sqrt {f} x\right )\right )\right ) \]
(2*d*f*Log[x]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n *Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(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) - ( (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])/x^2 - d*f*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12 *a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(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)*Log[1 + d*f*x^2] + 6*b*d*f*n*(2*a^2 + 2*a*b*n + b^2*n^2 + 4*a*b*(- (n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*(-(n *Log[x]) + Log[c*x^n])^2)*(Log[x]*(Log[x] - Log[1 - I*Sqrt[d]*Sqrt[f]*x] - Log[1 + I*Sqrt[d]*Sqrt[f]*x]) - PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - Poly Log[2, I*Sqrt[d]*Sqrt[f]*x]) + 12*b^2*d*f*n^2*(2*a + b*n - 2*b*n*Log[x] + 2*b*Log[c*x^n])*(Log[x]^3/3 - (Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x])/2 - (Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f]*x])/2 - Log[x]*PolyLog[2, (-I)*Sqrt[d] *Sqrt[f]*x] - Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + PolyLog[3, (-I)*Sqr t[d]*Sqrt[f]*x] + PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]) + 2*b^3*d*f*n^3*(Log[x] ^4 - 2*Log[x]^3*Log[1 - I*Sqrt[d]*Sqrt[f]*x] - 2*Log[x]^3*Log[1 + I*Sqr...
Time = 0.81 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 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.
\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx\) |
\(\Big \downarrow \) 2825 |
\(\displaystyle -2 f \int \left (-\frac {3 b^3 d n^3}{8 x \left (d f x^2+1\right )}-\frac {3 b^2 d \left (a+b \log \left (c x^n\right )\right ) n^2}{4 x \left (d f x^2+1\right )}-\frac {3 b d \left (a+b \log \left (c x^n\right )\right )^2 n}{4 x \left (d f x^2+1\right )}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{2 x \left (d f x^2+1\right )}\right )dx-\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{8 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-2 f \left (-\frac {3}{8} b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{8} b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3}{8} b^2 d n^2 \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{8} b d n \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3}{8} b d n \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} d \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{16} b^3 d n^3 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )-\frac {3}{16} b^3 d n^3 \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right )-\frac {3}{16} b^3 d n^3 \operatorname {PolyLog}\left (4,-\frac {1}{d f x^2}\right )+\frac {3}{16} b^3 d n^3 \log \left (d f x^2+1\right )-\frac {3}{8} b^3 d n^3 \log (x)\right )-\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{8 x^2}\) |
(-3*b^3*n^3*Log[1 + d*f*x^2])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[ 1 + d*f*x^2])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*x ^2) - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(2*x^2) - 2*f*((-3*b^3*d*n^3 *Log[x])/8 + (3*b^2*d*n^2*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n]))/8 + (3* b*d*n*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n])^2)/8 + (d*Log[1 + 1/(d*f*x^2 )]*(a + b*Log[c*x^n])^3)/4 + (3*b^3*d*n^3*Log[1 + d*f*x^2])/16 - (3*b^3*d* n^3*PolyLog[2, -(1/(d*f*x^2))])/16 - (3*b^2*d*n^2*(a + b*Log[c*x^n])*PolyL og[2, -(1/(d*f*x^2))])/8 - (3*b*d*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(d *f*x^2))])/8 - (3*b^3*d*n^3*PolyLog[3, -(1/(d*f*x^2))])/16 - (3*b^2*d*n^2* (a + b*Log[c*x^n])*PolyLog[3, -(1/(d*f*x^2))])/8 - (3*b^3*d*n^3*PolyLog[4, -(1/(d*f*x^2))])/16)
3.1.43.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* (a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r] u, x] - Simp[f*m*r Int[x^(m - 1)/(e + f*x^m) u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m , n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 70.05 (sec) , antiderivative size = 1171, normalized size of antiderivative = 2.76
-1/2*b^3/x^2*ln(d*f*x^2+1)*ln(x^n)^3-3/4*b^3*n/x^2*ln(d*f*x^2+1)*ln(x^n)^2 +b^3*d*f*ln(x)*ln(x^n)^3-3/4*b^3*n^2/x^2*ln(d*f*x^2+1)*ln(x^n)+b^3*d*f*ln( x)^3*ln(x^n)*n^2-3/2*b^3*d*f*ln(x)^2*ln(x^n)^2*n-3/4*b^3*n^2*d*f*polylog(2 ,-d*f*x^2)*ln(x^n)+3/4*b^3*n^2*d*f*polylog(3,-d*f*x^2)*ln(x^n)-1/2*b^3*d*f *ln(d*f*x^2+1)*ln(x^n)^3-3/8*b^3*n^3*ln(d*f*x^2+1)/x^2+1/2*b^3*n^3*d*f*ln( x)^3-1/4*b^3*n^3*d*f*ln(x)^4-3/8*b^3*n^3*d*f*polylog(2,-d*f*x^2)+3/8*b^3*n ^3*d*f*polylog(3,-d*f*x^2)-3/8*b^3*n^3*d*f*polylog(4,-d*f*x^2)-3/4*b^3*n^3 *d*f*ln(x)^2+3/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn( I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^ n)^3+2*b*ln(c)+2*a)^2*b*((ln(x^n)-n*ln(x))*(-1/2/x^2*ln(d*f*x^2+1)+d*f*(ln (x)-1/2*ln(d*f*x^2+1)))+n*((-1/4-1/2*ln(x))/x^2*ln(d*f*x^2+1)+1/2*d*f*ln(x )-1/4*d*f*ln(d*f*x^2+1)+1/2*d*f*ln(x)^2-1/2*d*f*ln(x)*ln(d*f*x^2+1)-1/4*d* f*polylog(2,-d*f*x^2)))+3/4*b^3*d*f*n^3*ln(x)-3/8*b^3*d*f*n^3*ln(d*f*x^2+1 )+1/8*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I *c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln (c)+2*a)^3*(-1/2/x^2*ln(d*f*x^2+1)+d*f*(ln(x)-1/2*ln(d*f*x^2+1)))+3/2*(-I* b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+ I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b ^2*((ln(x^n)-n*ln(x))^2*(-1/2/x^2*ln(d*f*x^2+1)+d*f*(ln(x)-1/2*ln(d*f*x^2+ 1)))+n^2*((-1/4-1/2*ln(x)-1/2*ln(x)^2)/x^2*ln(d*f*x^2+1)+1/2*d*f*ln(x)-...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
integral((b^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*log(d*f*x^2 + 1)*log (c*x^n)^2 + 3*a^2*b*log(d*f*x^2 + 1)*log(c*x^n) + a^3*log(d*f*x^2 + 1))/x^ 3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
-1/8*(4*b^3*log(x^n)^3 + 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 + 6*(b^3*(n + 2*log(c)) + 2*a*b^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)*log(x^n))*log(d*f* x^2 + 1)/x^2 + integrate(1/4*(4*b^3*d*f*log(x^n)^3 + 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*log(c) + 6*d*f*n*log(c)^2 + 4*d*f*log(c)^3)*b^3 + 6*(2*a*b^2*d*f + (d*f*n + 2*d*f*log(c))*b^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)*log(x^n))/(d*f*x^3 + x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]