Integrand size = 25, antiderivative size = 938 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=-24 a b^2 n^2 x+36 b^3 n^3 x-12 b^2 n^2 (a-b n) x+\frac {12 b^2 n^2 (a-b n) \arctan \left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}-36 b^3 n^2 x \log \left (c x^n\right )+\frac {12 b^3 n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right )}{\sqrt {d} \sqrt {f}}+12 b n x \left (a+b \log \left (c x^n\right )\right )^2-2 x \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+6 a b^2 n^2 x \log \left (1+d f x^2\right )-6 b^3 n^3 x \log \left (1+d f x^2\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (1+d f x^2\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 i b^3 n^3 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {6 i b^3 n^3 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{\sqrt {d} \sqrt {f}}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,\sqrt {-d} \sqrt {f} x\right )}{\sqrt {-d} \sqrt {f}} \]
-(a+b*ln(c*x^n))^3*ln(1-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+(a+b*ln(c *x^n))^3*ln(1+x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+x*(a+b*ln(c*x^n))^3 *ln(d*f*x^2+1)-24*a*b^2*n^2*x-36*b^3*n^2*x*ln(c*x^n)+12*b*n*x*(a+b*ln(c*x^ n))^2-12*b^2*n^2*(-b*n+a)*x-6*b^3*n^3*x*ln(d*f*x^2+1)-3*b*n*(a+b*ln(c*x^n) )^2*ln(1+x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^2*n^2*(a+b*ln(c*x^n) )*polylog(2,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+3*b*n*(a+b*ln(c*x^n) )^2*polylog(2,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+6*b^2*n^2*(a+b*ln( c*x^n))*polylog(2,x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-3*b*n*(a+b*ln(c *x^n))^2*polylog(2,x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^2*n^2*(a+b *ln(c*x^n))*polylog(3,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+6*b^2*n^2* (a+b*ln(c*x^n))*polylog(3,x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+12*b^2* n^2*(-b*n+a)*arctan(x*d^(1/2)*f^(1/2))/d^(1/2)/f^(1/2)+12*b^3*n^2*arctan(x *d^(1/2)*f^(1/2))*ln(c*x^n)/d^(1/2)/f^(1/2)-6*I*b^3*n^3*polylog(2,-I*x*d^( 1/2)*f^(1/2))/d^(1/2)/f^(1/2)-2*x*(a+b*ln(c*x^n))^3+6*I*b^3*n^3*polylog(2, I*x*d^(1/2)*f^(1/2))/d^(1/2)/f^(1/2)+36*b^3*n^3*x+3*b*n*(a+b*ln(c*x^n))^2* ln(1-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+6*a*b^2*n^2*x*ln(d*f*x^2+1)+ 6*b^3*n^2*x*ln(c*x^n)*ln(d*f*x^2+1)-3*b*n*x*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1 )+6*b^3*n^3*polylog(3,-x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^3*n^3* polylog(3,x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+6*b^3*n^3*polylog(4,-x* (-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)-6*b^3*n^3*polylog(4,x*(-d)^(1/2)...
Time = 0.49 (sec) , antiderivative size = 1027, normalized size of antiderivative = 1.09 \[ \int \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 {-2 \sqrt {d} \sqrt {f} x \left (a^3-3 a^2 b n+6 a b^2 n^2-6 b^3 n^3+6 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+3 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 )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a^3-3 a^2 b n+6 a b^2 n^2-6 b^3 n^3+6 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+3 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 )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+\sqrt {d} \sqrt {f} x \left (a^3-3 a^2 b n+6 a b^2 n^2-6 b^3 n^3+3 b \left (a^2-2 a b n+2 b^2 n^2\right ) \log \left (c x^n\right )+3 b^2 (a-b n) \log ^2\left (c x^n\right )+b^3 \log ^3\left (c x^n\right )\right ) \log \left (1+d f x^2\right )+3 b n \left (a^2-2 a b n+2 b^2 n^2+2 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+2 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (-2 \sqrt {d} \sqrt {f} x (-1+\log (x))-i \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )\right )+i \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )\right )-6 b^2 n^2 \left (a-b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\sqrt {d} \sqrt {f} x \left (2-2 \log (x)+\log ^2(x)\right )+\frac {1}{2} i \left (\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )\right )+2 b^3 n^3 \left (-\sqrt {d} \sqrt {f} x \left (-6+6 \log (x)-3 \log ^2(x)+\log ^3(x)\right )-\frac {1}{2} i \left (\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,-i \sqrt {d} \sqrt {f} x\right )\right )+\frac {1}{2} i \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,i \sqrt {d} \sqrt {f} x\right )\right )\right )}{\sqrt {d} \sqrt {f}} \]
(-2*Sqrt[d]*Sqrt[f]*x*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 6*a*b^2 *n*(n*Log[x] - Log[c*x^n]) + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^ 2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 3*b^ 3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 2*A rcTan[Sqrt[d]*Sqrt[f]*x]*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 6*a* b^2*n*(n*Log[x] - Log[c*x^n]) + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3 *n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 3 *b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) + Sqrt[d]*Sqrt[f]*x*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 3*b*(a^2 - 2*a*b*n + 2*b^2*n^2)*Log[c*x^n] + 3*b^2*(a - b*n)*Log[c*x^n]^2 + b^3*Log[c *x^n]^3)*Log[1 + d*f*x^2] + 3*b*n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b^2*n*(n* Log[x] - Log[c*x^n]) + 2*a*b*(-(n*Log[x]) + Log[c*x^n]) + b^2*(-(n*Log[x]) + Log[c*x^n])^2)*(-2*Sqrt[d]*Sqrt[f]*x*(-1 + Log[x]) - I*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x]) + I*(Log[x]*Log [1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])) - 6*b^2*n^2* (a - b*n - b*n*Log[x] + b*Log[c*x^n])*(Sqrt[d]*Sqrt[f]*x*(2 - 2*Log[x] + L og[x]^2) + (I/2)*(Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog [2, (-I)*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x]) - (I/2 )*(Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, I*Sqrt[d]*S qrt[f]*x] - 2*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x])) + 2*b^3*n^3*(-(Sqrt[d]*...
Time = 1.68 (sec) , antiderivative size = 972, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2818, 6, 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.
| \(\displaystyle \int \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2818 |
| \(\displaystyle -2 f \int \left (\frac {6 d n^2 x^2 \log \left (c x^n\right ) b^3}{d f x^2+1}-\frac {6 d n^3 x^2 b^3}{d f x^2+1}+\frac {6 a d n^2 x^2 b^2}{d f x^2+1}-\frac {3 d n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b}{d f x^2+1}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{d f x^2+1}\right )dx+6 a b^2 n^2 x \log \left (d f x^2+1\right )-3 b n x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right )-6 b^3 n^3 x \log \left (d f x^2+1\right )\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle -2 f \int \left (\frac {6 d n^2 x^2 \log \left (c x^n\right ) b^3}{d f x^2+1}-\frac {3 d n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b}{d f x^2+1}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{d f x^2+1}+\frac {d \left (6 a b^2 n^2-6 b^3 n^3\right ) x^2}{d f x^2+1}\right )dx+6 a b^2 n^2 x \log \left (d f x^2+1\right )-3 b n x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right )-6 b^3 n^3 x \log \left (d f x^2+1\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 n^3 x \log \left (d f x^2+1\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right ) b^3+6 a n^2 x \log \left (d f x^2+1\right ) b^2-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) b+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d f x^2+1\right )-2 f \left (-\frac {18 n^3 x b^3}{f}+\frac {18 n^2 x \log \left (c x^n\right ) b^3}{f}-\frac {6 n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right ) b^3}{\sqrt {d} f^{3/2}}+\frac {3 i n^3 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} f^{3/2}}-\frac {3 i n^3 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} f^{3/2}}-\frac {3 n^3 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}+\frac {3 n^3 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}-\frac {3 n^3 \operatorname {PolyLog}\left (4,-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}+\frac {3 n^3 \operatorname {PolyLog}\left (4,\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}+\frac {12 a n^2 x b^2}{f}+\frac {6 n^2 (a-b n) x b^2}{f}-\frac {6 n^2 (a-b n) \arctan \left (\sqrt {d} \sqrt {f} x\right ) b^2}{\sqrt {d} f^{3/2}}+\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}-\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}+\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}-\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}-\frac {6 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{f}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right ) b}{2 \sqrt {-d} f^{3/2}}+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt {-d} \sqrt {f} x+1\right ) b}{2 \sqrt {-d} f^{3/2}}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) b}{2 \sqrt {-d} f^{3/2}}+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) b}{2 \sqrt {-d} f^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{f}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{2 \sqrt {-d} f^{3/2}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\sqrt {-d} \sqrt {f} x+1\right )}{2 \sqrt {-d} f^{3/2}}\right )\) |
6*a*b^2*n^2*x*Log[1 + d*f*x^2] - 6*b^3*n^3*x*Log[1 + d*f*x^2] + 6*b^3*n^2* x*Log[c*x^n]*Log[1 + d*f*x^2] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[1 + d*f*x ^2] + x*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2] - 2*f*((12*a*b^2*n^2*x)/f - (18*b^3*n^3*x)/f + (6*b^2*n^2*(a - b*n)*x)/f - (6*b^2*n^2*(a - b*n)*ArcTan [Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*f^(3/2)) + (18*b^3*n^2*x*Log[c*x^n])/f - (6* b^3*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x]*Log[c*x^n])/(Sqrt[d]*f^(3/2)) - (6*b*n*x *(a + b*Log[c*x^n])^2)/f + (x*(a + b*Log[c*x^n])^3)/f - (3*b*n*(a + b*Log[ c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + ((a + b*Log[ c*x^n])^3*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) - ((a + b*Log[c*x^n])^3*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + (3*b^2 *n^2*(a + b*Log[c*x^n])*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*f^(3/ 2)) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(2*Sq rt[-d]*f^(3/2)) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f ]*x])/(Sqrt[-d]*f^(3/2)) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, Sqrt[-d] *Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + ((3*I)*b^3*n^3*PolyLog[2, (-I)*Sqrt[d] *Sqrt[f]*x])/(Sqrt[d]*f^(3/2)) - ((3*I)*b^3*n^3*PolyLog[2, I*Sqrt[d]*Sqrt[ f]*x])/(Sqrt[d]*f^(3/2)) - (3*b^3*n^3*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/( Sqrt[-d]*f^(3/2)) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(Sqrt[-d]*Sq rt[f]*x)])/(Sqrt[-d]*f^(3/2)) + (3*b^3*n^3*PolyLog[3, Sqrt[-d]*Sqrt[f]*...
3.1.44.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[(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, r, m, n}, x] && IGtQ[p, 0] && Inte gerQ[m]
\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )d x\]
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
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)
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\text {Timed out} \]
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
(b^3*x*log(x^n)^3 - 3*(b^3*(n - log(c)) - a*b^2)*x*log(x^n)^2 + 3*((2*n^2 - 2*n*log(c) + log(c)^2)*b^3 - 2*a*b^2*(n - log(c)) + a^2*b)*x*log(x^n) + (3*(2*n^2 - 2*n*log(c) + log(c)^2)*a*b^2 - (6*n^3 - 6*n^2*log(c) + 3*n*log (c)^2 - log(c)^3)*b^3 - 3*a^2*b*(n - log(c)) + a^3)*x)*log(d*f*x^2 + 1) - integrate(2*(b^3*d*f*x^2*log(x^n)^3 + 3*(a*b^2*d*f - (d*f*n - d*f*log(c))* b^3)*x^2*log(x^n)^2 + 3*(a^2*b*d*f - 2*(d*f*n - d*f*log(c))*a*b^2 + (2*d*f *n^2 - 2*d*f*n*log(c) + d*f*log(c)^2)*b^3)*x^2*log(x^n) + (a^3*d*f - 3*(d* f*n - d*f*log(c))*a^2*b + 3*(2*d*f*n^2 - 2*d*f*n*log(c) + d*f*log(c)^2)*a* b^2 - (6*d*f*n^3 - 6*d*f*n^2*log(c) + 3*d*f*n*log(c)^2 - d*f*log(c)^3)*b^3 )*x^2)/(d*f*x^2 + 1), x)
\[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int \ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]