Integrand size = 26, antiderivative size = 211 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=-\frac {8 b d f n}{9 x}-\frac {2}{9} b d^{3/2} f^{3/2} n \arctan \left (\sqrt {d} \sqrt {f} x\right )-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {2}{3} d^{3/2} f^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f x^2\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{3 x^3}+\frac {1}{3} i b d^{3/2} f^{3/2} n \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{3} i b d^{3/2} f^{3/2} n \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right ) \]
-8/9*b*d*f*n/x-2/9*b*d^(3/2)*f^(3/2)*n*arctan(x*d^(1/2)*f^(1/2))-2/3*d*f*( a+b*ln(c*x^n))/x-2/3*d^(3/2)*f^(3/2)*arctan(x*d^(1/2)*f^(1/2))*(a+b*ln(c*x ^n))-1/9*b*n*ln(d*f*x^2+1)/x^3-1/3*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/x^3+1/3*I *b*d^(3/2)*f^(3/2)*n*polylog(2,-I*x*d^(1/2)*f^(1/2))-1/3*I*b*d^(3/2)*f^(3/ 2)*n*polylog(2,I*x*d^(1/2)*f^(1/2))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.15 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=-\frac {2 a d f \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-d f x^2\right )}{3 x}-\frac {2}{9} b d^{3/2} f^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )-\frac {2 b \left (d f n+3 d f \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{9 x}-\frac {a \log \left (1+d f x^2\right )}{3 x^3}-\frac {b \left (n+3 n \log (x)+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )}{9 x^3}+\frac {2}{3} b d f n \left (-\frac {1}{x}-\frac {\log (x)}{x}+\frac {1}{2} i \sqrt {d} \sqrt {f} \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \sqrt {d} \sqrt {f} \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )\right ) \]
(-2*a*d*f*Hypergeometric2F1[-1/2, 1, 1/2, -(d*f*x^2)])/(3*x) - (2*b*d^(3/2 )*f^(3/2)*ArcTan[Sqrt[d]*Sqrt[f]*x]*(n + 3*(-(n*Log[x]) + Log[c*x^n])))/9 - (2*b*(d*f*n + 3*d*f*(-(n*Log[x]) + Log[c*x^n])))/(9*x) - (a*Log[1 + d*f* x^2])/(3*x^3) - (b*(n + 3*n*Log[x] + 3*(-(n*Log[x]) + Log[c*x^n]))*Log[1 + d*f*x^2])/(9*x^3) + (2*b*d*f*n*(-x^(-1) - Log[x]/x + (I/2)*Sqrt[d]*Sqrt[f ]*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x ]) - (I/2)*Sqrt[d]*Sqrt[f]*(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[ 2, I*Sqrt[d]*Sqrt[f]*x])))/3
Time = 0.34 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2823, 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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (-\frac {2 d^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right ) f^{3/2}}{3 x}-\frac {2 d f}{3 x^2}-\frac {\log \left (d f x^2+1\right )}{3 x^4}\right )dx-\frac {2}{3} d^{3/2} f^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{3} d^{3/2} f^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )}{3 x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-b n \left (\frac {2}{9} d^{3/2} f^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right )-\frac {1}{3} i d^{3/2} f^{3/2} \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+\frac {1}{3} i d^{3/2} f^{3/2} \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+\frac {\log \left (d f x^2+1\right )}{9 x^3}+\frac {8 d f}{9 x}\right )\) |
(-2*d*f*(a + b*Log[c*x^n]))/(3*x) - (2*d^(3/2)*f^(3/2)*ArcTan[Sqrt[d]*Sqrt [f]*x]*(a + b*Log[c*x^n]))/3 - ((a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(3*x^ 3) - b*n*((8*d*f)/(9*x) + (2*d^(3/2)*f^(3/2)*ArcTan[Sqrt[d]*Sqrt[f]*x])/9 + Log[1 + d*f*x^2]/(9*x^3) - (I/3)*d^(3/2)*f^(3/2)*PolyLog[2, (-I)*Sqrt[d] *Sqrt[f]*x] + (I/3)*d^(3/2)*f^(3/2)*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])
3.1.31.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 7.84 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.75
| method | result | size |
| risch | \(-\frac {b \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right )}{3 x^{3}}-\frac {2 b d f \ln \left (x^{n}\right )}{3 x}+\frac {2 b \,d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{3 \sqrt {d f}}-\frac {2 b \,d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{3 \sqrt {d f}}-\frac {b n \ln \left (d f \,x^{2}+1\right )}{9 x^{3}}-\frac {8 b d f n}{9 x}-\frac {2 b n \,d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{9 \sqrt {d f}}+\frac {b n d f \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )}{3}-\frac {b n d f \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )}{3}+\frac {b n d f \sqrt {-d f}\, \operatorname {dilog}\left (1+x \sqrt {-d f}\right )}{3}-\frac {b n d f \sqrt {-d f}\, \operatorname {dilog}\left (1-x \sqrt {-d f}\right )}{3}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (d f \,x^{2}+1\right )}{3 x^{3}}+\frac {2 d f \left (-\frac {1}{x}-\frac {d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}\right )}{3}\right )\) | \(369\) |
-1/3*b/x^3*ln(d*f*x^2+1)*ln(x^n)-2/3*b*d*f/x*ln(x^n)+2/3*b*d^2*f^2/(d*f)^( 1/2)*arctan(x*d*f/(d*f)^(1/2))*n*ln(x)-2/3*b*d^2*f^2/(d*f)^(1/2)*arctan(x* d*f/(d*f)^(1/2))*ln(x^n)-1/9*b*n*ln(d*f*x^2+1)/x^3-8/9*b*d*f*n/x-2/9*b*n*d ^2*f^2/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))+1/3*b*n*d*f*(-d*f)^(1/2)*ln(x )*ln(1+x*(-d*f)^(1/2))-1/3*b*n*d*f*(-d*f)^(1/2)*ln(x)*ln(1-x*(-d*f)^(1/2)) +1/3*b*n*d*f*(-d*f)^(1/2)*dilog(1+x*(-d*f)^(1/2))-1/3*b*n*d*f*(-d*f)^(1/2) *dilog(1-x*(-d*f)^(1/2))+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+ 1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^ 2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(-1/3/x^3*ln(d*f*x^2+1)+2/3*d*f*(- 1/x-d*f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \]
-1/9*(b*(n + 3*log(c)) + 3*b*log(x^n) + 3*a)*log(d*f*x^2 + 1)/x^3 + integr ate(2/9*(3*b*d*f*log(x^n) + 3*a*d*f + (d*f*n + 3*d*f*log(c))*b)/(d*f*x^4 + x^2), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]