Integrand size = 26, antiderivative size = 141 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\frac {1}{2} b d f n \log (x)-\frac {1}{2} b d f n \log ^2(x)+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d f n \log \left (1+d f x^2\right )-\frac {b n \log \left (1+d f x^2\right )}{4 x^2}-\frac {1}{2} d f \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac {1}{4} b d f n \operatorname {PolyLog}\left (2,-d f x^2\right ) \] Output:
1/2*b*d*f*n*ln(x)-1/2*b*d*f*n*ln(x)^2+d*f*ln(x)*(a+b*ln(c*x^n))-1/4*b*d*f* n*ln(d*f*x^2+1)-1/4*b*n*ln(d*f*x^2+1)/x^2-1/2*d*f*(a+b*ln(c*x^n))*ln(d*f*x ^2+1)-1/2*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/x^2-1/4*b*d*f*n*polylog(2,-d*f*x^2 )
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\frac {1}{2} b d f \log (x) \left (n+2 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+a d f \left (\log (x)-\frac {1}{2} \log \left (1+d f x^2\right )\right )-\frac {a \log \left (1+d f x^2\right )}{2 x^2}-\frac {1}{4} b d f \left (n+2 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )-\frac {b \left (n+2 n \log (x)+2 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}+b d f n \left (\frac {\log ^2(x)}{2}+\frac {1}{2} \left (-\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )\right )+\frac {1}{2} \left (-\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )\right ) \] Input:
Integrate[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)])/x^3,x]
Output:
(b*d*f*Log[x]*(n + 2*(-(n*Log[x]) + Log[c*x^n])))/2 + a*d*f*(Log[x] - Log[ 1 + d*f*x^2]/2) - (a*Log[1 + d*f*x^2])/(2*x^2) - (b*d*f*(n + 2*(-(n*Log[x] ) + Log[c*x^n]))*Log[1 + d*f*x^2])/4 - (b*(n + 2*n*Log[x] + 2*(-(n*Log[x]) + Log[c*x^n]))*Log[1 + d*f*x^2])/(4*x^2) + b*d*f*n*(Log[x]^2/2 + (-(Log[x ]*Log[1 + I*Sqrt[d]*Sqrt[f]*x]) - PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/2 + (-(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x]) - PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) /2)
Time = 0.37 (sec) , antiderivative size = 136, 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 = {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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (\frac {d f \log (x)}{x}-\frac {d f \log \left (d f x^2+1\right )}{2 x}-\frac {\log \left (d f x^2+1\right )}{2 x^3}\right )dx+d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} d f \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d f \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} d f \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-b n \left (\frac {1}{4} d f \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {1}{4} d f \log \left (d f x^2+1\right )+\frac {\log \left (d f x^2+1\right )}{4 x^2}+\frac {1}{2} d f \log ^2(x)-\frac {1}{2} d f \log (x)\right )\) |
Input:
Int[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)])/x^3,x]
Output:
d*f*Log[x]*(a + b*Log[c*x^n]) - (d*f*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/ 2 - ((a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(2*x^2) - b*n*(-1/2*(d*f*Log[x]) + (d*f*Log[x]^2)/2 + (d*f*Log[1 + d*f*x^2])/4 + Log[1 + d*f*x^2]/(4*x^2) + (d*f*PolyLog[2, -(d*f*x^2)])/4)
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 = 7.54 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.20
| method | result | size |
| risch | \(\left (-\frac {b \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )}{2 x^{2}}+b f d \ln \left (x \right )-\frac {b f d \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )}{2}\right ) \ln \left (x^{n}\right )+\frac {n b \ln \left (x \right ) \ln \left (d f \,x^{2}+1\right ) d f}{2}-\frac {n b \ln \left (1+x \sqrt {-d f}\right ) \ln \left (x \right ) d f}{2}-\frac {n b \ln \left (1-x \sqrt {-d f}\right ) \ln \left (x \right ) d f}{2}-\frac {n b \operatorname {dilog}\left (1+x \sqrt {-d f}\right ) d f}{2}-\frac {n b \operatorname {dilog}\left (1-x \sqrt {-d f}\right ) d f}{2}-\frac {b n \ln \left (d f \,x^{2}+1\right )}{4 x^{2}}+\frac {b d f n \ln \left (x \right )}{2}-\frac {b d f n \ln \left (d f \,x^{2}+1\right )}{4}-\frac {b d f n \ln \left (x \right )^{2}}{2}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (d f \,x^{2}+1\right )}{2 x^{2}}+d f \left (\ln \left (x \right )-\frac {\ln \left (d f \,x^{2}+1\right )}{2}\right )\right )\) | \(310\) |
Input:
int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^2))/x^3,x,method=_RETURNVERBOSE)
Output:
(-1/2*b/x^2*ln(d*(1/d+f*x^2))+b*f*d*ln(x)-1/2*b*f*d*ln(d*(1/d+f*x^2)))*ln( x^n)+1/2*n*b*ln(x)*ln(d*f*x^2+1)*d*f-1/2*n*b*ln(1+x*(-d*f)^(1/2))*ln(x)*d* f-1/2*n*b*ln(1-x*(-d*f)^(1/2))*ln(x)*d*f-1/2*n*b*dilog(1+x*(-d*f)^(1/2))*d *f-1/2*n*b*dilog(1-x*(-d*f)^(1/2))*d*f-1/4*b*n*ln(d*f*x^2+1)/x^2+1/2*b*d*f *n*ln(x)-1/4*b*d*f*n*ln(d*f*x^2+1)-1/2*b*d*f*n*ln(x)^2+(1/2*I*Pi*b*csgn(I* x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I* Pi*b*csgn(I*c*x^n)^3+1/2*I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+b*ln(c)+a)*(-1/2 /x^2*ln(d*f*x^2+1)+d*f*(ln(x)-1/2*ln(d*f*x^2+1)))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x^3,x, algorithm="fricas")
Output:
integral((b*log(d*f*x^2 + 1)*log(c*x^n) + a*log(d*f*x^2 + 1))/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*x**n))*ln(d*(1/d+f*x**2))/x**3,x)
Output:
Timed out
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x^3,x, algorithm="maxima")
Output:
-1/4*(b*(n + 2*log(c)) + 2*b*log(x^n) + 2*a)*log(d*f*x^2 + 1)/x^2 + integr ate(1/2*(2*b*d*f*log(x^n) + 2*a*d*f + (d*f*n + 2*d*f*log(c))*b)/(d*f*x^3 + x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x^3,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*log((f*x^2 + 1/d)*d)/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \] Input:
int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)))/x^3,x)
Output:
int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)))/x^3, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\frac {-4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d f \,x^{5}+x^{3}}d x \right ) b \,x^{2}-2 \,\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right ) b -2 \,\mathrm {log}\left (d f \,x^{2}+1\right ) a d f \,x^{2}-2 \,\mathrm {log}\left (d f \,x^{2}+1\right ) a -\mathrm {log}\left (d f \,x^{2}+1\right ) b d f n \,x^{2}-\mathrm {log}\left (d f \,x^{2}+1\right ) b n -2 \,\mathrm {log}\left (x^{n} c \right ) b +4 \,\mathrm {log}\left (x \right ) a d f \,x^{2}+2 \,\mathrm {log}\left (x \right ) b d f n \,x^{2}-b n}{4 x^{2}} \] Input:
int((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x^3,x)
Output:
( - 4*int(log(x**n*c)/(d*f*x**5 + x**3),x)*b*x**2 - 2*log(d*f*x**2 + 1)*lo g(x**n*c)*b - 2*log(d*f*x**2 + 1)*a*d*f*x**2 - 2*log(d*f*x**2 + 1)*a - log (d*f*x**2 + 1)*b*d*f*n*x**2 - log(d*f*x**2 + 1)*b*n - 2*log(x**n*c)*b + 4* log(x)*a*d*f*x**2 + 2*log(x)*b*d*f*n*x**2 - b*n)/(4*x**2)