Integrand size = 26, antiderivative size = 195 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\frac {b f m n \log (x)}{2 e}-\frac {b f m n \log ^2(x)}{2 e}+\frac {f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b f m n \log \left (e+f x^2\right )}{4 e}+\frac {b f m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{4 e}-\frac {f m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{2 e}-\frac {b n \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {b f m n \operatorname {PolyLog}\left (2,1+\frac {f x^2}{e}\right )}{4 e} \]
1/2*b*f*m*n*ln(x)/e-1/2*b*f*m*n*ln(x)^2/e+f*m*ln(x)*(a+b*ln(c*x^n))/e-1/4* b*f*m*n*ln(f*x^2+e)/e+1/4*b*f*m*n*ln(-f*x^2/e)*ln(f*x^2+e)/e-1/2*f*m*(a+b* ln(c*x^n))*ln(f*x^2+e)/e-1/4*b*n*ln(d*(f*x^2+e)^m)/x^2-1/2*(a+b*ln(c*x^n)) *ln(d*(f*x^2+e)^m)/x^2+1/4*b*f*m*n*polylog(2,1+f*x^2/e)/e
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=-\frac {-4 a f m x^2 \log (x)-2 b f m n x^2 \log (x)+2 b f m n x^2 \log ^2(x)-4 b f m x^2 \log (x) \log \left (c x^n\right )+2 b f m n x^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b f m n x^2 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 a f m x^2 \log \left (e+f x^2\right )+b f m n x^2 \log \left (e+f x^2\right )-2 b f m n x^2 \log (x) \log \left (e+f x^2\right )+2 b f m x^2 \log \left (c x^n\right ) \log \left (e+f x^2\right )+2 a e \log \left (d \left (e+f x^2\right )^m\right )+b e n \log \left (d \left (e+f x^2\right )^m\right )+2 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b f m n x^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b f m n x^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{4 e x^2} \]
-1/4*(-4*a*f*m*x^2*Log[x] - 2*b*f*m*n*x^2*Log[x] + 2*b*f*m*n*x^2*Log[x]^2 - 4*b*f*m*x^2*Log[x]*Log[c*x^n] + 2*b*f*m*n*x^2*Log[x]*Log[1 - (I*Sqrt[f]* x)/Sqrt[e]] + 2*b*f*m*n*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 2*a*f* m*x^2*Log[e + f*x^2] + b*f*m*n*x^2*Log[e + f*x^2] - 2*b*f*m*n*x^2*Log[x]*L og[e + f*x^2] + 2*b*f*m*x^2*Log[c*x^n]*Log[e + f*x^2] + 2*a*e*Log[d*(e + f *x^2)^m] + b*e*n*Log[d*(e + f*x^2)^m] + 2*b*e*Log[c*x^n]*Log[d*(e + f*x^2) ^m] + 2*b*f*m*n*x^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 2*b*f*m*n*x^2*P olyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(e*x^2)
Time = 0.41 (sec) , antiderivative size = 188, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (\frac {f m \log (x)}{e x}-\frac {f m \log \left (f x^2+e\right )}{2 e x}-\frac {\log \left (d \left (f x^2+e\right )^m\right )}{2 x^3}\right )dx-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {f m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {f m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}-b n \left (\frac {\log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {f m \operatorname {PolyLog}\left (2,\frac {f x^2}{e}+1\right )}{4 e}+\frac {f m \log \left (e+f x^2\right )}{4 e}-\frac {f m \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{4 e}+\frac {f m \log ^2(x)}{2 e}-\frac {f m \log (x)}{2 e}\right )\) |
(f*m*Log[x]*(a + b*Log[c*x^n]))/e - (f*m*(a + b*Log[c*x^n])*Log[e + f*x^2] )/(2*e) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(2*x^2) - b*n*(-1/2*(f *m*Log[x])/e + (f*m*Log[x]^2)/(2*e) + (f*m*Log[e + f*x^2])/(4*e) - (f*m*Lo g[-((f*x^2)/e)]*Log[e + f*x^2])/(4*e) + Log[d*(e + f*x^2)^m]/(4*x^2) - (f* m*PolyLog[2, 1 + (f*x^2)/e])/(4*e))
3.1.93.3.1 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 = 17.23 (sec) , antiderivative size = 862, normalized size of antiderivative = 4.42
(-1/2*b/x^2*ln(x^n)-1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*P i*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)+b*n+2*a)/x^2)*ln((f*x^2+e)^m)+(1/4*I*Pi*csgn(I*(f*x^ 2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2-1/4*I*Pi*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x ^2+e)^m)*csgn(I*d)-1/4*I*Pi*csgn(I*d*(f*x^2+e)^m)^3+1/4*I*Pi*csgn(I*d*(f*x ^2+e)^m)^2*csgn(I*d)+1/2*ln(d))*(-1/2*(-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)/x^2-b/x^2*ln(x^n)-1/2*b*n/x^2)-1/ 4*I*m*f/e*ln(f*x^2+e)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*m*f/e*ln(x)*b *Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*m*f/e*ln(x)*b*Pi*csgn(I*c*x^ n)^3+1/2*I*m*f/e*ln(x)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*m*f/e*ln(f*x^2+e )*b*ln(c)-1/4*b*f*m*n*ln(f*x^2+e)/e-1/2*m*f/e*ln(f*x^2+e)*a+1/4*I*m*f/e*ln (f*x^2+e)*b*Pi*csgn(I*c*x^n)^3-1/4*I*m*f/e*ln(f*x^2+e)*b*Pi*csgn(I*c)*csgn (I*c*x^n)^2+1/4*I*m*f/e*ln(f*x^2+e)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^ n)+1/2*I*m*f/e*ln(x)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+m*f/e*ln(x)*b*ln(c)+ 1/2*b*f*m*n*ln(x)/e+m*f/e*ln(x)*a-1/2*m*f*b*ln(x^n)/e*ln(f*x^2+e)+m*f*b*ln (x^n)/e*ln(x)-1/2*b*f*m*n*ln(x)^2/e-1/2*m*f*b*n/e*ln(x)*ln((-f*x+(-e*f)^(1 /2))/(-e*f)^(1/2))-1/2*m*f*b*n/e*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2)) +1/2*b*f*m*n*ln(x)/e*ln(f*x^2+e)-1/2*m*f*b*n/e*dilog((-f*x+(-e*f)^(1/2))/( -e*f)^(1/2))-1/2*m*f*b*n/e*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}} \,d x } \]
-1/4*(b*(n + 2*log(c)) + 2*b*log(x^n) + 2*a)*log((f*x^2 + e)^m)/x^2 + inte grate(1/2*(2*b*e*log(c)*log(d) + (2*(f*m + f*log(d))*a + (f*m*n + 2*(f*m + f*log(d))*log(c))*b)*x^2 + 2*a*e*log(d) + 2*((f*m + f*log(d))*b*x^2 + b*e *log(d))*log(x^n))/(f*x^5 + e*x^3), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]