Integrand size = 28, antiderivative size = 478 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\frac {4 b^2 \sqrt {f} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {4 b \sqrt {f} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {\sqrt {f} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b \sqrt {f} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}+\frac {2 b \sqrt {f} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 i b^2 \sqrt {f} m n^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 i b^2 \sqrt {f} m n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {2 b^2 \sqrt {f} m n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}}-\frac {2 b^2 \sqrt {f} m n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e}} \]
-2*b^2*n^2*ln(d*(f*x^2+e)^m)/x-2*b*n*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x-( a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x+m*(a+b*ln(c*x^n))^2*ln(1-x*f^(1/2)/(- e)^(1/2))*f^(1/2)/(-e)^(1/2)-m*(a+b*ln(c*x^n))^2*ln(1+x*f^(1/2)/(-e)^(1/2) )*f^(1/2)/(-e)^(1/2)-2*b*m*n*(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/ 2))*f^(1/2)/(-e)^(1/2)+2*b*m*n*(a+b*ln(c*x^n))*polylog(2,x*f^(1/2)/(-e)^(1 /2))*f^(1/2)/(-e)^(1/2)+2*b^2*m*n^2*polylog(3,-x*f^(1/2)/(-e)^(1/2))*f^(1/ 2)/(-e)^(1/2)-2*b^2*m*n^2*polylog(3,x*f^(1/2)/(-e)^(1/2))*f^(1/2)/(-e)^(1/ 2)+4*b^2*m*n^2*arctan(x*f^(1/2)/e^(1/2))*f^(1/2)/e^(1/2)+4*b*m*n*arctan(x* f^(1/2)/e^(1/2))*(a+b*ln(c*x^n))*f^(1/2)/e^(1/2)-2*I*b^2*m*n^2*polylog(2,- I*x*f^(1/2)/e^(1/2))*f^(1/2)/e^(1/2)+2*I*b^2*m*n^2*polylog(2,I*x*f^(1/2)/e ^(1/2))*f^(1/2)/e^(1/2)
Time = 0.26 (sec) , antiderivative size = 917, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\frac {2 a^2 \sqrt {f} m x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+4 a b \sqrt {f} m n x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+4 b^2 \sqrt {f} m n^2 x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-4 a b \sqrt {f} m n x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-4 b^2 \sqrt {f} m n^2 x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+2 b^2 \sqrt {f} m n^2 x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)+4 a b \sqrt {f} m x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )+4 b^2 \sqrt {f} m n x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-4 b^2 \sqrt {f} m n x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) \log \left (c x^n\right )+2 b^2 \sqrt {f} m x \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )+2 i a b \sqrt {f} m n x \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {f} m n^2 x \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-i b^2 \sqrt {f} m n^2 x \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {f} m n x \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i a b \sqrt {f} m n x \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {f} m n^2 x \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b^2 \sqrt {f} m n^2 x \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {f} m n x \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-a^2 \sqrt {e} \log \left (d \left (e+f x^2\right )^m\right )-2 a b \sqrt {e} n \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 \sqrt {e} n^2 \log \left (d \left (e+f x^2\right )^m\right )-2 a b \sqrt {e} \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 \sqrt {e} n \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-b^2 \sqrt {e} \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 i b \sqrt {f} m n x \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b \sqrt {f} m n x \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 i b^2 \sqrt {f} m n^2 x \operatorname {PolyLog}\left (3,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 i b^2 \sqrt {f} m n^2 x \operatorname {PolyLog}\left (3,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} x} \]
(2*a^2*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 4*a*b*Sqrt[f]*m*n*x*ArcTa n[(Sqrt[f]*x)/Sqrt[e]] + 4*b^2*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*a*b*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 4*b^2*Sqrt[f]* m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 2*b^2*Sqrt[f]*m*n^2*x*ArcTan[ (Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 4*a*b*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[ e]]*Log[c*x^n] + 4*b^2*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n ] - 4*b^2*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 2* b^2*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + (2*I)*a*b*Sqrt[ f]*m*n*x*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (2*I)*b^2*Sqrt[f]*m*n^2*x *Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - I*b^2*Sqrt[f]*m*n^2*x*Log[x]^2*Lo g[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (2*I)*b^2*Sqrt[f]*m*n*x*Log[x]*Log[c*x^n]*L og[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (2*I)*a*b*Sqrt[f]*m*n*x*Log[x]*Log[1 + (I* Sqrt[f]*x)/Sqrt[e]] - (2*I)*b^2*Sqrt[f]*m*n^2*x*Log[x]*Log[1 + (I*Sqrt[f]* x)/Sqrt[e]] + I*b^2*Sqrt[f]*m*n^2*x*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e] ] - (2*I)*b^2*Sqrt[f]*m*n*x*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e ]] - a^2*Sqrt[e]*Log[d*(e + f*x^2)^m] - 2*a*b*Sqrt[e]*n*Log[d*(e + f*x^2)^ m] - 2*b^2*Sqrt[e]*n^2*Log[d*(e + f*x^2)^m] - 2*a*b*Sqrt[e]*Log[c*x^n]*Log [d*(e + f*x^2)^m] - 2*b^2*Sqrt[e]*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] - b^2* Sqrt[e]*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] - (2*I)*b*Sqrt[f]*m*n*x*(a + b*n + b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + (2*I)*b*Sqrt[f]...
Time = 0.74 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f m \int \left (-\frac {2 b^2 n^2}{f x^2+e}-\frac {2 b \left (a+b \log \left (c x^n\right )\right ) n}{f x^2+e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{f x^2+e}\right )dx-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 f m \left (-\frac {2 b n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {f}}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-e} \sqrt {f}}-\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-e} \sqrt {f}}-\frac {\log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {-e} \sqrt {f}}+\frac {\log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {-e} \sqrt {f}}-\frac {2 b^2 n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f}}+\frac {i b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f}}-\frac {i b^2 n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f}}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e} \sqrt {f}}+\frac {b^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{\sqrt {-e} \sqrt {f}}\right )-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}\) |
(-2*b^2*n^2*Log[d*(e + f*x^2)^m])/x - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/x - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x - 2*f*m*((-2 *b^2*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]) - (2*b*n*ArcTan[(S qrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(Sqrt[e]*Sqrt[f]) - ((a + b*Log[c*x ^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(2*Sqrt[-e]*Sqrt[f]) + ((a + b*Log[c *x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(2*Sqrt[-e]*Sqrt[f]) + (b*n*(a + b *Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(Sqrt[-e]*Sqrt[f]) - (b* n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(Sqrt[-e]*Sqrt[f]) + (I*b^2*n^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]) - (I* b^2*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]) - (b^2*n^2*Po lyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(Sqrt[-e]*Sqrt[f]) + (b^2*n^2*PolyLog[3 , (Sqrt[f]*x)/Sqrt[-e]])/(Sqrt[-e]*Sqrt[f]))
3.2.6.3.1 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]
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{2}}d x\]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \]