Integrand size = 28, antiderivative size = 459 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=4 b^2 \sqrt {d} \sqrt {f} n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right )+4 b \sqrt {d} \sqrt {f} n \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )-\sqrt {-d} \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x}-2 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )+2 b \sqrt {-d} \sqrt {f} n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )-2 i b^2 \sqrt {d} \sqrt {f} n^2 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+2 i b^2 \sqrt {d} \sqrt {f} n^2 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+2 b^2 \sqrt {-d} \sqrt {f} n^2 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )-2 b^2 \sqrt {-d} \sqrt {f} n^2 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) \]
-2*b^2*n^2*ln(d*f*x^2+1)/x-2*b*n*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/x-(a+b*ln(c *x^n))^2*ln(d*f*x^2+1)/x+(a+b*ln(c*x^n))^2*ln(1-x*(-d)^(1/2)*f^(1/2))*(-d) ^(1/2)*f^(1/2)-(a+b*ln(c*x^n))^2*ln(1+x*(-d)^(1/2)*f^(1/2))*(-d)^(1/2)*f^( 1/2)-2*b*n*(a+b*ln(c*x^n))*polylog(2,-x*(-d)^(1/2)*f^(1/2))*(-d)^(1/2)*f^( 1/2)+2*b*n*(a+b*ln(c*x^n))*polylog(2,x*(-d)^(1/2)*f^(1/2))*(-d)^(1/2)*f^(1 /2)+2*b^2*n^2*polylog(3,-x*(-d)^(1/2)*f^(1/2))*(-d)^(1/2)*f^(1/2)-2*b^2*n^ 2*polylog(3,x*(-d)^(1/2)*f^(1/2))*(-d)^(1/2)*f^(1/2)+4*b^2*n^2*arctan(x*d^ (1/2)*f^(1/2))*d^(1/2)*f^(1/2)+4*b*n*arctan(x*d^(1/2)*f^(1/2))*(a+b*ln(c*x ^n))*d^(1/2)*f^(1/2)-2*I*b^2*n^2*polylog(2,-I*x*d^(1/2)*f^(1/2))*d^(1/2)*f ^(1/2)+2*I*b^2*n^2*polylog(2,I*x*d^(1/2)*f^(1/2))*d^(1/2)*f^(1/2)
Time = 0.22 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=2 \sqrt {d} \sqrt {f} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a^2+2 a b n+2 b^2 n^2+2 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )-\frac {\left (a^2+2 a b n+2 b^2 n^2+2 b (a+b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x}+2 i b \sqrt {d} \sqrt {f} n \left (a+b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )-\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )+i b^2 \sqrt {d} \sqrt {f} n^2 \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-2 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )+2 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right ) \]
2*Sqrt[d]*Sqrt[f]*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2 *a*b*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + b^2 *(-(n*Log[x]) + Log[c*x^n])^2) - ((a^2 + 2*a*b*n + 2*b^2*n^2 + 2*b*(a + b* n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*Log[1 + d*f*x^2])/x + (2*I)*b*Sqrt[d]*Sq rt[f]*n*(a + b*n - b*n*Log[x] + b*Log[c*x^n])*(Log[x]*(Log[1 - I*Sqrt[d]*S qrt[f]*x] - Log[1 + I*Sqrt[d]*Sqrt[f]*x]) - PolyLog[2, (-I)*Sqrt[d]*Sqrt[f ]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) + I*b^2*Sqrt[d]*Sqrt[f]*n^2*(Log[x ]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] - Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f]*x] - 2*Log[x]*PolyLog[2, (-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] - 2*PolyLog[3, I*Sqrt [d]*Sqrt[f]*x])
Time = 0.76 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f \int \left (-\frac {2 b^2 d n^2}{d f x^2+1}-\frac {2 b d \left (a+b \log \left (c x^n\right )\right ) n}{d f x^2+1}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{d f x^2+1}\right )dx-\frac {2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b^2 n^2 \log \left (d f x^2+1\right )}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 f \left (-\frac {2 b \sqrt {d} n \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+\frac {b \sqrt {-d} n \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-\frac {b \sqrt {-d} n \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-\frac {\sqrt {-d} \log \left (1-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {f}}+\frac {\sqrt {-d} \log \left (\sqrt {-d} \sqrt {f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {f}}-\frac {2 b^2 \sqrt {d} n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right )}{\sqrt {f}}+\frac {i b^2 \sqrt {d} n^2 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{\sqrt {f}}-\frac {i b^2 \sqrt {d} n^2 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{\sqrt {f}}-\frac {b^2 \sqrt {-d} n^2 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )}{\sqrt {f}}+\frac {b^2 \sqrt {-d} n^2 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )}{\sqrt {f}}\right )-\frac {2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b^2 n^2 \log \left (d f x^2+1\right )}{x}\) |
(-2*b^2*n^2*Log[1 + d*f*x^2])/x - (2*b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^ 2])/x - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/x - 2*f*((-2*b^2*Sqrt[d]*n ^2*ArcTan[Sqrt[d]*Sqrt[f]*x])/Sqrt[f] - (2*b*Sqrt[d]*n*ArcTan[Sqrt[d]*Sqrt [f]*x]*(a + b*Log[c*x^n]))/Sqrt[f] - (Sqrt[-d]*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[f]) + (Sqrt[-d]*(a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[f]) + (b*Sqrt[-d]*n*(a + b*Log[c*x^n])*Poly Log[2, -(Sqrt[-d]*Sqrt[f]*x)])/Sqrt[f] - (b*Sqrt[-d]*n*(a + b*Log[c*x^n])* PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/Sqrt[f] + (I*b^2*Sqrt[d]*n^2*PolyLog[2, (- I)*Sqrt[d]*Sqrt[f]*x])/Sqrt[f] - (I*b^2*Sqrt[d]*n^2*PolyLog[2, I*Sqrt[d]*S qrt[f]*x])/Sqrt[f] - (b^2*Sqrt[-d]*n^2*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/ Sqrt[f] + (b^2*Sqrt[-d]*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/Sqrt[f])
3.1.38.3.1 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]
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )}{x^{2}}d x\]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \]
integral((b^2*log(d*f*x^2 + 1)*log(c*x^n)^2 + 2*a*b*log(d*f*x^2 + 1)*log(c *x^n) + a^2*log(d*f*x^2 + 1))/x^2, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2} \log {\left (d f x^{2} + 1 \right )}}{x^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \]
-(b^2*log(x^n)^2 + (2*n^2 + 2*n*log(c) + log(c)^2)*b^2 + 2*a*b*(n + log(c) ) + a^2 + 2*(b^2*(n + log(c)) + a*b)*log(x^n))*log(d*f*x^2 + 1)/x + integr ate(2*(b^2*d*f*log(x^n)^2 + a^2*d*f + 2*(d*f*n + d*f*log(c))*a*b + (2*d*f* n^2 + 2*d*f*n*log(c) + d*f*log(c)^2)*b^2 + 2*(a*b*d*f + (d*f*n + d*f*log(c ))*b^2)*log(x^n))/(d*f*x^2 + 1), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \]