Integrand size = 26, antiderivative size = 241 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3+\frac {2 b n \arctan \left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}} \]
-8/9*b*n*x/d/f+4/27*b*n*x^3+2/9*b*n*arctan(x*d^(1/2)*f^(1/2))/d^(3/2)/f^(3 /2)+2/3*x*(a+b*ln(c*x^n))/d/f-2/9*x^3*(a+b*ln(c*x^n))-2/3*arctan(x*d^(1/2) *f^(1/2))*(a+b*ln(c*x^n))/d^(3/2)/f^(3/2)-1/9*b*n*x^3*ln(d*f*x^2+1)+1/3*x^ 3*(a+b*ln(c*x^n))*ln(d*f*x^2+1)+1/3*I*b*n*polylog(2,-I*x*d^(1/2)*f^(1/2))/ d^(3/2)/f^(3/2)-1/3*I*b*n*polylog(2,I*x*d^(1/2)*f^(1/2))/d^(3/2)/f^(3/2)
Time = 0.08 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.51 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\frac {2 a x}{3 d f}-\frac {2 a x^3}{9}-\frac {2 a \arctan \left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {2 b x \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{9 d f}-\frac {2}{27} b x^3 \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )-\frac {2 b \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {1}{3} a x^3 \log \left (1+d f x^2\right )+\frac {1}{9} b x^3 \left (-n+3 n \log (x)+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )-\frac {2}{3} b d f n \left (-\frac {x (-1+\log (x))}{d^2 f^2}+\frac {-\frac {x^3}{9}+\frac {1}{3} x^3 \log (x)}{d f}-\frac {i \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}+\frac {i \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}\right ) \]
(2*a*x)/(3*d*f) - (2*a*x^3)/9 - (2*a*ArcTan[Sqrt[d]*Sqrt[f]*x])/(3*d^(3/2) *f^(3/2)) + (2*b*x*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(9*d*f) - (2*b*x^3 *(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/27 - (2*b*ArcTan[Sqrt[d]*Sqrt[f]*x]* (-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(9*d^(3/2)*f^(3/2)) + (a*x^3*Log[1 + d*f*x^2])/3 + (b*x^3*(-n + 3*n*Log[x] + 3*(-(n*Log[x]) + Log[c*x^n]))*Log[ 1 + d*f*x^2])/9 - (2*b*d*f*n*(-((x*(-1 + Log[x]))/(d^2*f^2)) + (-1/9*x^3 + (x^3*Log[x])/3)/(d*f) - ((I/2)*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + Pol yLog[2, (-I)*Sqrt[d]*Sqrt[f]*x]))/(d^(5/2)*f^(5/2)) + ((I/2)*(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]))/(d^(5/2)*f^(5/ 2))))/3
Time = 0.40 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.97, 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.
\(\displaystyle \int x^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2823 |
\(\displaystyle -b n \int \left (\frac {1}{3} \log \left (d f x^2+1\right ) x^2-\frac {2 x^2}{9}+\frac {2}{3 d f}-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2} x}\right )dx-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-b n \left (-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {i \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {i \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{9} x^3 \log \left (d f x^2+1\right )+\frac {8 x}{9 d f}-\frac {4 x^3}{27}\right )\) |
(2*x*(a + b*Log[c*x^n]))/(3*d*f) - (2*x^3*(a + b*Log[c*x^n]))/9 - (2*ArcTa n[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(3*d^(3/2)*f^(3/2)) + (x^3*(a + b *Log[c*x^n])*Log[1 + d*f*x^2])/3 - b*n*((8*x)/(9*d*f) - (4*x^3)/27 - (2*Ar cTan[Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) + (x^3*Log[1 + d*f*x^2])/9 - ((I/3)*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2)) + ((I/3)*Poly Log[2, I*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2)))
3.1.28.3.1 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 = 16.61 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.75
method | result | size |
risch | \(\frac {b \,x^{3} \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right )}{3}-\frac {2 b \,x^{3} \ln \left (x^{n}\right )}{9}+\frac {2 b x \ln \left (x^{n}\right )}{3 d f}+\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{3 d f \sqrt {d f}}-\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{3 d f \sqrt {d f}}-\frac {b n \,x^{3} \ln \left (d f \,x^{2}+1\right )}{9}+\frac {4 b n \,x^{3}}{27}-\frac {8 b n x}{9 d f}+\frac {2 b n \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{9 d f \sqrt {d f}}+\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )}{3 d^{2} f^{2}}-\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )}{3 d^{2} f^{2}}+\frac {b n \sqrt {-d f}\, \operatorname {dilog}\left (1+x \sqrt {-d f}\right )}{3 d^{2} f^{2}}-\frac {b n \sqrt {-d f}\, \operatorname {dilog}\left (1-x \sqrt {-d f}\right )}{3 d^{2} f^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x^{3} \ln \left (d f \,x^{2}+1\right )}{3}-\frac {2 d f \left (\frac {\frac {1}{3} x^{3} d f -x}{d^{2} f^{2}}+\frac {\arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{d^{2} f^{2} \sqrt {d f}}\right )}{3}\right )\) | \(422\) |
1/3*b*x^3*ln(d*f*x^2+1)*ln(x^n)-2/9*b*x^3*ln(x^n)+2/3*b/d/f*x*ln(x^n)+2/3* b/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))*n*ln(x)-2/3*b/d/f/(d*f)^(1/2)* arctan(x*d*f/(d*f)^(1/2))*ln(x^n)-1/9*b*n*x^3*ln(d*f*x^2+1)+4/27*b*n*x^3-8 /9*b*n*x/d/f+2/9*b*n/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))+1/3*b*n/d^2 /f^2*(-d*f)^(1/2)*ln(x)*ln(1+x*(-d*f)^(1/2))-1/3*b*n/d^2/f^2*(-d*f)^(1/2)* ln(x)*ln(1-x*(-d*f)^(1/2))+1/3*b*n/d^2/f^2*(-d*f)^(1/2)*dilog(1+x*(-d*f)^( 1/2))-1/3*b*n/d^2/f^2*(-d*f)^(1/2)*dilog(1-x*(-d*f)^(1/2))+(-1/2*I*b*Pi*cs gn(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/d^2/f^2*(1/3*x^3*d*f-x)+1/d^2/f^2/(d*f)^ (1/2)*arctan(x*d*f/(d*f)^(1/2))))
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\text {Timed out} \]
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
1/9*(3*b*x^3*log(x^n) - (b*(n - 3*log(c)) - 3*a)*x^3)*log(d*f*x^2 + 1) - i ntegrate(2/9*(3*b*d*f*x^4*log(x^n) + (3*a*d*f - (d*f*n - 3*d*f*log(c))*b)* x^4)/(d*f*x^2 + 1), x)
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int x^2\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]