Integrand size = 26, antiderivative size = 310 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=-\frac {3}{4} b^2 m n^2 x^2+b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 e m n^2 \log \left (e+f x^2\right )}{4 f}+\frac {1}{4} b^2 n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {b e m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 f}-\frac {b^2 e m n^2 \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{4 f}+\frac {b e m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{2 f}-\frac {b^2 e m n^2 \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{4 f} \]
-3/4*b^2*m*n^2*x^2+b*m*n*x^2*(a+b*ln(c*x^n))-1/2*m*x^2*(a+b*ln(c*x^n))^2+1 /4*b^2*e*m*n^2*ln(f*x^2+e)/f+1/4*b^2*n^2*x^2*ln(d*(f*x^2+e)^m)-1/2*b*n*x^2 *(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)+1/2*x^2*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e )^m)-1/2*b*e*m*n*(a+b*ln(c*x^n))*ln(1+f*x^2/e)/f+1/2*e*m*(a+b*ln(c*x^n))^2 *ln(1+f*x^2/e)/f-1/4*b^2*e*m*n^2*polylog(2,-f*x^2/e)/f+1/2*b*e*m*n*(a+b*ln (c*x^n))*polylog(2,-f*x^2/e)/f-1/4*b^2*e*m*n^2*polylog(3,-f*x^2/e)/f
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 814, normalized size of antiderivative = 2.63 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\frac {-2 a^2 f m x^2+4 a b f m n x^2-3 b^2 f m n^2 x^2-4 a b f m x^2 \log \left (c x^n\right )+4 b^2 f m n x^2 \log \left (c x^n\right )-2 b^2 f m x^2 \log ^2\left (c x^n\right )+4 a b e m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 a b e m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 a^2 e m \log \left (e+f x^2\right )-2 a b e m n \log \left (e+f x^2\right )+b^2 e m n^2 \log \left (e+f x^2\right )-4 a b e m n \log (x) \log \left (e+f x^2\right )+2 b^2 e m n^2 \log (x) \log \left (e+f x^2\right )+2 b^2 e m n^2 \log ^2(x) \log \left (e+f x^2\right )+4 a b e m \log \left (c x^n\right ) \log \left (e+f x^2\right )-2 b^2 e m n \log \left (c x^n\right ) \log \left (e+f x^2\right )-4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (e+f x^2\right )+2 b^2 e m \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )+2 a^2 f x^2 \log \left (d \left (e+f x^2\right )^m\right )-2 a b f n x^2 \log \left (d \left (e+f x^2\right )^m\right )+b^2 f n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )+4 a b f x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 f n x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 f x^2 \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b e m n \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b e m n \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )-4 b^2 e m n^2 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-4 b^2 e m n^2 \operatorname {PolyLog}\left (3,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{4 f} \]
(-2*a^2*f*m*x^2 + 4*a*b*f*m*n*x^2 - 3*b^2*f*m*n^2*x^2 - 4*a*b*f*m*x^2*Log[ c*x^n] + 4*b^2*f*m*n*x^2*Log[c*x^n] - 2*b^2*f*m*x^2*Log[c*x^n]^2 + 4*a*b*e *m*n*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt [e]] + 4*b^2*e*m*n*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 4*a* b*e*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/S qrt[e]] + 4*b^2*e*m*n*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 2 *a^2*e*m*Log[e + f*x^2] - 2*a*b*e*m*n*Log[e + f*x^2] + b^2*e*m*n^2*Log[e + f*x^2] - 4*a*b*e*m*n*Log[x]*Log[e + f*x^2] + 2*b^2*e*m*n^2*Log[x]*Log[e + f*x^2] + 2*b^2*e*m*n^2*Log[x]^2*Log[e + f*x^2] + 4*a*b*e*m*Log[c*x^n]*Log [e + f*x^2] - 2*b^2*e*m*n*Log[c*x^n]*Log[e + f*x^2] - 4*b^2*e*m*n*Log[x]*L og[c*x^n]*Log[e + f*x^2] + 2*b^2*e*m*Log[c*x^n]^2*Log[e + f*x^2] + 2*a^2*f *x^2*Log[d*(e + f*x^2)^m] - 2*a*b*f*n*x^2*Log[d*(e + f*x^2)^m] + b^2*f*n^2 *x^2*Log[d*(e + f*x^2)^m] + 4*a*b*f*x^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 2*b^2*f*n*x^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 2*b^2*f*x^2*Log[c*x^n]^2*L og[d*(e + f*x^2)^m] + 2*b*e*m*n*(2*a - b*n + 2*b*Log[c*x^n])*PolyLog[2, (( -I)*Sqrt[f]*x)/Sqrt[e]] + 2*b*e*m*n*(2*a - b*n + 2*b*Log[c*x^n])*PolyLog[2 , (I*Sqrt[f]*x)/Sqrt[e]] - 4*b^2*e*m*n^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[ e]] - 4*b^2*e*m*n^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]])/(4*f)
Time = 0.71 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f m \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2 x^3}{2 \left (f x^2+e\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) x^3}{2 \left (f x^2+e\right )}+\frac {b^2 n^2 x^3}{4 \left (f x^2+e\right )}\right )dx+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} b^2 n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 f m \left (-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {b e n \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac {e \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 f}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac {b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{8 f^2}+\frac {b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{8 f^2}-\frac {b^2 e n^2 \log \left (e+f x^2\right )}{8 f^2}+\frac {3 b^2 n^2 x^2}{8 f}\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} b^2 n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )\) |
(b^2*n^2*x^2*Log[d*(e + f*x^2)^m])/4 - (b*n*x^2*(a + b*Log[c*x^n])*Log[d*( e + f*x^2)^m])/2 + (x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/2 - 2*f *m*((3*b^2*n^2*x^2)/(8*f) - (b*n*x^2*(a + b*Log[c*x^n]))/(2*f) + (x^2*(a + b*Log[c*x^n])^2)/(4*f) - (b^2*e*n^2*Log[e + f*x^2])/(8*f^2) + (b*e*n*(a + b*Log[c*x^n])*Log[1 + (f*x^2)/e])/(4*f^2) - (e*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(4*f^2) + (b^2*e*n^2*PolyLog[2, -((f*x^2)/e)])/(8*f^2) - (b *e*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)])/(4*f^2) + (b^2*e*n^2*Pol yLog[3, -((f*x^2)/e)])/(8*f^2))
3.1.100.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 174.33 (sec) , antiderivative size = 4839, normalized size of antiderivative = 15.61
-1/2*I*m/f*e*ln(f*x^2+e)*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I* m/f*ln(x^n)*e*ln(f*x^2+e)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I *m/f*n*e*ln(x)*ln(f*x^2+e)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m/f*n*e* ln(x)*ln(f*x^2+e)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*m/f*e*ln(f*x^2+ e)*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/4*I*m/f*e*ln(f*x^2+e )*Pi*b^2*n*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*m/f*n*e*ln(x)*ln((-f* x+(-e*f)^(1/2))/(-e*f)^(1/2))*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*m/f*n *e*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b^2*Pi*csgn(I*x^n)*csgn(I*c* x^n)^2+1/2*I*m/f*n*e*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))*b^2*Pi*csgn (I*c)*csgn(I*c*x^n)^2-m*ln(x^n)*x^2*b^2*ln(c)+m*n*x^2*b^2*ln(c)-m*b*ln(x^n )*x^2*a+m*b*n*x^2*a+1/8*m*x^2*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-1/4*m*x ^2*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+1/8*m*x^2*Pi^2*b^2*csgn(I*x^n)^2*csg n(I*c*x^n)^4-1/4*m*x^2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+(1/2*x^2*b^2*l n(x^n)^2+1/2*b*x^2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csg n(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)*ln(x^n)+1/8*x^2*(4*a^2+2*I*Pi*b^2*n*csgn(I*c)*cs gn(I*x^n)*csgn(I*c*x^n)+2*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+4*I*ln(c)*Pi* b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+2*b ^2*n^2-4*I*ln(c)*Pi*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-2*I*Pi*b^2*n*c sgn(I*c)*csgn(I*c*x^n)^2+8*ln(c)*a*b+4*ln(c)^2*b^2-4*b^2*ln(c)*n-4*a*b*...
\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Timed out} \]
\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]
1/4*(2*b^2*x^2*log(x^n)^2 - 2*(b^2*(n - 2*log(c)) - 2*a*b)*x^2*log(x^n) + ((n^2 - 2*n*log(c) + 2*log(c)^2)*b^2 - 2*a*b*(n - 2*log(c)) + 2*a^2)*x^2)* log((f*x^2 + e)^m) + integrate(-1/2*((2*(f*m - f*log(d))*a^2 - 2*(f*m*n - 2*(f*m - f*log(d))*log(c))*a*b + (f*m*n^2 - 2*f*m*n*log(c) + 2*(f*m - f*lo g(d))*log(c)^2)*b^2)*x^3 + 2*((f*m - f*log(d))*b^2*x^3 - b^2*e*x*log(d))*l og(x^n)^2 - 2*(b^2*e*log(c)^2*log(d) + 2*a*b*e*log(c)*log(d) + a^2*e*log(d ))*x + 2*((2*(f*m - f*log(d))*a*b - (f*m*n - 2*(f*m - f*log(d))*log(c))*b^ 2)*x^3 - 2*(b^2*e*log(c)*log(d) + a*b*e*log(d))*x)*log(x^n))/(f*x^2 + e), x)
\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f x^{2} + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\int x\,\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]