\(\int x^3 (a+b \log (c x^n))^3 \log (d (\frac {1}{d}+f x^2)) \, dx\) [46]

Optimal result

Integrand size = 28, antiderivative size = 591 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=-\frac {45 b^3 n^3 x^2}{128 d f}+\frac {3}{64} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{128 d^2 f^2}-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (2,-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,-d f x^2\right )}{16 d^2 f^2} \] Output:

-45/128*b^3*n^3*x^2/d/f+3/64*b^3*n^3*x^4+21/32*b^2*n^2*x^2*(a+b*ln(c*x^n)) 
/d/f-9/64*b^2*n^2*x^4*(a+b*ln(c*x^n))-9/16*b*n*x^2*(a+b*ln(c*x^n))^2/d/f+3 
/16*b*n*x^4*(a+b*ln(c*x^n))^2+1/4*x^2*(a+b*ln(c*x^n))^3/d/f-1/8*x^4*(a+b*l 
n(c*x^n))^3+3/128*b^3*n^3*ln(d*f*x^2+1)/d^2/f^2-3/128*b^3*n^3*x^4*ln(d*f*x 
^2+1)-3/32*b^2*n^2*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/d^2/f^2+3/32*b^2*n^2*x^4* 
(a+b*ln(c*x^n))*ln(d*f*x^2+1)+3/16*b*n*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)/d^2 
/f^2-3/16*b*n*x^4*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)-1/4*(a+b*ln(c*x^n))^3*ln 
(d*f*x^2+1)/d^2/f^2+1/4*x^4*(a+b*ln(c*x^n))^3*ln(d*f*x^2+1)-3/64*b^3*n^3*p 
olylog(2,-d*f*x^2)/d^2/f^2+3/16*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-d*f*x^2 
)/d^2/f^2-3/8*b*n*(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^2)/d^2/f^2-3/32*b^3*n 
^3*polylog(3,-d*f*x^2)/d^2/f^2+3/8*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f* 
x^2)/d^2/f^2-3/16*b^3*n^3*polylog(4,-d*f*x^2)/d^2/f^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.77 (sec) , antiderivative size = 1234, normalized size of antiderivative = 2.09 \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx =\text {Too large to display} \] Input:

Integrate[x^3*(a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)],x]
 

Output:

-1/256*(-2*d*f*x^2*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a* 
b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*b 
^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 
- 24*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n] 
)^3) + d^2*f^2*x^4*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a* 
b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*b 
^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 
- 24*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n] 
)^3) - 2*d^2*f^2*x^4*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 12* 
b*(8*a^2 - 4*a*b*n + b^2*n^2)*Log[c*x^n] - 24*b^2*(-4*a + b*n)*Log[c*x^n]^ 
2 + 32*b^3*Log[c*x^n]^3)*Log[1 + d*f*x^2] + 2*(32*a^3 - 24*a^2*b*n + 12*a* 
b^2*n^2 - 3*b^3*n^3 + 48*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*b*(-(n*L 
og[x]) + Log[c*x^n]) + 12*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(- 
(n*Log[x]) + Log[c*x^n])^2 - 24*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 32*b^ 
3*(-(n*Log[x]) + Log[c*x^n])^3)*Log[1 + d*f*x^2] + 24*b*n*(8*a^2 - 4*a*b*n 
 + b^2*n^2 + 4*b^2*n*(n*Log[x] - Log[c*x^n]) + 16*a*b*(-(n*Log[x]) + Log[c 
*x^n]) + 8*b^2*(-(n*Log[x]) + Log[c*x^n])^2)*((d*f*x^2)/2 - (d^2*f^2*x^4)/ 
8 - d*f*x^2*Log[x] + (d^2*f^2*x^4*Log[x])/2 + Log[x]*Log[1 - I*Sqrt[d]*Sqr 
t[f]*x] + Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sq 
rt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) - 96*b^2*n^2*(4*a - b*n - 4...
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 562, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2824, 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.

\(\displaystyle \int x^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\)

\(\Big \downarrow \) 2824

\(\displaystyle -3 b n \int \left (-\frac {1}{8} \left (a+b \log \left (c x^n\right )\right )^2 x^3+\frac {1}{4} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) x^3+\frac {\left (a+b \log \left (c x^n\right )\right )^2 x}{4 d f}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right )}{4 d^2 f^2 x}\right )dx-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 b n \left (\frac {\operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {b n \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{16 d^2 f^2}-\frac {b n \operatorname {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{16 d^2 f^2}+\frac {b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{32 d^2 f^2}+\frac {3 x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}-\frac {7 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}+\frac {1}{16} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{32} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3}{64} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-d f x^2\right )}{64 d^2 f^2}+\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-d f x^2\right )}{32 d^2 f^2}+\frac {b^2 n^2 \operatorname {PolyLog}\left (4,-d f x^2\right )}{16 d^2 f^2}-\frac {b^2 n^2 \log \left (d f x^2+1\right )}{128 d^2 f^2}+\frac {15 b^2 n^2 x^2}{128 d f}+\frac {1}{128} b^2 n^2 x^4 \log \left (d f x^2+1\right )-\frac {1}{64} b^2 n^2 x^4\right )-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3\)

Input:

Int[x^3*(a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)],x]
 

Output:

(x^2*(a + b*Log[c*x^n])^3)/(4*d*f) - (x^4*(a + b*Log[c*x^n])^3)/8 - ((a + 
b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(4*d^2*f^2) + (x^4*(a + b*Log[c*x^n])^3* 
Log[1 + d*f*x^2])/4 - 3*b*n*((15*b^2*n^2*x^2)/(128*d*f) - (b^2*n^2*x^4)/64 
 - (7*b*n*x^2*(a + b*Log[c*x^n]))/(32*d*f) + (3*b*n*x^4*(a + b*Log[c*x^n]) 
)/64 + (3*x^2*(a + b*Log[c*x^n])^2)/(16*d*f) - (x^4*(a + b*Log[c*x^n])^2)/ 
16 - (b^2*n^2*Log[1 + d*f*x^2])/(128*d^2*f^2) + (b^2*n^2*x^4*Log[1 + d*f*x 
^2])/128 + (b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(32*d^2*f^2) - (b*n*x 
^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/32 - ((a + b*Log[c*x^n])^2*Log[1 + 
 d*f*x^2])/(16*d^2*f^2) + (x^4*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/16 + 
 (b^2*n^2*PolyLog[2, -(d*f*x^2)])/(64*d^2*f^2) - (b*n*(a + b*Log[c*x^n])*P 
olyLog[2, -(d*f*x^2)])/(16*d^2*f^2) + ((a + b*Log[c*x^n])^2*PolyLog[2, -(d 
*f*x^2)])/(8*d^2*f^2) + (b^2*n^2*PolyLog[3, -(d*f*x^2)])/(32*d^2*f^2) - (b 
*n*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*x^2)])/(8*d^2*f^2) + (b^2*n^2*PolyL 
og[4, -(d*f*x^2)])/(16*d^2*f^2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2824

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ 
.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)], x]}, Simp[(a + b*Log[c*x^n])^p   u, x] - Simp[b*n*p   Int[(a 
+ b*Log[c*x^n])^(p - 1)/x   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, 
 q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ 
[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[ 
(q + 1)/m] && EqQ[d*e, 1]))
 

Maple [F]

\[\int x^{3} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )d x\]

Input:

int(x^3*(a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2)),x)
 

Output:

int(x^3*(a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2)),x)
 

Fricas [F]

\[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="fricas")
 

Output:

integral(b^3*x^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*x^3*log(d*f*x^2 + 
 1)*log(c*x^n)^2 + 3*a^2*b*x^3*log(d*f*x^2 + 1)*log(c*x^n) + a^3*x^3*log(d 
*f*x^2 + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\text {Timed out} \] Input:

integrate(x**3*(a+b*ln(c*x**n))**3*ln(d*(1/d+f*x**2)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="maxima")
 

Output:

1/128*(32*b^3*x^4*log(x^n)^3 - 24*(b^3*(n - 4*log(c)) - 4*a*b^2)*x^4*log(x 
^n)^2 + 12*((n^2 - 4*n*log(c) + 8*log(c)^2)*b^3 - 4*a*b^2*(n - 4*log(c)) + 
 8*a^2*b)*x^4*log(x^n) + (12*(n^2 - 4*n*log(c) + 8*log(c)^2)*a*b^2 - (3*n^ 
3 - 12*n^2*log(c) + 24*n*log(c)^2 - 32*log(c)^3)*b^3 - 24*a^2*b*(n - 4*log 
(c)) + 32*a^3)*x^4)*log(d*f*x^2 + 1) - integrate(1/64*(32*b^3*d*f*x^5*log( 
x^n)^3 + 24*(4*a*b^2*d*f - (d*f*n - 4*d*f*log(c))*b^3)*x^5*log(x^n)^2 + 12 
*(8*a^2*b*d*f - 4*(d*f*n - 4*d*f*log(c))*a*b^2 + (d*f*n^2 - 4*d*f*n*log(c) 
 + 8*d*f*log(c)^2)*b^3)*x^5*log(x^n) + (32*a^3*d*f - 24*(d*f*n - 4*d*f*log 
(c))*a^2*b + 12*(d*f*n^2 - 4*d*f*n*log(c) + 8*d*f*log(c)^2)*a*b^2 - (3*d*f 
*n^3 - 12*d*f*n^2*log(c) + 24*d*f*n*log(c)^2 - 32*d*f*log(c)^3)*b^3)*x^5)/ 
(d*f*x^2 + 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int x^3\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \] Input:

int(x^3*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3,x)
 

Output:

int(x^3*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3, x)
 

Reduce [F]

\[ \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx =\text {Too large to display} \] Input:

int(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x)
 

Output:

(64*int(log(x**n*c)**3/(d*f*x**3 + x),x)*b**3*n + 192*int(log(x**n*c)**2/( 
d*f*x**3 + x),x)*a*b**2*n - 48*int(log(x**n*c)**2/(d*f*x**3 + x),x)*b**3*n 
**2 + 192*int(log(x**n*c)/(d*f*x**3 + x),x)*a**2*b*n - 96*int(log(x**n*c)/ 
(d*f*x**3 + x),x)*a*b**2*n**2 + 24*int(log(x**n*c)/(d*f*x**3 + x),x)*b**3* 
n**3 + 32*log(d*f*x**2 + 1)*log(x**n*c)**3*b**3*d**2*f**2*n*x**4 + 96*log( 
d*f*x**2 + 1)*log(x**n*c)**2*a*b**2*d**2*f**2*n*x**4 - 24*log(d*f*x**2 + 1 
)*log(x**n*c)**2*b**3*d**2*f**2*n**2*x**4 + 96*log(d*f*x**2 + 1)*log(x**n* 
c)*a**2*b*d**2*f**2*n*x**4 - 48*log(d*f*x**2 + 1)*log(x**n*c)*a*b**2*d**2* 
f**2*n**2*x**4 + 12*log(d*f*x**2 + 1)*log(x**n*c)*b**3*d**2*f**2*n**3*x**4 
 + 32*log(d*f*x**2 + 1)*a**3*d**2*f**2*n*x**4 - 32*log(d*f*x**2 + 1)*a**3* 
n - 24*log(d*f*x**2 + 1)*a**2*b*d**2*f**2*n**2*x**4 + 24*log(d*f*x**2 + 1) 
*a**2*b*n**2 + 12*log(d*f*x**2 + 1)*a*b**2*d**2*f**2*n**3*x**4 - 12*log(d* 
f*x**2 + 1)*a*b**2*n**3 - 3*log(d*f*x**2 + 1)*b**3*d**2*f**2*n**4*x**4 + 3 
*log(d*f*x**2 + 1)*b**3*n**4 - 16*log(x**n*c)**4*b**3 - 64*log(x**n*c)**3* 
a*b**2 - 16*log(x**n*c)**3*b**3*d**2*f**2*n*x**4 + 32*log(x**n*c)**3*b**3* 
d*f*n*x**2 + 16*log(x**n*c)**3*b**3*n - 96*log(x**n*c)**2*a**2*b - 48*log( 
x**n*c)**2*a*b**2*d**2*f**2*n*x**4 + 96*log(x**n*c)**2*a*b**2*d*f*n*x**2 + 
 48*log(x**n*c)**2*a*b**2*n + 24*log(x**n*c)**2*b**3*d**2*f**2*n**2*x**4 - 
 72*log(x**n*c)**2*b**3*d*f*n**2*x**2 - 12*log(x**n*c)**2*b**3*n**2 - 48*l 
og(x**n*c)*a**2*b*d**2*f**2*n*x**4 + 96*log(x**n*c)*a**2*b*d*f*n*x**2 +...