\(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x} \, dx\) [115]

Optimal result

Integrand size = 28, antiderivative size = 181 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x^2}{e}\right )}{4 b n}-\frac {1}{2} m \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )+\frac {3}{4} b m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right )-\frac {3}{4} b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (4,-\frac {f x^2}{e}\right )+\frac {3}{8} b^3 m n^3 \operatorname {PolyLog}\left (5,-\frac {f x^2}{e}\right ) \] Output:

1/4*(a+b*ln(c*x^n))^4*ln(d*(f*x^2+e)^m)/b/n-1/4*m*(a+b*ln(c*x^n))^4*ln(1+f 
*x^2/e)/b/n-1/2*m*(a+b*ln(c*x^n))^3*polylog(2,-f*x^2/e)+3/4*b*m*n*(a+b*ln( 
c*x^n))^2*polylog(3,-f*x^2/e)-3/4*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(4,-f*x 
^2/e)+3/8*b^3*m*n^3*polylog(5,-f*x^2/e)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 1348, normalized size of antiderivative = 7.45 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x,x]
 

Output:

-(a^3*m*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]]) + (3*a^2*b*m*n*Log[x]^2*Log 
[1 - (I*Sqrt[f]*x)/Sqrt[e]])/2 - a*b^2*m*n^2*Log[x]^3*Log[1 - (I*Sqrt[f]*x 
)/Sqrt[e]] + (b^3*m*n^3*Log[x]^4*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]])/4 - 3*a^2 
*b*m*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 3*a*b^2*m*n*Log[x] 
^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - b^3*m*n^2*Log[x]^3*Log[c*x^ 
n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 3*a*b^2*m*Log[x]*Log[c*x^n]^2*Log[1 - 
(I*Sqrt[f]*x)/Sqrt[e]] + (3*b^3*m*n*Log[x]^2*Log[c*x^n]^2*Log[1 - (I*Sqrt[ 
f]*x)/Sqrt[e]])/2 - b^3*m*Log[x]*Log[c*x^n]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e 
]] - a^3*m*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (3*a^2*b*m*n*Log[x]^2*L 
og[1 + (I*Sqrt[f]*x)/Sqrt[e]])/2 - a*b^2*m*n^2*Log[x]^3*Log[1 + (I*Sqrt[f] 
*x)/Sqrt[e]] + (b^3*m*n^3*Log[x]^4*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]])/4 - 3*a 
^2*b*m*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 3*a*b^2*m*n*Log[ 
x]^2*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - b^3*m*n^2*Log[x]^3*Log[c* 
x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 3*a*b^2*m*Log[x]*Log[c*x^n]^2*Log[1 
+ (I*Sqrt[f]*x)/Sqrt[e]] + (3*b^3*m*n*Log[x]^2*Log[c*x^n]^2*Log[1 + (I*Sqr 
t[f]*x)/Sqrt[e]])/2 - b^3*m*Log[x]*Log[c*x^n]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt 
[e]] + a^3*Log[x]*Log[d*(e + f*x^2)^m] - (3*a^2*b*n*Log[x]^2*Log[d*(e + f* 
x^2)^m])/2 + a*b^2*n^2*Log[x]^3*Log[d*(e + f*x^2)^m] - (b^3*n^3*Log[x]^4*L 
og[d*(e + f*x^2)^m])/4 + 3*a^2*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 
3*a*b^2*n*Log[x]^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] + b^3*n^2*Log[x]^3*L...
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2822, 2775, 2821, 2830, 2830, 7143}

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.

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

\(\Big \downarrow \) 2822

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

\(\Big \downarrow \) 2775

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (e+f x^2\right )^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{2 f}-\frac {2 b n \left (\frac {3}{2} b n \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-b n \left (\frac {1}{2} \operatorname {PolyLog}\left (4,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \operatorname {PolyLog}\left (5,-\frac {f x^2}{e}\right )\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{2 b n}\)

Input:

Int[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x,x]
 

Output:

((a + b*Log[c*x^n])^4*Log[d*(e + f*x^2)^m])/(4*b*n) - (f*m*(((a + b*Log[c* 
x^n])^4*Log[1 + (f*x^2)/e])/(2*f) - (2*b*n*(-1/2*((a + b*Log[c*x^n])^3*Pol 
yLog[2, -((f*x^2)/e)]) + (3*b*n*(((a + b*Log[c*x^n])^2*PolyLog[3, -((f*x^2 
)/e)])/2 - b*n*(((a + b*Log[c*x^n])*PolyLog[4, -((f*x^2)/e)])/2 - (b*n*Pol 
yLog[5, -((f*x^2)/e)])/4)))/2))/f))/(2*b*n)
 

Defintions of rubi rules used

rule 2775

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) 
+ (e_.)*(x_)^(r_)), x_Symbol] :> Simp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c* 
x^n])^p/(e*r)), x] - Simp[b*f^m*n*(p/(e*r))   Int[Log[1 + e*(x^r/d)]*((a + 
b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] & 
& EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2822

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_ 
.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp[Log[d*(e + f*x^m)^r]*((a + b*Log[ 
c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[f*m*(r/(b*n*(p + 1)))   Int[x^(m 
- 1)*((a + b*Log[c*x^n])^(p + 1)/(e + f*x^m)), x], x] /; FreeQ[{a, b, c, d, 
 e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[d*e, 1]
 

rule 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.03 (sec) , antiderivative size = 5812, normalized size of antiderivative = 32.11

\[\text {output too large to display}\]

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/4*(b^3*n^3*log(x)^4 - 4*b^3*log(x)*log(x^n)^3 - 4*(b^3*n^2*log(c) + a*b 
^2*n^2)*log(x)^3 + 6*(b^3*n*log(c)^2 + 2*a*b^2*n*log(c) + a^2*b*n)*log(x)^ 
2 + 6*(b^3*n*log(x)^2 - 2*(b^3*log(c) + a*b^2)*log(x))*log(x^n)^2 - 4*(b^3 
*n^2*log(x)^3 - 3*(b^3*n*log(c) + a*b^2*n)*log(x)^2 + 3*(b^3*log(c)^2 + 2* 
a*b^2*log(c) + a^2*b)*log(x))*log(x^n) - 4*(b^3*log(c)^3 + 3*a*b^2*log(c)^ 
2 + 3*a^2*b*log(c) + a^3)*log(x))*log((f*x^2 + e)^m) - integrate(-1/2*(b^3 
*f*m*n^3*x^2*log(x)^4 + 2*b^3*e*log(c)^3*log(d) + 6*a*b^2*e*log(c)^2*log(d 
) + 6*a^2*b*e*log(c)*log(d) - 4*(b^3*f*m*n^2*log(c) + a*b^2*f*m*n^2)*x^2*l 
og(x)^3 + 2*a^3*e*log(d) + 6*(b^3*f*m*n*log(c)^2 + 2*a*b^2*f*m*n*log(c) + 
a^2*b*f*m*n)*x^2*log(x)^2 - 4*(b^3*f*m*log(c)^3 + 3*a*b^2*f*m*log(c)^2 + 3 
*a^2*b*f*m*log(c) + a^3*f*m)*x^2*log(x) - 2*(2*b^3*f*m*x^2*log(x) - b^3*f* 
x^2*log(d) - b^3*e*log(d))*log(x^n)^3 + 2*(b^3*f*log(c)^3*log(d) + 3*a*b^2 
*f*log(c)^2*log(d) + 3*a^2*b*f*log(c)*log(d) + a^3*f*log(d))*x^2 + 6*(b^3* 
f*m*n*x^2*log(x)^2 + b^3*e*log(c)*log(d) + a*b^2*e*log(d) - 2*(b^3*f*m*log 
(c) + a*b^2*f*m)*x^2*log(x) + (b^3*f*log(c)*log(d) + a*b^2*f*log(d))*x^2)* 
log(x^n)^2 - 2*(2*b^3*f*m*n^2*x^2*log(x)^3 - 3*b^3*e*log(c)^2*log(d) - 6*a 
*b^2*e*log(c)*log(d) - 3*a^2*b*e*log(d) - 6*(b^3*f*m*n*log(c) + a*b^2*f*m* 
n)*x^2*log(x)^2 + 6*(b^3*f*m*log(c)^2 + 2*a*b^2*f*m*log(c) + a^2*b*f*m)*x^ 
2*log(x) - 3*(b^3*f*log(c)^2*log(d) + 2*a*b^2*f*log(c)*log(d) + a^2*b*f*lo 
g(d))*x^2)*log(x^n))/(f*x^3 + e*x), x)
 

Giac [F]

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

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

Output:

integrate((b*log(c*x^n) + a)^3*log((f*x^2 + e)^m*d)/x, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right )}{f \,x^{3}+e x}d x \right ) a^{3} e m +4 \left (\int \frac {\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )^{3}}{x}d x \right ) b^{3} m +12 \left (\int \frac {\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )^{2}}{x}d x \right ) a \,b^{2} m +12 \left (\int \frac {\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )}{x}d x \right ) a^{2} b m +{\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right )}^{2} a^{3}}{4 m} \] Input:

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

Output:

(4*int(log((e + f*x**2)**m*d)/(e*x + f*x**3),x)*a**3*e*m + 4*int((log((e + 
 f*x**2)**m*d)*log(x**n*c)**3)/x,x)*b**3*m + 12*int((log((e + f*x**2)**m*d 
)*log(x**n*c)**2)/x,x)*a*b**2*m + 12*int((log((e + f*x**2)**m*d)*log(x**n* 
c))/x,x)*a**2*b*m + log((e + f*x**2)**m*d)**2*a**3)/(4*m)