\(\int \frac {\log ^2(c (d+e x^3)^p)}{x} \, dx\) [131]

Optimal result

Integrand size = 18, antiderivative size = 77 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x} \, dx=\frac {1}{3} \log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {2}{3} p \log \left (c \left (d+e x^3\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^3}{d}\right )-\frac {2}{3} p^2 \operatorname {PolyLog}\left (3,1+\frac {e x^3}{d}\right ) \] Output:

1/3*ln(-e*x^3/d)*ln(c*(e*x^3+d)^p)^2+2/3*p*ln(c*(e*x^3+d)^p)*polylog(2,1+e 
*x^3/d)-2/3*p^2*polylog(3,1+e*x^3/d)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.37 (sec) , antiderivative size = 2965, normalized size of antiderivative = 38.51 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x} \, dx=\text {Result too large to show} \] Input:

Integrate[Log[c*(d + e*x^3)^p]^2/x,x]
 

Output:

Log[x]*(-(p*Log[d + e*x^3]) + Log[c*(d + e*x^3)^p])^2 + 2*p*(-(p*Log[d + e 
*x^3]) + Log[c*(d + e*x^3)^p])*(Log[x]*(Log[d + e*x^3] - Log[1 + (e*x^3)/d 
]) - PolyLog[2, -((e*x^3)/d)]/3) + p^2*(Log[-((e^(1/3)*x)/d^(1/3))]*Log[d^ 
(1/3)/e^(1/3) + x]^2 + 2*Log[-((e^(1/3)*x)/d^(1/3))]*Log[d^(1/3)/e^(1/3) + 
 x]*Log[-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x] + Log[-(((-1)^(2/3)*e^(1/3)*x 
)/d^(1/3))]*Log[-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x]^2 + 2*Log[-((e^(1/3)* 
x)/d^(1/3))]*Log[d^(1/3)/e^(1/3) + x]*Log[((-1)^(2/3)*d^(1/3))/e^(1/3) + x 
] + 2*Log[-(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))]*Log[-(((-1)^(1/3)*d^(1/3))/e^ 
(1/3)) + x]*Log[((-1)^(2/3)*d^(1/3))/e^(1/3) + x] + Log[((-1)^(1/3)*e^(1/3 
)*x)/d^(1/3)]*Log[((-1)^(2/3)*d^(1/3))/e^(1/3) + x]^2 + Log[((-1)^(2/3)*(( 
(-1)^(2/3)*d^(1/3))/e^(1/3) + x))/(-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x)]^2 
*(Log[-(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))] + Log[(I*Sqrt[3]*d^(1/3))/((-1)^( 
1/3)*d^(1/3) - e^(1/3)*x)] - Log[((-1)^(2/3)*(1 + (-1)^(1/3))*e^(1/3)*x)/( 
(-1)^(1/3)*d^(1/3) - e^(1/3)*x)]) + (Log[-((e^(1/3)*x)/d^(1/3))] + Log[-(( 
(-1 + (-1)^(2/3))*d^(1/3))/(d^(1/3) + e^(1/3)*x))] - Log[((1 + (-1)^(1/3)) 
*e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)])*Log[(d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/( 
d^(1/3) + e^(1/3)*x)]^2 + (Log[2] + Log[-((e^(1/3)*x)/d^(1/3))] + Log[((1 
+ (-1)^(1/3))*d^(1/3))/(d^(1/3) + e^(1/3)*x)] - Log[((3 - I*Sqrt[3])*e^(1/ 
3)*x)/(d^(1/3) + e^(1/3)*x)])*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/(d^(1/3 
) + e^(1/3)*x)]^2 + 2*(Log[((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - Log[-(((-1...
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2904, 2843, 2881, 2821, 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.

\(\displaystyle \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle \frac {1}{3} \int \frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{x^3}dx^3\)

\(\Big \downarrow \) 2843

\(\displaystyle \frac {1}{3} \left (\log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-2 e p \int \frac {\log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )}{e x^3+d}dx^3\right )\)

\(\Big \downarrow \) 2881

\(\displaystyle \frac {1}{3} \left (\log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-2 p \int \frac {\log \left (-\frac {e x^3}{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )}{x^3}d\left (e x^3+d\right )\right )\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {1}{3} \left (\log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-2 p \left (p \int \frac {\operatorname {PolyLog}\left (2,\frac {e x^3+d}{d}\right )}{x^3}d\left (e x^3+d\right )-\operatorname {PolyLog}\left (2,\frac {e x^3+d}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{3} \left (\log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-2 p \left (p \operatorname {PolyLog}\left (3,\frac {e x^3+d}{d}\right )-\operatorname {PolyLog}\left (2,\frac {e x^3+d}{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )\right )\right )\)

Input:

Int[Log[c*(d + e*x^3)^p]^2/x,x]
 

Output:

(Log[-((e*x^3)/d)]*Log[c*(d + e*x^3)^p]^2 - 2*p*(-(Log[c*(d + e*x^3)^p]*Po 
lyLog[2, (d + e*x^3)/d]) + p*PolyLog[3, (d + e*x^3)/d]))/3
 

Defintions of rubi rules used

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 2843

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

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* 
((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, 
 f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, 
 x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & 
&  !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 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 [F]

\[\int \frac {{\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )}^{2}}{x}d x\]

Input:

int(ln(c*(e*x^3+d)^p)^2/x,x)
 

Output:

int(ln(c*(e*x^3+d)^p)^2/x,x)
 

Fricas [F]

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

integrate(log(c*(e*x^3+d)^p)^2/x,x, algorithm="fricas")
 

Output:

integral(log((e*x^3 + d)^p*c)^2/x, x)
 

Sympy [F]

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

integrate(ln(c*(e*x**3+d)**p)**2/x,x)
 

Output:

Integral(log(c*(d + e*x**3)**p)**2/x, x)
 

Maxima [F]

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

integrate(log(c*(e*x^3+d)^p)^2/x,x, algorithm="maxima")
 

Output:

integrate(log((e*x^3 + d)^p*c)^2/x, x)
 

Giac [F]

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

integrate(log(c*(e*x^3+d)^p)^2/x,x, algorithm="giac")
 

Output:

integrate(log((e*x^3 + d)^p*c)^2/x, x)
 

Mupad [F(-1)]

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

int(log(c*(d + e*x^3)^p)^2/x,x)
 

Output:

int(log(c*(d + e*x^3)^p)^2/x, x)
 

Reduce [F]

\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x} \, dx=\frac {9 \left (\int \frac {{\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{2}}{e \,x^{4}+d x}d x \right ) d p +{\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{3}}{9 p} \] Input:

int(log(c*(e*x^3+d)^p)^2/x,x)
 

Output:

(9*int(log((d + e*x**3)**p*c)**2/(d*x + e*x**4),x)*d*p + log((d + e*x**3)* 
*p*c)**3)/(9*p)