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

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2504, 2443, 2481, 2421, 6724} \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x} \, dx=\frac {2}{3} p \operatorname {PolyLog}\left (2,\frac {e x^3}{d}+1\right ) \log \left (c \left (d+e x^3\right )^p\right )+\frac {1}{3} \log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {2}{3} p^2 \operatorname {PolyLog}\left (3,\frac {e x^3}{d}+1\right ) \]

[In]

Int[Log[c*(d + e*x^3)^p]^2/x,x]

[Out]

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

Rule 2421

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] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[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 2443

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] - Dist[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 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[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 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\log ^2\left (c (d+e x)^p\right )}{x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {1}{3} (2 e p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {1}{3} (2 p) \text {Subst}\left (\int \frac {\log \left (c x^p\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x^3\right ) \\ & = \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 ) \text {Li}_2\left (1+\frac {e x^3}{d}\right )-\frac {1}{3} \left (2 p^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^3\right ) \\ & = \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 ) \text {Li}_2\left (1+\frac {e x^3}{d}\right )-\frac {2}{3} p^2 \text {Li}_3\left (1+\frac {e x^3}{d}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 2965, normalized size of antiderivative = 38.51 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x} \, dx=\text {Result too large to show} \]

[In]

Integrate[Log[c*(d + e*x^3)^p]^2/x,x]

[Out]

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*Sq
rt[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*(L
og[((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - Log[-(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])*Log[((-1)^(2/3)*(((-1)^(2/3)*d^(1
/3))/e^(1/3) + x))/(-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x)]*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + 2*(-Log[-(
(e^(1/3)*x)/d^(1/3))] + Log[((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])*Log[(d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/(d^(1/3) + e
^(1/3)*x)]*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + (Log[-((e^(1/3)*x)/d^(1/3))] - Log[((-1)^(1/3)*e^(1/3)*x)
/d^(1/3)])*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]*(-2*Log[d^(1/3)/e^(1/3) + x] + Log[1 - ((-1)^(1/3)*e^(1/3)*
x)/d^(1/3)]) + (-Log[((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + Log[-(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])*Log[1 - ((-1)^(
1/3)*e^(1/3)*x)/d^(1/3)]*(-2*Log[-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x] + Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)
]) + 2*(-Log[-((e^(1/3)*x)/d^(1/3))] + Log[-(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])*Log[(d^(1/3) + (-1)^(2/3)*e^(1/
3)*x)/(d^(1/3) + e^(1/3)*x)]*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + (Log[-((e^(1/3)*x)/d^(1/3))] - Log[-(((
-1)^(2/3)*e^(1/3)*x)/d^(1/3))])*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)]*(-2*Log[d^(1/3)/e^(1/3) + x] + Log[1 +
((-1)^(2/3)*e^(1/3)*x)/d^(1/3)]) + Log[x]*(Log[d^(1/3)/e^(1/3) + x] + Log[-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x
] + Log[((-1)^(2/3)*d^(1/3))/e^(1/3) + x] - Log[d + e*x^3])^2 - 2*(Log[d^(1/3)/e^(1/3) + x] + Log[-(((-1)^(1/3
)*d^(1/3))/e^(1/3)) + x] + Log[((-1)^(2/3)*d^(1/3))/e^(1/3) + x] - Log[d + e*x^3])*(Log[x]*Log[d^(1/3)/e^(1/3)
+ x] + Log[x]*Log[-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x] + Log[x]*Log[((-1)^(2/3)*d^(1/3))/e^(1/3) + x] - Log[x
]*Log[1 + (e^(1/3)*x)/d^(1/3)] - Log[x]*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - Log[x]*Log[1 + ((-1)^(2/3)*e
^(1/3)*x)/d^(1/3)] - PolyLog[2, -((e^(1/3)*x)/d^(1/3))] - PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - PolyLog
[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))]) + 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)]*(-PolyLog[2, ((-1)^(2/3)*(((-1)^(2/3)*d^(1/3))/e^(1/3) + x))/(-(((-1)^(1/3)*d^(1/3))/
e^(1/3)) + x)] + PolyLog[2, ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/(-((-1)^(1/3)*d^(1/3)) + e^(1/3)*x)]) + 2*Log[(d^
(1/3) - (-1)^(1/3)*e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)]*(PolyLog[2, ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/(d^(1/3) + e
^(1/3)*x)] - PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)]) + 2*Log[(d^(1/3) + (-1)^(2/3)
*e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)]*(PolyLog[2, (-((-1)^(1/3)*d^(1/3)) + e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)] - Po
lyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)]) + 2*Log[d^(1/3)/e^(1/3) + x]*PolyLog[2, 1 +
(e^(1/3)*x)/d^(1/3)] + 2*(Log[((-1)^(2/3)*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)])*PolyLog[2, 1 + (e^(1/3)*x)/d^(1/3)] + 2*(Log[-(((-1)^(1/3)*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)])*PolyLog[2, 1 + (e^(1/3)*x)/d^(1/3)] + 2*Log[((-1)^(2/3)*d^
(1/3))/e^(1/3) + x]*PolyLog[2, 1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + 2*(Log[-(((-1)^(1/3)*d^(1/3))/e^(1/3)) +
x] + Log[((-1)^(2/3)*(((-1)^(2/3)*d^(1/3))/e^(1/3) + x))/(-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x)])*PolyLog[2, 1
- ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + 2*(Log[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)])*PolyLog[2, 1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + 2*Log[-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x]*P
olyLog[2, 1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + 2*(Log[((-1)^(2/3)*d^(1/3))/e^(1/3) + x] - Log[((-1)^(2/3)*(((
-1)^(2/3)*d^(1/3))/e^(1/3) + x))/(-(((-1)^(1/3)*d^(1/3))/e^(1/3)) + x)])*PolyLog[2, 1 + ((-1)^(2/3)*e^(1/3)*x)
/d^(1/3)] + 2*(Log[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)])*PolyLog
[2, 1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + 2*PolyLog[3, ((-1)^(2/3)*(((-1)^(2/3)*d^(1/3))/e^(1/3) + x))/(-(((-1
)^(1/3)*d^(1/3))/e^(1/3)) + x)] - 2*PolyLog[3, (-((-1)^(1/3)*d^(1/3)) + e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)] - 2*
PolyLog[3, ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)] - 2*PolyLog[3, ((-1)^(2/3)*d^(1/3) + e^(1/3
)*x)/(-((-1)^(1/3)*d^(1/3)) + e^(1/3)*x)] + 2*PolyLog[3, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/(d^(1/3) + e^(1/3)*x
)] + 2*PolyLog[3, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/(d^(1/3) + e^(1/3)*x)] - 6*PolyLog[3, 1 + (e^(1/3)*x)/d^(1/
3)] - 6*PolyLog[3, 1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] - 6*PolyLog[3, 1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(log((e*x^3 + d)^p*c)^2/x, x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(log(c*(d + e*x^3)^p)^2/x,x)

[Out]

int(log(c*(d + e*x^3)^p)^2/x, x)