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

Optimal. Leaf size=1137 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.32774, antiderivative size = 1143, normalized size of antiderivative = 1.01, number of steps used = 39, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {2457, 2476, 2462, 260, 2416, 2390, 2301, 2394, 2393, 2391, 12} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^3)^p]^2/x^2,x]

[Out]

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

Rule 2457

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

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2462

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int \frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx &=-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+(6 e p) \int \frac{x \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+(6 e p) \int \left (-\frac{\log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac{(-1)^{2/3} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac{\sqrt [3]{-1} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx\\ &=-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}-\frac{\left (2 e^{2/3} p\right ) \int \frac{\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 \sqrt [3]{-1} e^{2/3} p\right ) \int \frac{\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac{\left (2 (-1)^{2/3} e^{2/3} p\right ) \int \frac{\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=-\frac{2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac{\left (6 e^{4/3} p^2\right ) \int \frac{x^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d+e x^3} \, dx}{\sqrt [3]{d}}-\frac{\left (6 \sqrt [3]{-1} e^{4/3} p^2\right ) \int \frac{x^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{d+e x^3} \, dx}{\sqrt [3]{d}}+\frac{\left (6 (-1)^{2/3} e^{4/3} p^2\right ) \int \frac{x^2 \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{d+e x^3} \, dx}{\sqrt [3]{d}}\\ &=-\frac{2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac{\left (6 e^{4/3} p^2\right ) \int \left (\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{\sqrt [3]{d}}-\frac{\left (6 \sqrt [3]{-1} e^{4/3} p^2\right ) \int \left (\frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{\sqrt [3]{d}}+\frac{\left (6 (-1)^{2/3} e^{4/3} p^2\right ) \int \left (\frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{\sqrt [3]{d}}\\ &=-\frac{2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac{\left (2 e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac{\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac{\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac{\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac{\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac{\left (2 \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{\left (2 \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{-1} \log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{\left (2 \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{(-1)^{2/3} \log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{\left (2 e^{2/3} p^2\right ) \int \frac{\log \left (\frac{\sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac{\left (2 e^{2/3} p^2\right ) \int \frac{\log \left (\frac{\sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac{\log \left (\frac{(-1)^{2/3} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac{\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac{\log \left (\frac{(-1)^{2/3} \sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac{\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac{\log \left (-\frac{\sqrt [3]{-1} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac{\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac{\log \left (-\frac{\sqrt [3]{-1} \sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=\frac{\sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}-\frac{\left (2 \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{\left (2 \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{\left (2 \sqrt [3]{-1} \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{\left (2 \sqrt [3]{-1} \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{\left (2 \sqrt [3]{-1} \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{\left (2 (-1)^{2/3} \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{\left (2 (-1)^{2/3} \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac{\log \left (\frac{\sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=\frac{\sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{\sqrt [3]{-1} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{(-1)^{2/3} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac{2 \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt{3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text{Li}_2\left (-\frac{\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{\left (2 (-1)^{2/3} \sqrt [3]{e} p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}\\ &=\frac{\sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{\sqrt [3]{-1} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac{(-1)^{2/3} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac{2 \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt{3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text{Li}_2\left (-\frac{\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac{2 (-1)^{2/3} \sqrt [3]{e} p^2 \text{Li}_2\left (\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}\\ \end{align*}

Mathematica [A]  time = 0.838462, size = 742, normalized size = 0.65 \[ -\frac{\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}-\frac{\sqrt [3]{e} p \left (\sqrt [3]{-1} p \left (2 \text{PolyLog}\left (2,\frac{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}}{\left ((-1)^{2/3}-1\right ) \sqrt [3]{d}}\right )+\log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \left (2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )+2 \log \left (\frac{(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left ((-1)^{2/3}-1\right ) \sqrt [3]{d}}\right )\right )\right )-(-1)^{2/3} p \left (2 \text{PolyLog}\left (2,\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )+\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \left (2 \log \left (\frac{(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left ((-1)^{2/3}-1\right ) \sqrt [3]{d}}\right )+2 \log \left (\frac{\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )\right )\right )-p \left (2 \text{PolyLog}\left (2,\frac{\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text{PolyLog}\left (2,\frac{2 i \left (\frac{\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{\sqrt{3}+3 i}\right )+\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \left (\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 \left (\log \left (\frac{\sqrt [3]{-1} \sqrt [3]{d}-\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (\frac{-\frac{2 i \sqrt [3]{e} x}{\sqrt [3]{d}}+\sqrt{3}+i}{\sqrt{3}+3 i}\right )\right )\right )\right )+2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-2 \sqrt [3]{-1} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )+2 (-1)^{2/3} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )\right )}{\sqrt [3]{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( e{x}^{3}+d \right ) ^{p} \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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