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

Optimal. Leaf size=77 \[ \frac {2}{3} p \text {Li}_2\left (\frac {e x^3}{d}+1\right ) \log \left (c \left (d+e x^3\right )^p\right )+\frac {1}{3} \log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {2}{3} p^2 \text {Li}_3\left (\frac {e x^3}{d}+1\right ) \]

[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)

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Rubi [A]  time = 0.11, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2454, 2396, 2433, 2374, 6589} \[ \frac {2}{3} p \text {PolyLog}\left (2,\frac {e x^3}{d}+1\right ) \log \left (c \left (d+e x^3\right )^p\right )-\frac {2}{3} p^2 \text {PolyLog}\left (3,\frac {e x^3}{d}+1\right )+\frac {1}{3} \log \left (-\frac {e x^3}{d}\right ) \log ^2\left (c \left (d+e x^3\right )^p\right ) \]

Antiderivative was successfully verified.

[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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*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 2396

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 2433

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 2454

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 6589

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*} \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x} \, dx &=\frac {1}{3} \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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*}

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Mathematica [B]  time = 0.10, size = 163, normalized size = 2.12 \[ 2 p \left (\log (x) \left (\log \left (d+e x^3\right )-\log \left (\frac {e x^3}{d}+1\right )\right )-\frac {1}{3} \text {Li}_2\left (-\frac {e x^3}{d}\right )\right ) \left (\log \left (c \left (d+e x^3\right )^p\right )-p \log \left (d+e x^3\right )\right )+\log (x) \left (\log \left (c \left (d+e x^3\right )^p\right )-p \log \left (d+e x^3\right )\right )^2+\frac {1}{3} p^2 \left (-2 \text {Li}_3\left (\frac {e x^3}{d}+1\right )+2 \text {Li}_2\left (\frac {e x^3}{d}+1\right ) \log \left (d+e x^3\right )+\log \left (-\frac {e x^3}{d}\right ) \log ^2\left (d+e x^3\right )\right ) \]

Antiderivative was successfully verified.

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

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^3)^p)^2/x,x)

[Out]

int(log(c*(d + e*x^3)^p)^2/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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