3.1.80 \(\int \frac {\log ^2(c (a+b x^2)^p)}{x} \, dx\) [80]

Optimal. Leaf size=72 \[ \frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-p^2 \text {Li}_3\left (1+\frac {b x^2}{a}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 72, 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 = {2504, 2443, 2481, 2421, 6724} \begin {gather*} p \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )+p^2 \left (-\text {PolyLog}\left (3,\frac {b x^2}{a}+1\right )\right )+\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]^2/x,x]

[Out]

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

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*} \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )-(b p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )-p \text {Subst}\left (\int \frac {\log \left (c x^p\right ) \log \left (-\frac {b \left (-\frac {a}{b}+\frac {x}{b}\right )}{a}\right )}{x} \, dx,x,a+b x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-p^2 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-p^2 \text {Li}_3\left (1+\frac {b x^2}{a}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(72)=144\).
time = 0.08, size = 163, normalized size = 2.26 \begin {gather*} \log (x) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+2 p \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (\log (x) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )-\frac {1}{2} \text {Li}_2\left (-\frac {b x^2}{a}\right )\right )+\frac {1}{2} p^2 \left (\log \left (-\frac {b x^2}{a}\right ) \log ^2\left (a+b x^2\right )+2 \log \left (a+b x^2\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-2 \text {Li}_3\left (1+\frac {b x^2}{a}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]^2/x,x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^2+a)^p)^2/x,x)

[Out]

int(ln(c*(b*x^2+a)^p)^2/x,x)

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 118, normalized size = 1.64 \begin {gather*} \frac {1}{2} \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} p^{2} + {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} p \log \left (c\right ) + \log \left (c\right )^{2} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^2/x,x, algorithm="maxima")

[Out]

1/2*(log(b*x^2 + a)^2*log(-(b*x^2 + a)/a + 1) + 2*dilog((b*x^2 + a)/a)*log(b*x^2 + a) - 2*polylog(3, (b*x^2 +
a)/a))*p^2 + (log(b*x^2 + a)*log(-(b*x^2 + a)/a + 1) + dilog((b*x^2 + a)/a))*p*log(c) + log(c)^2*log(x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^2/x,x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)^2/x, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x**2+a)**p)**2/x,x)

[Out]

Integral(log(c*(a + b*x**2)**p)**2/x, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^2/x,x, algorithm="giac")

[Out]

integrate(log((b*x^2 + a)^p*c)^2/x, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x^2)^p)^2/x,x)

[Out]

int(log(c*(a + b*x^2)^p)^2/x, x)

________________________________________________________________________________________