3.80 \(\int \frac{\log ^2(c (a+b x^2)^p)}{x} \, dx\)

Optimal. Leaf size=72 \[ p \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )+p^2 \left (-\text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )\right )+\frac{1}{2} \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right ) \]

[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]

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Rubi [A]  time = 0.112751, antiderivative size = 72, normalized size of antiderivative = 1., 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} \[ p \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )+p^2 \left (-\text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )\right )+\frac{1}{2} \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right ) \]

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 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 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 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 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 (a+b x^2\right )^p\right )}{x} \, dx &=\frac{1}{2} \operatorname{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) \operatorname{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 \operatorname{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 \operatorname{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]  time = 0.060482, size = 163, normalized size = 2.26 \[ 2 p \left (\log (x) \left (\log \left (a+b x^2\right )-\log \left (\frac{b x^2}{a}+1\right )\right )-\frac{1}{2} \text{PolyLog}\left (2,-\frac{b x^2}{a}\right )\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )+\frac{1}{2} p^2 \left (-2 \text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )+2 \log \left (a+b x^2\right ) \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )+\log \left (-\frac{b x^2}{a}\right ) \log ^2\left (a+b x^2\right )\right )+\log (x) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2 \]

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

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Maple [F]  time = 0.849, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}}{x}}\, dx \end{align*}

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)

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

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]

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

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

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)

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

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)

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

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)