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

Optimal. Leaf size=67 \[ \frac{8 b \text{PolyLog}\left (2,1-\frac{b}{a x+b}\right )}{a}+x \log ^2\left (\frac{c x^2}{(a x+b)^2}\right )+\frac{4 b \log \left (\frac{b}{a x+b}\right ) \log \left (\frac{c x^2}{(a x+b)^2}\right )}{a} \]

[Out]

x*Log[(c*x^2)/(b + a*x)^2]^2 + (4*b*Log[(c*x^2)/(b + a*x)^2]*Log[b/(b + a*x)])/a + (8*b*PolyLog[2, 1 - b/(b +
a*x)])/a

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Rubi [A]  time = 0.154865, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2486, 2488, 2411, 2343, 2333, 2315} \[ \frac{8 b \text{PolyLog}\left (2,1-\frac{b}{a x+b}\right )}{a}+x \log ^2\left (\frac{c x^2}{(a x+b)^2}\right )+\frac{4 b \log \left (\frac{b}{a x+b}\right ) \log \left (\frac{c x^2}{(a x+b)^2}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[Log[(c*x^2)/(b + a*x)^2]^2,x]

[Out]

x*Log[(c*x^2)/(b + a*x)^2]^2 + (4*b*Log[(c*x^2)/(b + a*x)^2]*Log[b/(b + a*x)])/a + (8*b*PolyLog[2, 1 - b/(b +
a*x)])/a

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 2488

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_)),
x_Symbol] :> -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p
*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a
+ b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2343

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Dist[1/n, Subst[Int[(a
+ b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]

Rule 2333

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \log ^2\left (\frac{c x^2}{(b+a x)^2}\right ) \, dx &=x \log ^2\left (\frac{c x^2}{(b+a x)^2}\right )-(4 b) \int \frac{\log \left (\frac{c x^2}{(b+a x)^2}\right )}{b+a x} \, dx\\ &=x \log ^2\left (\frac{c x^2}{(b+a x)^2}\right )+\frac{4 b \log \left (\frac{c x^2}{(b+a x)^2}\right ) \log \left (\frac{b}{b+a x}\right )}{a}-\frac{\left (8 b^2\right ) \int \frac{\log \left (\frac{b}{b+a x}\right )}{x (b+a x)} \, dx}{a}\\ &=x \log ^2\left (\frac{c x^2}{(b+a x)^2}\right )+\frac{4 b \log \left (\frac{c x^2}{(b+a x)^2}\right ) \log \left (\frac{b}{b+a x}\right )}{a}-\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{b}{x}\right )}{x \left (-\frac{b}{a}+\frac{x}{a}\right )} \, dx,x,b+a x\right )}{a^2}\\ &=x \log ^2\left (\frac{c x^2}{(b+a x)^2}\right )+\frac{4 b \log \left (\frac{c x^2}{(b+a x)^2}\right ) \log \left (\frac{b}{b+a x}\right )}{a}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (b x)}{\left (-\frac{b}{a}+\frac{1}{a x}\right ) x} \, dx,x,\frac{1}{b+a x}\right )}{a^2}\\ &=x \log ^2\left (\frac{c x^2}{(b+a x)^2}\right )+\frac{4 b \log \left (\frac{c x^2}{(b+a x)^2}\right ) \log \left (\frac{b}{b+a x}\right )}{a}+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (b x)}{\frac{1}{a}-\frac{b x}{a}} \, dx,x,\frac{1}{b+a x}\right )}{a^2}\\ &=x \log ^2\left (\frac{c x^2}{(b+a x)^2}\right )+\frac{4 b \log \left (\frac{c x^2}{(b+a x)^2}\right ) \log \left (\frac{b}{b+a x}\right )}{a}+\frac{8 b \text{Li}_2\left (\frac{a x}{b+a x}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0141174, size = 106, normalized size = 1.58 \[ \frac{8 b \text{PolyLog}\left (2,\frac{a x+b}{b}\right )}{a}+x \log ^2\left (\frac{c x^2}{(a x+b)^2}\right )+\frac{4 b \log \left (\frac{b}{a x+b}\right ) \log \left (\frac{c x^2}{(a x+b)^2}\right )}{a}-\frac{4 b \log ^2\left (\frac{b}{a x+b}\right )}{a}-\frac{8 b \log \left (-\frac{a x}{b}\right ) \log \left (\frac{b}{a x+b}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(c*x^2)/(b + a*x)^2]^2,x]

[Out]

x*Log[(c*x^2)/(b + a*x)^2]^2 - (8*b*Log[-((a*x)/b)]*Log[b/(b + a*x)])/a + (4*b*Log[(c*x^2)/(b + a*x)^2]*Log[b/
(b + a*x)])/a - (4*b*Log[b/(b + a*x)]^2)/a + (8*b*PolyLog[2, (b + a*x)/b])/a

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Maple [F]  time = 0.94, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ({\frac{c{x}^{2}}{ \left ( ax+b \right ) ^{2}}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1449, size = 159, normalized size = 2.37 \begin{align*} x \log \left (\frac{c x^{2}}{{\left (a x + b\right )}^{2}}\right )^{2} - \frac{4 \, b \log \left (a x + b\right ) \log \left (\frac{c x^{2}}{{\left (a x + b\right )}^{2}}\right )}{a} + \frac{4 \,{\left ({\left (\frac{c \log \left (a x + b\right )^{2}}{a} - \frac{2 \,{\left (\log \left (\frac{a x}{b} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{a x}{b}\right )\right )} c}{a}\right )} b - \frac{2 \,{\left (c \log \left (a x + b\right ) - c \log \left (x\right )\right )} b \log \left (a x + b\right )}{a}\right )}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(c*x^2/(a*x + b)^2)^2 - 4*b*log(a*x + b)*log(c*x^2/(a*x + b)^2)/a + 4*((c*log(a*x + b)^2/a - 2*(log(a*x/b
+ 1)*log(x) + dilog(-a*x/b))*c/a)*b - 2*(c*log(a*x + b) - c*log(x))*b*log(a*x + b)/a)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - 4 b \int \frac{\log{\left (\frac{c x^{2}}{a^{2} x^{2} + 2 a b x + b^{2}} \right )}}{a x + b}\, dx + x \log{\left (\frac{c x^{2}}{\left (a x + b\right )^{2}} \right )}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*b*Integral(log(c*x**2/(a**2*x**2 + 2*a*b*x + b**2))/(a*x + b), x) + x*log(c*x**2/(a*x + b)**2)**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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