3.330 \(\int \frac{\log ^2(c (a+b x)^n)}{d+e x^2} \, dx\)

Optimal. Leaf size=347 \[ -\frac{n \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}} \]

[Out]

(Log[c*(a + b*x)^n]^2*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*
(a + b*x)^n]^2*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*Log[c*(a +
b*x)^n]*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) + (n*Log[c*(a + b*x)^n
]*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + (n^2*PolyLog[3, -((Sqrt[e]*(a
+ b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) - (n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*
Sqrt[e])])/(Sqrt[-d]*Sqrt[e])

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Rubi [A]  time = 0.316108, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {2409, 2396, 2433, 2374, 6589} \[ -\frac{n \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^n]^2/(d + e*x^2),x]

[Out]

(Log[c*(a + b*x)^n]^2*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*
(a + b*x)^n]^2*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*Log[c*(a +
b*x)^n]*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) + (n*Log[c*(a + b*x)^n
]*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + (n^2*PolyLog[3, -((Sqrt[e]*(a
+ b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) - (n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*
Sqrt[e])])/(Sqrt[-d]*Sqrt[e])

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 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 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 (a+b x)^n\right )}{d+e x^2} \, dx &=\int \left (\frac{\sqrt{-d} \log ^2\left (c (a+b x)^n\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \log ^2\left (c (a+b x)^n\right )}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\log ^2\left (c (a+b x)^n\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{\int \frac{\log ^2\left (c (a+b x)^n\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}\\ &=\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{b \sqrt{-d}+a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{(b n) \int \frac{\log \left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{b \sqrt{-d}+a \sqrt{e}}\right )}{a+b x} \, dx}{\sqrt{-d} \sqrt{e}}+\frac{(b n) \int \frac{\log \left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{a+b x} \, dx}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{b \sqrt{-d}+a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right ) \log \left (\frac{b \left (\frac{b \sqrt{-d}+a \sqrt{e}}{b}-\frac{\sqrt{e} x}{b}\right )}{b \sqrt{-d}+a \sqrt{e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt{-d} \sqrt{e}}+\frac{n \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right ) \log \left (\frac{b \left (\frac{b \sqrt{-d}-a \sqrt{e}}{b}+\frac{\sqrt{e} x}{b}\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{b \sqrt{-d}+a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (c (a+b x)^n\right ) \text{Li}_2\left (-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n \log \left (c (a+b x)^n\right ) \text{Li}_2\left (\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}+a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{e} x}{b \sqrt{-d}-a \sqrt{e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt{-d} \sqrt{e}}-\frac{n^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{b \sqrt{-d}+a \sqrt{e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{b \sqrt{-d}+a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (c (a+b x)^n\right ) \text{Li}_2\left (-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n \log \left (c (a+b x)^n\right ) \text{Li}_2\left (\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}+a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{n^2 \text{Li}_3\left (-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{n^2 \text{Li}_3\left (\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}+a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}\\ \end{align*}

Mathematica [C]  time = 0.153351, size = 488, normalized size = 1.41 \[ \frac{2 i n \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )-2 i n \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )-2 i n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )+2 i n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )+2 i n \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )-2 i n \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log ^2\left (c (a+b x)^n\right )-4 n \log (a+b x) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c (a+b x)^n\right )-i n^2 \log ^2(a+b x) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )+i n^2 \log ^2(a+b x) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )+2 n^2 \log ^2(a+b x) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*n^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[a + b*x]^2 - 4*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[a + b*x]*Log[c*(a + b*
x)^n] + 2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(a + b*x)^n]^2 - I*n^2*Log[a + b*x]^2*Log[1 - (Sqrt[e]*(a + b*x))/
((-I)*b*Sqrt[d] + a*Sqrt[e])] + (2*I)*n*Log[a + b*x]*Log[c*(a + b*x)^n]*Log[1 - (Sqrt[e]*(a + b*x))/((-I)*b*Sq
rt[d] + a*Sqrt[e])] + I*n^2*Log[a + b*x]^2*Log[1 - (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] - (2*I)*n*Lo
g[a + b*x]*Log[c*(a + b*x)^n]*Log[1 - (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] + (2*I)*n*Log[c*(a + b*x)
^n]*PolyLog[2, (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a*Sqrt[e])] - (2*I)*n*Log[c*(a + b*x)^n]*PolyLog[2, (Sqrt
[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])] - (2*I)*n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/((-I)*b*Sqrt[d] + a*Sqrt[
e])] + (2*I)*n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/(I*b*Sqrt[d] + a*Sqrt[e])])/(2*Sqrt[d]*Sqrt[e])

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Maple [F]  time = 4.311, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( bx+a \right ) ^{n} \right ) \right ) ^{2}}{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(ln(c*(b*x+a)^n)^2/(e*x^2+d),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*(b*x+a)^n)^2/(e*x^2+d),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 (b x + a\right )}^{n} c\right )^{2}}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((b*x + a)^n*c)^2/(e*x^2 + d), 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\right )^{n} \right )}^{2}}{d + e x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*(a + b*x)**n)**2/(d + e*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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