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

Optimal. Leaf size=291 \[ \frac{n \text{PolyLog}\left (2,\frac{(a+b x) \left (d \sqrt{f}-c \sqrt{g}\right )}{(c+d x) \left (b \sqrt{f}-a \sqrt{g}\right )}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \text{PolyLog}\left (2,\frac{(a+b x) \left (c \sqrt{g}+d \sqrt{f}\right )}{(c+d x) \left (a \sqrt{g}+b \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac{(a+b x) \left (d \sqrt{f}-c \sqrt{g}\right )}{(c+d x) \left (b \sqrt{f}-a \sqrt{g}\right )}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac{(a+b x) \left (c \sqrt{g}+d \sqrt{f}\right )}{(c+d x) \left (a \sqrt{g}+b \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}} \]

[Out]

(Log[e*((a + b*x)/(c + d*x))^n]*Log[1 - ((d*Sqrt[f] - c*Sqrt[g])*(a + b*x))/((b*Sqrt[f] - a*Sqrt[g])*(c + d*x)
)])/(2*Sqrt[f]*Sqrt[g]) - (Log[e*((a + b*x)/(c + d*x))^n]*Log[1 - ((d*Sqrt[f] + c*Sqrt[g])*(a + b*x))/((b*Sqrt
[f] + a*Sqrt[g])*(c + d*x))])/(2*Sqrt[f]*Sqrt[g]) + (n*PolyLog[2, ((d*Sqrt[f] - c*Sqrt[g])*(a + b*x))/((b*Sqrt
[f] - a*Sqrt[g])*(c + d*x))])/(2*Sqrt[f]*Sqrt[g]) - (n*PolyLog[2, ((d*Sqrt[f] + c*Sqrt[g])*(a + b*x))/((b*Sqrt
[f] + a*Sqrt[g])*(c + d*x))])/(2*Sqrt[f]*Sqrt[g])

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Rubi [A]  time = 0.317435, antiderivative size = 468, normalized size of antiderivative = 1.61, number of steps used = 18, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2513, 2409, 2394, 2393, 2391, 208} \[ \frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{\sqrt{f} \sqrt{g}}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((ArcTanh[(Sqrt[g]*x)/Sqrt[f]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(Sqrt[f]*S
qrt[g])) - (n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*Sqrt[f]*Sqrt[g]) + (n*Lo
g[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*Sqrt[f]*Sqrt[g]) + (n*Log[a + b*x]*Log[(
b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*Sqrt[f]*Sqrt[g]) - (n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqr
t[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*Sqrt[f]*Sqrt[g]) + (n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sq
rt[g]))])/(2*Sqrt[f]*Sqrt[g]) - (n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*Sqrt[f]*Sqrt[g]
) - (n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*Sqrt[f]*Sqrt[g]) + (n*PolyLog[2, (Sqrt[g
]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*Sqrt[f]*Sqrt[g])

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

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 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx &=n \int \frac{\log (a+b x)}{f-g x^2} \, dx-n \int \frac{\log (c+d x)}{f-g x^2} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac{1}{f-g x^2} \, dx\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{\sqrt{f} \sqrt{g}}+n \int \left (\frac{\log (a+b x)}{2 \sqrt{f} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (a+b x)}{2 \sqrt{f} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx-n \int \left (\frac{\log (c+d x)}{2 \sqrt{f} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (c+d x)}{2 \sqrt{f} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{\sqrt{f} \sqrt{g}}+\frac{n \int \frac{\log (a+b x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 \sqrt{f}}+\frac{n \int \frac{\log (a+b x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 \sqrt{f}}-\frac{n \int \frac{\log (c+d x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 \sqrt{f}}-\frac{n \int \frac{\log (c+d x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 \sqrt{f}}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{\sqrt{f} \sqrt{g}}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{(b n) \int \frac{\log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{a+b x} \, dx}{2 \sqrt{f} \sqrt{g}}-\frac{(b n) \int \frac{\log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{a+b x} \, dx}{2 \sqrt{f} \sqrt{g}}-\frac{(d n) \int \frac{\log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{c+d x} \, dx}{2 \sqrt{f} \sqrt{g}}+\frac{(d n) \int \frac{\log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{c+d x} \, dx}{2 \sqrt{f} \sqrt{g}}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{\sqrt{f} \sqrt{g}}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{b \sqrt{f}-a \sqrt{g}}\right )}{x} \, dx,x,a+b x\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{b \sqrt{f}+a \sqrt{g}}\right )}{x} \, dx,x,a+b x\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{d \sqrt{f}-c \sqrt{g}}\right )}{x} \, dx,x,c+d x\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{d \sqrt{f}+c \sqrt{g}}\right )}{x} \, dx,x,c+d x\right )}{2 \sqrt{f} \sqrt{g}}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{\sqrt{f} \sqrt{g}}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \text{Li}_2\left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \text{Li}_2\left (\frac{\sqrt{g} (a+b x)}{b \sqrt{f}+a \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{n \text{Li}_2\left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{n \text{Li}_2\left (\frac{\sqrt{g} (c+d x)}{d \sqrt{f}+c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}}\\ \end{align*}

Mathematica [A]  time = 0.101591, size = 421, normalized size = 1.45 \[ \frac{n \text{PolyLog}\left (2,\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )-n \text{PolyLog}\left (2,\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )-n \text{PolyLog}\left (2,\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )+n \text{PolyLog}\left (2,\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )-\log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )-n \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )-n \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )+n \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 \sqrt{f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.474, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-g{x}^{2}+f}\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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