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

Optimal. Leaf size=291 \[ \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) \left (d \sqrt {f}-c \sqrt {g}\right )}{(c+d x) \left (b \sqrt {f}-a \sqrt {g}\right )}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) \left (c \sqrt {g}+d \sqrt {f}\right )}{(c+d x) \left (a \sqrt {g}+b \sqrt {f}\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {n \text {Li}_2\left (\frac {\left (d \sqrt {f}-c \sqrt {g}\right ) (a+b x)}{\left (b \sqrt {f}-a \sqrt {g}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {n \text {Li}_2\left (\frac {\left (\sqrt {g} c+d \sqrt {f}\right ) (a+b x)}{\left (\sqrt {g} a+b \sqrt {f}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}} \]

[Out]

1/2*ln(e*((b*x+a)/(d*x+c))^n)*ln(1-(b*x+a)*(d*f^(1/2)-c*g^(1/2))/(d*x+c)/(b*f^(1/2)-a*g^(1/2)))/f^(1/2)/g^(1/2
)-1/2*ln(e*((b*x+a)/(d*x+c))^n)*ln(1-(b*x+a)*(d*f^(1/2)+c*g^(1/2))/(d*x+c)/(b*f^(1/2)+a*g^(1/2)))/f^(1/2)/g^(1
/2)+1/2*n*polylog(2,(b*x+a)*(d*f^(1/2)-c*g^(1/2))/(d*x+c)/(b*f^(1/2)-a*g^(1/2)))/f^(1/2)/g^(1/2)-1/2*n*polylog
(2,(b*x+a)*(d*f^(1/2)+c*g^(1/2))/(d*x+c)/(b*f^(1/2)+a*g^(1/2)))/f^(1/2)/g^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.32, 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 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]

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 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 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 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 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]]

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 = 0.11, size = 421, normalized size = 1.45 \[ \frac {-\log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \text {Li}_2\left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} a+b \sqrt {f}}\right )-n \text {Li}_2\left (\frac {b \left (\sqrt {g} x+\sqrt {f}\right )}{b \sqrt {f}-a \sqrt {g}}\right )+n \log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (\frac {\sqrt {g} (a+b x)}{a \sqrt {g}+b \sqrt {f}}\right )-n \log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )-n \text {Li}_2\left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} c+d \sqrt {f}}\right )+n \text {Li}_2\left (\frac {d \left (\sqrt {g} x+\sqrt {f}\right )}{d \sqrt {f}-c \sqrt {g}}\right )-n \log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (\frac {\sqrt {g} (c+d x)}{c \sqrt {g}+d \sqrt {f}}\right )+n \log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 \sqrt {f} \sqrt {g}} \]

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])

________________________________________________________________________________________

fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f}, x\right ) \]

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)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

________________________________________________________________________________________

maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{-g \,x^{2}+f}\, dx \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.84, size = 349, normalized size = 1.20 \[ \frac {{\left (\log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}} + 1\right ) - \log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}} + 1\right ) - \log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}} + 1\right ) + \log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}} + 1\right ) + {\rm Li}_2\left (-\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}}\right ) - {\rm Li}_2\left (\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}}\right ) - {\rm Li}_2\left (-\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}}\right ) + {\rm Li}_2\left (\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}}\right )\right )} n}{2 \, \sqrt {f g}} - \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) \log \left (\frac {g x - \sqrt {f g}}{g x + \sqrt {f g}}\right )}{2 \, \sqrt {f g}} \]

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]

1/2*(log(sqrt(g)*x - sqrt(f))*log((b*sqrt(g)*x - b*sqrt(f))/(b*sqrt(f) + a*sqrt(g)) + 1) - log(sqrt(g)*x + sqr
t(f))*log(-(b*sqrt(g)*x + b*sqrt(f))/(b*sqrt(f) - a*sqrt(g)) + 1) - log(sqrt(g)*x - sqrt(f))*log((d*sqrt(g)*x
- d*sqrt(f))/(d*sqrt(f) + c*sqrt(g)) + 1) + log(sqrt(g)*x + sqrt(f))*log(-(d*sqrt(g)*x + d*sqrt(f))/(d*sqrt(f)
- c*sqrt(g)) + 1) + dilog(-(b*sqrt(g)*x - b*sqrt(f))/(b*sqrt(f) + a*sqrt(g))) - dilog((b*sqrt(g)*x + b*sqrt(f
))/(b*sqrt(f) - a*sqrt(g))) - dilog(-(d*sqrt(g)*x - d*sqrt(f))/(d*sqrt(f) + c*sqrt(g))) + dilog((d*sqrt(g)*x +
d*sqrt(f))/(d*sqrt(f) - c*sqrt(g))))*n/sqrt(f*g) - 1/2*log(e*((b*x + a)/(d*x + c))^n)*log((g*x - sqrt(f*g))/(
g*x + sqrt(f*g)))/sqrt(f*g)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f-g\,x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________