\(\int \frac {\log (e (\frac {a+b x}{c+d x})^n)}{f-g x^2} \, dx\) [79]

Optimal result

Integrand size = 29, antiderivative size = 291 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {\left (d \sqrt {f}-c \sqrt {g}\right ) (a+b x)}{\left (b \sqrt {f}-a \sqrt {g}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {\left (d \sqrt {f}+c \sqrt {g}\right ) (a+b x)}{\left (b \sqrt {f}+a \sqrt {g}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {n \operatorname {PolyLog}\left (2,\frac {\left (d \sqrt {f}-c \sqrt {g}\right ) (a+b x)}{\left (b \sqrt {f}-a \sqrt {g}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {n \operatorname {PolyLog}\left (2,\frac {\left (d \sqrt {f}+c \sqrt {g}\right ) (a+b x)}{\left (b \sqrt {f}+a \sqrt {g}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}} \]

[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] (verified)

Time = 0.24 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2576, 2404, 2354, 2438} \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) \left (d \sqrt {f}-c \sqrt {g}\right )}{(c+d x) \left (b \sqrt {f}-a \sqrt {g}\right )}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) \left (c \sqrt {g}+d \sqrt {f}\right )}{(c+d x) \left (a \sqrt {g}+b \sqrt {f}\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {n \operatorname {PolyLog}\left (2,\frac {\left (d \sqrt {f}-c \sqrt {g}\right ) (a+b x)}{\left (b \sqrt {f}-a \sqrt {g}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {n \operatorname {PolyLog}\left (2,\frac {\left (\sqrt {g} c+d \sqrt {f}\right ) (a+b x)}{\left (\sqrt {g} a+b \sqrt {f}\right ) (c+d x)}\right )}{2 \sqrt {f} \sqrt {g}} \]

[In]

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

[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])

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2438

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 2576

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*(P2x_)^(m_.), x_Symbol]
:> With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, Dist[b*c - a*d, Subst[Int[(b^2*f
- a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^2*h)*x^2)^m*((A + B*Log[e*x^n])^p
/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/(c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x,
x, 2] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.45 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\frac {n \log \left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )-n \log \left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )-n \log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+n \log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+n \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )-n \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )-n \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )+n \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 \sqrt {f} \sqrt {g}} \]

[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])

Maple [F]

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

[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)

Fricas [F]

\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int { -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.20 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\frac {{\left (\log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}} + 1\right ) - \log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}} + 1\right ) - \log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}} + 1\right ) + \log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}} + 1\right ) + {\rm Li}_2\left (-\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}}\right ) - {\rm Li}_2\left (\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}}\right ) - {\rm Li}_2\left (-\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}}\right ) + {\rm Li}_2\left (\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}}\right )\right )} n}{2 \, \sqrt {f g}} - \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) \log \left (\frac {g x - \sqrt {f g}}{g x + \sqrt {f g}}\right )}{2 \, \sqrt {f g}} \]

[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)

Giac [F]

\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int { -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f-g\,x^2} \,d x \]

[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)