\(\int \frac {\log (c (d+\frac {e}{x^2})^p)}{f+g x^2} \, dx\) [265]

Optimal result

Integrand size = 22, antiderivative size = 597 \[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \] Output:

arctan(g^(1/2)*x/f^(1/2))*ln(c*(d+e/x^2)^p)/f^(1/2)/g^(1/2)+2*p*arctan(g^( 
1/2)*x/f^(1/2))*ln(2*f^(1/2)/(f^(1/2)-I*g^(1/2)*x))/f^(1/2)/g^(1/2)-p*arct 
an(g^(1/2)*x/f^(1/2))*ln(-2*f^(1/2)*g^(1/2)*(e^(1/2)-(-d)^(1/2)*x)/(I*(-d) 
^(1/2)*f^(1/2)-e^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(1/2)/g^(1/2)-p*a 
rctan(g^(1/2)*x/f^(1/2))*ln(2*f^(1/2)*g^(1/2)*(e^(1/2)+(-d)^(1/2)*x)/(I*(- 
d)^(1/2)*f^(1/2)+e^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(1/2)/g^(1/2)+I 
*p*polylog(2,-I*g^(1/2)*x/f^(1/2))/f^(1/2)/g^(1/2)-I*p*polylog(2,I*g^(1/2) 
*x/f^(1/2))/f^(1/2)/g^(1/2)-I*p*polylog(2,1-2*f^(1/2)/(f^(1/2)-I*g^(1/2)*x 
))/f^(1/2)/g^(1/2)+1/2*I*p*polylog(2,1+2*f^(1/2)*g^(1/2)*(e^(1/2)-(-d)^(1/ 
2)*x)/(I*(-d)^(1/2)*f^(1/2)-e^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(1/2 
)/g^(1/2)+1/2*I*p*polylog(2,1-2*f^(1/2)*g^(1/2)*(e^(1/2)+(-d)^(1/2)*x)/(I* 
(-d)^(1/2)*f^(1/2)+e^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(1/2)/g^(1/2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.18 \[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )+2 p \log \left (\frac {\sqrt {g} x}{\sqrt {-f}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left (-\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-\log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-2 p \log \left (\frac {f \sqrt {g} x}{(-f)^{3/2}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (\frac {\sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (-\frac {\sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right )-2 p \operatorname {PolyLog}\left (2,1+\frac {\sqrt {g} x}{\sqrt {-f}}\right )+2 p \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {g} x}{(-f)^{3/2}}\right )}{2 \sqrt {-f} \sqrt {g}} \] Input:

Integrate[Log[c*(d + e/x^2)^p]/(f + g*x^2),x]
 

Output:

(Log[c*(d + e/x^2)^p]*Log[Sqrt[-f] - Sqrt[g]*x] + 2*p*Log[(Sqrt[g]*x)/Sqrt 
[-f]]*Log[Sqrt[-f] - Sqrt[g]*x] - p*Log[(Sqrt[g]*(-Sqrt[e] + Sqrt[-d]*x))/ 
(Sqrt[-d]*Sqrt[-f] - Sqrt[e]*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*x] - p*Log[( 
Sqrt[g]*(Sqrt[e] + Sqrt[-d]*x))/(Sqrt[-d]*Sqrt[-f] + Sqrt[e]*Sqrt[g])]*Log 
[Sqrt[-f] - Sqrt[g]*x] - Log[c*(d + e/x^2)^p]*Log[Sqrt[-f] + Sqrt[g]*x] - 
2*p*Log[(f*Sqrt[g]*x)/(-f)^(3/2)]*Log[Sqrt[-f] + Sqrt[g]*x] + p*Log[(Sqrt[ 
g]*(Sqrt[e] - Sqrt[-d]*x))/(Sqrt[-d]*Sqrt[-f] + Sqrt[e]*Sqrt[g])]*Log[Sqrt 
[-f] + Sqrt[g]*x] + p*Log[-((Sqrt[g]*(Sqrt[e] + Sqrt[-d]*x))/(Sqrt[-d]*Sqr 
t[-f] - Sqrt[e]*Sqrt[g]))]*Log[Sqrt[-f] + Sqrt[g]*x] - p*PolyLog[2, (Sqrt[ 
-d]*(Sqrt[-f] - Sqrt[g]*x))/(Sqrt[-d]*Sqrt[-f] - Sqrt[e]*Sqrt[g])] - p*Pol 
yLog[2, (Sqrt[-d]*(Sqrt[-f] - Sqrt[g]*x))/(Sqrt[-d]*Sqrt[-f] + Sqrt[e]*Sqr 
t[g])] + p*PolyLog[2, (Sqrt[-d]*(Sqrt[-f] + Sqrt[g]*x))/(Sqrt[-d]*Sqrt[-f] 
 - Sqrt[e]*Sqrt[g])] + p*PolyLog[2, (Sqrt[-d]*(Sqrt[-f] + Sqrt[g]*x))/(Sqr 
t[-d]*Sqrt[-f] + Sqrt[e]*Sqrt[g])] - 2*p*PolyLog[2, 1 + (Sqrt[g]*x)/Sqrt[- 
f]] + 2*p*PolyLog[2, 1 + (f*Sqrt[g]*x)/(-f)^(3/2)])/(2*Sqrt[-f]*Sqrt[g])
 

Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 557, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2920, 27, 2005, 5463, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx\)

\(\Big \downarrow \) 2920

\(\displaystyle 2 e p \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (d+\frac {e}{x^2}\right ) x^3}dx+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 e p \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (d+\frac {e}{x^2}\right ) x^3}dx}{\sqrt {f} \sqrt {g}}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}\)

\(\Big \downarrow \) 2005

\(\displaystyle \frac {2 e p \int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x \left (d x^2+e\right )}dx}{\sqrt {f} \sqrt {g}}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}\)

\(\Big \downarrow \) 5463

\(\displaystyle \frac {2 e p \int \left (\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e x}-\frac {d x \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e \left (d x^2+e\right )}\right )dx}{\sqrt {f} \sqrt {g}}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {2 e p \left (-\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{2 e}-\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d} x+\sqrt {e}\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{2 e}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{e}+\frac {i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{4 e}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d} x+\sqrt {e}\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 e}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 e}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 e}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 e}\right )}{\sqrt {f} \sqrt {g}}\)

Input:

Int[Log[c*(d + e/x^2)^p]/(f + g*x^2),x]
 

Output:

(ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e/x^2)^p])/(Sqrt[f]*Sqrt[g]) + (2* 
e*p*((ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)] 
)/e - (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(Sqrt[e] - Sqrt 
[-d]*x))/((I*Sqrt[-d]*Sqrt[f] - Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))] 
)/(2*e) - (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[e] + S 
qrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] + Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x 
))])/(2*e) + ((I/2)*PolyLog[2, ((-I)*Sqrt[g]*x)/Sqrt[f]])/e - ((I/2)*PolyL 
og[2, (I*Sqrt[g]*x)/Sqrt[f]])/e - ((I/2)*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[ 
f] - I*Sqrt[g]*x)])/e + ((I/4)*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[e] 
- Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] - Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g 
]*x))])/e + ((I/4)*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[e] + Sqrt[-d]*x 
))/((I*Sqrt[-d]*Sqrt[f] + Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/e))/ 
(Sqrt[f]*Sqrt[g])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2920

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.) 
*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* 
Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p   Int[u*(x^(n - 1)/(d + e*x^n)), x 
], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
 

rule 5463

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Int[ExpandIntegrand[a + b*ArcTan[c*x], x^m/(d + e*x^2), x], x] 
 /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a, 0])
 

Maple [F]

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

Input:

int(ln(c*(d+e/x^2)^p)/(g*x^2+f),x)
 

Output:

int(ln(c*(d+e/x^2)^p)/(g*x^2+f),x)
 

Fricas [F]

\[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \] Input:

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

Output:

integral(log(c*((d*x^2 + e)/x^2)^p)/(g*x^2 + f), x)
 

Sympy [F(-1)]

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

integrate(ln(c*(d+e/x**2)**p)/(g*x**2+f),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \] Input:

integrate(log(c*(d+e/x^2)^p)/(g*x^2+f),x, algorithm="maxima")
 

Output:

integrate(log(c*(d + e/x^2)^p)/(g*x^2 + f), x)
 

Giac [F]

\[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \] Input:

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

Output:

integrate(log(c*(d + e/x^2)^p)/(g*x^2 + f), x)
 

Mupad [F(-1)]

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

int(log(c*(d + e/x^2)^p)/(f + g*x^2),x)
 

Output:

int(log(c*(d + e/x^2)^p)/(f + g*x^2), x)
 

Reduce [F]

\[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int \frac {\mathrm {log}\left (\frac {\left (d \,x^{2}+e \right )^{p} c}{x^{2 p}}\right )}{g \,x^{2}+f}d x \] Input:

int(log(c*(d+e/x^2)^p)/(g*x^2+f),x)
 

Output:

int(log(((d*x**2 + e)**p*c)/x**(2*p))/(f + g*x**2),x)