\(\int \frac {\log (c (d+e x^2)^p)}{(f+g x^2)^2} \, dx\) [272]

Optimal result

Integrand size = 22, antiderivative size = 751 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f (e f-d g)}-\frac {e p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{3/2} \sqrt {g}}+\frac {e p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^{3/2} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{3/2} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 f^{3/2} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 f^{3/2} \sqrt {g}} \] Output:

d^(1/2)*e^(1/2)*p*arctan(e^(1/2)*x/d^(1/2))/f/(-d*g+e*f)-1/2*e*p*ln((-f)^( 
1/2)-g^(1/2)*x)/(-f)^(1/2)/g^(1/2)/(-d*g+e*f)+p*arctan(g^(1/2)*x/f^(1/2))* 
ln(2*f^(1/2)/(f^(1/2)-I*g^(1/2)*x))/f^(3/2)/g^(1/2)-1/2*p*arctan(g^(1/2)*x 
/f^(1/2))*ln(-2*f^(1/2)*g^(1/2)*((-d)^(1/2)-e^(1/2)*x)/(I*e^(1/2)*f^(1/2)- 
(-d)^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(3/2)/g^(1/2)-1/2*p*arctan(g^ 
(1/2)*x/f^(1/2))*ln(2*f^(1/2)*g^(1/2)*((-d)^(1/2)+e^(1/2)*x)/(I*e^(1/2)*f^ 
(1/2)+(-d)^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(3/2)/g^(1/2)+1/2*e*p*l 
n((-f)^(1/2)+g^(1/2)*x)/(-f)^(1/2)/g^(1/2)/(-d*g+e*f)-1/4*ln(c*(e*x^2+d)^p 
)/f/g^(1/2)/((-f)^(1/2)-g^(1/2)*x)+1/4*ln(c*(e*x^2+d)^p)/f/g^(1/2)/((-f)^( 
1/2)+g^(1/2)*x)+1/2*arctan(g^(1/2)*x/f^(1/2))*ln(c*(e*x^2+d)^p)/f^(3/2)/g^ 
(1/2)-1/2*I*p*polylog(2,1-2*f^(1/2)/(f^(1/2)-I*g^(1/2)*x))/f^(3/2)/g^(1/2) 
+1/4*I*p*polylog(2,1+2*f^(1/2)*g^(1/2)*((-d)^(1/2)-e^(1/2)*x)/(I*e^(1/2)*f 
^(1/2)-(-d)^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(3/2)/g^(1/2)+1/4*I*p* 
polylog(2,1-2*f^(1/2)*g^(1/2)*((-d)^(1/2)+e^(1/2)*x)/(I*e^(1/2)*f^(1/2)+(- 
d)^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/f^(3/2)/g^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.47 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.17 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {-d} \sqrt {e} \sqrt {f} p \log \left (\sqrt {-d}-\sqrt {e} x\right )}{-e f+d g}+\frac {2 \sqrt {-d} \sqrt {e} \sqrt {f} p \log \left (\sqrt {-d}+\sqrt {e} x\right )}{e f-d g}+\frac {2 e \sqrt {-f^2} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} (e f-d g)}+\frac {2 e \sqrt {-f^2} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {g} (-e f+d g)}-\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {i p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}+\frac {\sqrt {f} \log \left (c \left (d+e x^2\right )^p\right )}{-\sqrt {-f} \sqrt {g}+g x}+\frac {\sqrt {f} \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f} \sqrt {g}+g x}+\frac {2 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )}{\sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )}{\sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )}{\sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )}{\sqrt {g}}}{4 f^{3/2}} \] Input:

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

Output:

((2*Sqrt[-d]*Sqrt[e]*Sqrt[f]*p*Log[Sqrt[-d] - Sqrt[e]*x])/(-(e*f) + d*g) + 
 (2*Sqrt[-d]*Sqrt[e]*Sqrt[f]*p*Log[Sqrt[-d] + Sqrt[e]*x])/(e*f - d*g) + (2 
*e*Sqrt[-f^2]*p*Log[Sqrt[-f] - Sqrt[g]*x])/(Sqrt[g]*(e*f - d*g)) + (2*e*Sq 
rt[-f^2]*p*Log[Sqrt[-f] + Sqrt[g]*x])/(Sqrt[g]*(-(e*f) + d*g)) - (I*p*Log[ 
(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*L 
og[1 - (I*Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] - (I*p*Log[(Sqrt[g]*(Sqrt[-d] + Sqr 
t[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/ 
Sqrt[f]])/Sqrt[g] + (I*p*Log[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((-I)*Sqrt[e 
]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + ( 
I*p*Log[(Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqr 
t[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + (Sqrt[f]*Log[c*(d + e*x^2 
)^p])/(-(Sqrt[-f]*Sqrt[g]) + g*x) + (Sqrt[f]*Log[c*(d + e*x^2)^p])/(Sqrt[- 
f]*Sqrt[g] + g*x) + (2*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/S 
qrt[g] - (I*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f 
] - I*Sqrt[-d]*Sqrt[g])])/Sqrt[g] - (I*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I* 
Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])])/Sqrt[g] + (I*p*PolyLo 
g[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[ 
g])])/Sqrt[g] + (I*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e] 
*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])])/Sqrt[g])/(4*f^(3/2))
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2921, 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.

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

\(\Big \downarrow \) 2921

\(\displaystyle \int \left (-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (-f g-g^2 x^2\right )}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 f \left (\sqrt {-f} \sqrt {g}-g x\right )^2}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 f \left (\sqrt {-f} \sqrt {g}+g x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^{3/2} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2} \sqrt {g}}+\frac {\sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f (e f-d g)}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{3/2} \sqrt {g}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {i p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{4 f^{3/2} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 f^{3/2} \sqrt {g}}-\frac {e p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {e p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{3/2} \sqrt {g}}\)

Input:

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

Output:

(Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(f*(e*f - d*g)) - (e*p*Log 
[Sqrt[-f] - Sqrt[g]*x])/(2*Sqrt[-f]*Sqrt[g]*(e*f - d*g)) + (p*ArcTan[(Sqrt 
[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(f^(3/2)*Sqrt[g] 
) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqr 
t[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))] 
)/(2*f^(3/2)*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt 
[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[ 
f] - I*Sqrt[g]*x))])/(2*f^(3/2)*Sqrt[g]) + (e*p*Log[Sqrt[-f] + Sqrt[g]*x]) 
/(2*Sqrt[-f]*Sqrt[g]*(e*f - d*g)) - Log[c*(d + e*x^2)^p]/(4*f*Sqrt[g]*(Sqr 
t[-f] - Sqrt[g]*x)) + Log[c*(d + e*x^2)^p]/(4*f*Sqrt[g]*(Sqrt[-f] + Sqrt[g 
]*x)) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*f^(3/2)*Sqrt 
[g]) - ((I/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(f^(3 
/2)*Sqrt[g]) + ((I/4)*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt 
[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))]) 
/(f^(3/2)*Sqrt[g]) + ((I/4)*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] 
+ Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g] 
*x))])/(f^(3/2)*Sqrt[g])
 

Defintions of rubi rules used

rule 2009

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

rule 2921

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + 
 (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{t = ExpandIntegrand[(a + b*Log[ 
c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, 
 b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ 
erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 
0] && LtQ[r, 0]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(log((e*x^2 + d)^p*c)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{g^{2} x^{4}+2 f g \,x^{2}+f^{2}}d x \] Input:

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

Output:

int(log((d + e*x**2)**p*c)/(f**2 + 2*f*g*x**2 + g**2*x**4),x)