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

Optimal. Leaf size=751 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^{3/2} \sqrt {g}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {i p \text {Li}_2\left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{4 f^{3/2} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 f^{3/2} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2} \sqrt {g}}-\frac {e p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {e p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f (e f-d g)}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{3/2} \sqrt {g}}+\frac {p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{f^{3/2} \sqrt {g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.82, antiderivative size = 751, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {2471, 2463, 801, 635, 205, 260, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac {i p \text {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{4 f^{3/2} \sqrt {g}}+\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{4 f^{3/2} \sqrt {g}}-\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{3/2} \sqrt {g}}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^{3/2} \sqrt {g}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 f \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 f^{3/2} \sqrt {g}}-\frac {e p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {e p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 \sqrt {-f} \sqrt {g} (e f-d g)}+\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f (e f-d g)}+\frac {p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{f^{3/2} \sqrt {g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 205

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2463

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

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[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 2471

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]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4928

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])

Rubi steps

\begin {align*} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx &=\int \left (-\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}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx\\ &=-\frac {g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt {-f} \sqrt {g}-g x\right )^2} \, dx}{4 f}-\frac {g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt {-f} \sqrt {g}+g x\right )^2} \, dx}{4 f}-\frac {g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{-f g-g^2 x^2} \, dx}{2 f}\\ &=-\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 {\tan ^{-1}\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 {(e p) \int \frac {x}{\left (\sqrt {-f} \sqrt {g}-g x\right ) \left (d+e x^2\right )} \, dx}{2 f}-\frac {(e p) \int \frac {x}{\left (\sqrt {-f} \sqrt {g}+g x\right ) \left (d+e x^2\right )} \, dx}{2 f}-\frac {(e g p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} g^{3/2} \left (d+e x^2\right )} \, dx}{f}\\ &=-\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 {\tan ^{-1}\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 {(e p) \int \left (\frac {\sqrt {-f}}{(e f-d g) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {-d \sqrt {g}-e \sqrt {-f} x}{\sqrt {g} (-e f+d g) \left (d+e x^2\right )}\right ) \, dx}{2 f}+\frac {(e p) \int \left (\frac {\sqrt {-f}}{(e f-d g) \left (-\sqrt {-f}+\sqrt {g} x\right )}-\frac {d \sqrt {g}-e \sqrt {-f} x}{\sqrt {g} (-e f+d g) \left (d+e x^2\right )}\right ) \, dx}{2 f}-\frac {(e p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{d+e x^2} \, dx}{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 {\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 {\tan ^{-1}\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 {(e p) \int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{f^{3/2} \sqrt {g}}-\frac {(e p) \int \frac {-d \sqrt {g}-e \sqrt {-f} x}{d+e x^2} \, dx}{2 f \sqrt {g} (e f-d g)}+\frac {(e p) \int \frac {d \sqrt {g}-e \sqrt {-f} x}{d+e x^2} \, dx}{2 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 {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 {\tan ^{-1}\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 {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 f^{3/2} \sqrt {g}}-\frac {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 f^{3/2} \sqrt {g}}+2 \frac {(d e p) \int \frac {1}{d+e x^2} \, dx}{2 f (e f-d g)}\\ &=\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 {\tan ^{-1}\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}}-2 \frac {p \int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{1+\frac {g x^2}{f}} \, dx}{2 f^2}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {f} \left (-i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{2 f^2}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {f} \left (i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{2 f^2}\\ &=\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 {\tan ^{-1}\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 \text {Li}_2\left (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 \text {Li}_2\left (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}}-2 \frac {(i p) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{2 f^{3/2} \sqrt {g}}\\ &=\frac {\sqrt {d} \sqrt {e} p \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 {\tan ^{-1}\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 \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{3/2} \sqrt {g}}+\frac {i p \text {Li}_2\left (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 \text {Li}_2\left (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}}\\ \end {align*}

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Mathematica [A]  time = 3.19, size = 1236, normalized size = 1.65 \[ \frac {1}{2} \left (\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )}{f^{3/2} \sqrt {g}}+\frac {x \left (\log \left (c \left (e x^2+d\right )^p\right )-p \log \left (e x^2+d\right )\right )}{f \left (g x^2+f\right )}+\frac {1}{2} p \left (\frac {i \left (\frac {\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )}{i \sqrt {g} x+\sqrt {f}}+\frac {\sqrt {e} \left (\log \left (i \sqrt {f}-\sqrt {g} x\right )-\log \left (i \sqrt {d}-\sqrt {e} x\right )\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )}{f \sqrt {g}}+\frac {i \left (\frac {\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )}{i \sqrt {g} x+\sqrt {f}}+\frac {\sqrt {e} \left (\log \left (i \sqrt {f}-\sqrt {g} x\right )-\log \left (\sqrt {e} x+i \sqrt {d}\right )\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )}{f \sqrt {g}}+\frac {\sqrt {e} \left (\sqrt {g} x+i \sqrt {f}\right ) \left (\log \left (i \sqrt {d}-\sqrt {e} x\right )-\log \left (\sqrt {g} x+i \sqrt {f}\right )\right )-i \left (\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}\right ) \log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )}{f \left (\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}\right ) \sqrt {g} \left (\sqrt {f}-i \sqrt {g} x\right )}-\frac {-\frac {\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )}{\sqrt {g} x+i \sqrt {f}}-\frac {i \sqrt {e} \left (\log \left (\sqrt {e} x+i \sqrt {d}\right )-\log \left (\sqrt {g} x+i \sqrt {f}\right )\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}}{f \sqrt {g}}+2 \left (\frac {x}{f^2+g x^2 f}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{f^{3/2} \sqrt {g}}\right ) \left (-\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right )-\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )+\log \left (e x^2+d\right )\right )+\frac {i \left (\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (-\frac {\sqrt {g} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )\right )}{f^{3/2} \sqrt {g}}-\frac {i \left (\log \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )\right )}{f^{3/2} \sqrt {g}}-\frac {i \left (\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (-\frac {\sqrt {g} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {e} \sqrt {f}-\sqrt {d} \sqrt {g}}\right )\right )}{f^{3/2} \sqrt {g}}+\frac {i \left (\log \left (x-\frac {i \sqrt {d}}{\sqrt {e}}\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {e} \sqrt {f}+\sqrt {d} \sqrt {g}}\right )\right )}{f^{3/2} \sqrt {g}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{2}+f \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*x^2+f)^2*ln(c*(e*x^2+d)^p),x)

[Out]

int(1/(g*x^2+f)^2*ln(c*(e*x^2+d)^p),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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