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

Optimal. Leaf size=597 \[ \frac{i p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e}-\sqrt{-d} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d} x+\sqrt{e}\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}}-\frac{i p \text{PolyLog}\left (2,\frac{i \sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}}-\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+\frac{e}{x^2}\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e}-\sqrt{-d} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d} x+\sqrt{e}\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{2 p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.83858, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {205, 2470, 12, 260, 6688, 4928, 4848, 2391, 4856, 2402, 2315, 2447} \[ \frac{i p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e}-\sqrt{-d} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d} x+\sqrt{e}\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}}-\frac{i p \text{PolyLog}\left (2,\frac{i \sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}}-\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+\frac{e}{x^2}\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{e}-\sqrt{-d} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d} x+\sqrt{e}\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{e} \sqrt{g}+i \sqrt{-d} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{2 p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 12

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

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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])

Rule 4848

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

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 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 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 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 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]]

Rubi steps

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

Mathematica [A]  time = 0.371281, size = 706, normalized size = 1.18 \[ \frac{-p \text{PolyLog}\left (2,\frac{\sqrt{-d} \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{-d} \sqrt{-f}-\sqrt{e} \sqrt{g}}\right )-p \text{PolyLog}\left (2,\frac{\sqrt{-d} \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{-d} \sqrt{-f}+\sqrt{e} \sqrt{g}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt{-d} \left (\sqrt{-f}+\sqrt{g} x\right )}{\sqrt{-d} \sqrt{-f}-\sqrt{e} \sqrt{g}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt{-d} \left (\sqrt{-f}+\sqrt{g} x\right )}{\sqrt{-d} \sqrt{-f}+\sqrt{e} \sqrt{g}}\right )-2 p \text{PolyLog}\left (2,\frac{\sqrt{g} x}{\sqrt{-f}}+1\right )+2 p \text{PolyLog}\left (2,\frac{f \sqrt{g} x}{(-f)^{3/2}}+1\right )+\log \left (\sqrt{-f}-\sqrt{g} x\right ) \log \left (c \left (d+\frac{e}{x^2}\right )^p\right )-\log \left (\sqrt{-f}+\sqrt{g} x\right ) \log \left (c \left (d+\frac{e}{x^2}\right )^p\right )-p \log \left (\sqrt{-f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d} x-\sqrt{e}\right )}{\sqrt{-d} \sqrt{-f}-\sqrt{e} \sqrt{g}}\right )-p \log \left (\sqrt{-f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d} x+\sqrt{e}\right )}{\sqrt{-d} \sqrt{-f}+\sqrt{e} \sqrt{g}}\right )+p \log \left (\sqrt{-f}+\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{e}-\sqrt{-d} x\right )}{\sqrt{-d} \sqrt{-f}+\sqrt{e} \sqrt{g}}\right )+p \log \left (\sqrt{-f}+\sqrt{g} x\right ) \log \left (-\frac{\sqrt{g} \left (\sqrt{-d} x+\sqrt{e}\right )}{\sqrt{-d} \sqrt{-f}-\sqrt{e} \sqrt{g}}\right )+2 p \log \left (\frac{\sqrt{g} x}{\sqrt{-f}}\right ) \log \left (\sqrt{-f}-\sqrt{g} x\right )-2 p \log \left (\frac{f \sqrt{g} x}{(-f)^{3/2}}\right ) \log \left (\sqrt{-f}+\sqrt{g} x\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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[(S
qrt[g]*(Sqrt[e] - Sqrt[-d]*x))/(Sqrt[-d]*Sqrt[-f] + Sqrt[e]*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*x] + p*Log[-((Sqr
t[g]*(Sqrt[e] + Sqrt[-d]*x))/(Sqrt[-d]*Sqrt[-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*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))/(Sqrt[-d]*
Sqrt[-f] - Sqrt[e]*Sqrt[g])] + p*PolyLog[2, (Sqrt[-d]*(Sqrt[-f] + Sqrt[g]*x))/(Sqrt[-d]*Sqrt[-f] + Sqrt[e]*Sqr
t[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])

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Maple [F]  time = 0.743, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{g{x}^{2}+f}\ln \left ( c \left ( d+{\frac{e}{{x}^{2}}} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c \left (\frac{d x^{2} + e}{x^{2}}\right )^{p}\right )}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (c{\left (d + \frac{e}{x^{2}}\right )}^{p}\right )}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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