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

Optimal. Leaf size=533 \[ \frac{i p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{2 p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.425435, antiderivative size = 533, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {205, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac{i p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}+\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 \sqrt{f} \sqrt{g}}-\frac{i p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}-\frac{p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{2 p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*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[-d] + Sqr
t[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[
g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(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[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f
] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + ((I/2)*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))])/(Sqrt[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 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 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+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+e x^2\right )^p\right )}{\sqrt{f} \sqrt{g}}-(2 e p) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g} \left (d+e x^2\right )} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{f} \sqrt{g}}-\frac{(2 e p) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{d+e x^2} \, dx}{\sqrt{f} \sqrt{g}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+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 )}{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}{\sqrt{f} \sqrt{g}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{f} \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}{\sqrt{f} \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}{\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{-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 )}{\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}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\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{-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}{f}+\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}{f}\\ &=\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{-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 )}{\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}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{f} \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 )}{2 \sqrt{f} \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 )}{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{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{-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 )}{\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}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{f} \sqrt{g}}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\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{-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 \sqrt{f} \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 )}{2 \sqrt{f} \sqrt{g}}\\ \end{align*}

Mathematica [A]  time = 0.158188, size = 564, normalized size = 1.06 \[ -\frac{i \left (p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )+p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )-p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )-p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )+2 i \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )+p \log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-p \log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-p \log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )\right )}{2 \sqrt{f} \sqrt{g}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.082, size = 504, normalized size = 1. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/2*p/g*sum(1/_alpha*(ln(x-_alpha)*ln(e*x^
2+d)-ln(x-_alpha)*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*
e*g+d*g-e*f,index=1))+ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alp
ha*e*g+d*g-e*f,index=2)))-dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*
_Z*_alpha*e*g+d*g-e*f,index=1))-dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)-x+_alpha)/RootOf(_Z^2*
e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2))),_alpha=RootOf(_Z^2*g+f))+1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*c
sgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*(e*x^2+d)^p)*cs
gn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x^2+d)^p)^3+1/2*I/(f*g)
^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+1/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*ln(c
)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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